Harmonic Analysis in Hypercomplex Systems

Harmonic Analysis in Hypercomplex Systems

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science. Amsterdam. The Netherlands

Volume 434

Harmonic Analysis in Hypercomplex Systems

by

Yu. M. Berezansky

and

A. A. Kalyuzhnyi

Department of Functional Analysis, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5022-9 ISBN 978-94-017-1758-8 (eBook) DOI 10.1007/978-94-017-1758-8

This is a completely revised and updated translation of the original Russian work of the same title, Kiev, Naukova Dumka, 1992. Copyright by the authors. Translated by P. Malyshev.

Printed on acid-free paper

All Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

conTEnTS

Preface to the English Edition

Introduction

Chapter 1. GENERAL THEORY OF HYPERCOMPLEX SYSTEMS

1. Fundamental Concepts of the Theory of Hypercomplex Systems with Locally Compact Basis

1.1. Definition of Hypercomplex Systems. Characters 1.2. Theorem of Existence of a Multiplicative Measure 1.3. Normal Hypercomplex Systems 1.4. Normal Hypercomplex Systems with Basis Unity 1.5. Even Subsystem of a Normal Hypercomplex System 1.6. The Case of a Discrete Basis 1.7. Hilbert Algebras 1 .8. Parameter Depending Measures

2. Hypercomplex Systems and Related Objects

ix

1

7

8

8 15 20 28 32 34 37 39

42

2.1. Generalized Translation Operators and Hypercomplex Systems 42 2.2. Strong Invariance of Invariant Measures 55 2.3. Hypergroups and Hypercomplex Systems 60 2.4. Hypercomplex Systems Whose Structure Measure Is Not

Necessarily Nonnegative 63 2.5. Convolution Algebras and Hypercomplex Systems 66

3. Elements of Harmonic Analysis for Normal Hypercomplex Systems with Basis Unity 69

3.1. Fourier Transformation and the Plancherel Theorem 69 3.2. Duality of Commutative Hypercomplex Systems 83

v

Vl Contents

3.3. The Case of Discrete Hypercomplex Systems 99 3.4. Representations of Hypercomplex Systems and Approximation

Theorem 105

4. Hypercomplex Subsystems and Homomorphisms 121

4.1. Definition of Hypercomplex Subsystems 4.2. Fundamental Properties of Hypercomplex Subsystems 4.3. Homomorphisms 4.4. Direct and Semidirect Products of Hypercomplex Systems.

Join of Hypercomplex Systems

5. Further Generalizations of Hypercomplex Systems

121 126 128

133

138

5.1. Properties of Hilbert Bialgebras 139 5.2. Quantized Hypercomplex Systems 142 5.3. Harmonic Analysis in Quantized Hypercomplex Systems

with One Commutative Operation 157 5.4. Real Hypercomplex Systems with Compact and Discrete Bases 161

Chapter 2. EXAMPLES OF HYPERCOMPLEX SYSTEMS 165

1. Centers of Group Algebras of Compact Groups 165

1.1. General Construction of Hypercomplex Systems Corresponding to Locally Compact Groups 166

1.2. Centers of Group Algebras of Compact Groups 167 1 .3. Elements of the Theory of Representations of Compact Groups 168 1.4. Peter-Weyl Theorem 174 1.5. Tannaka-M Krein Duality Theorem 180 1 .6. Elements of the Theory of Semisimple Groups and Lie Algebras 189 1 .7. Center of the Group Algebra of Compact Semisimple Lie Groups 202

1 .8. Algebra lJ of Equivalence Classes of Irreducible Representations

of a Compact Semisimple Lie Group G 217

2. Gelfand Pairs 223

2.1. Definition of Gelfand Pairs 223 2.2. Spherical Functions 227 2.3. Representations of Class I 232 2.4. Harmonic Analysis on Gelfand Pairs 237 2.5. Hypercomplex Systems Associated With the Delsarte Generalized

Translation Operators 246 2.6. Center of Group Algebra as a Gelfand Pair 249

Contents vii

3. Orthogonal Polynomials 250

3.1. Discrete Hypercomplex Systems Associated with Orthogonal Systems of Polynomials 250

3.2. Jacobian Matrices and Generalized Translation Operators 254 3.3. Characterization of Hypercomplex Systems Associated with

Orthogonal Polynomials 256 3.4. Another Method for the Construction of a Hypercomplex System

Associated with Orthogonal Polynomials. Examples 259 3.5. Compact Hypercomplex Systems Associated with Orthogonal

Polynomials 265 3.6. The Case of Not Necessarily Nonnegative Structure Constants 272 3.7. Examples of Hypercomplex Systems with Real Structure Constants 275 3.8. Transmutation Operators 280

4. Hypercomplex Systems Constructed for the Sturm - Liouville Equation 284

4.1. Riemann Function 284 4.2. Hypercomplex Systems Constructed for the Sturm-Liouville

Equation 289 4.3. Structure Measure Which Is Not Necessarily Nonnegative 295 4.4 Set of Characters of the Hypercomplex System Associated with the

Sturm-Liouville Equation 302 4.5. Survey of Related Results 311

Chapter 3. ELEMENTS OF LIE THEORY FOR GENERALIZED TRANSLATION OPERATORS 315

1. Basic Concepts 316

1.1. Hypergroup Algebra of Infinitely Differentiable Generalized Translation Operators 317

1.2. Topological Bialgebras 322 1.3. Infinitesimal Object for Generalized Translation Operators 324 1.4. General Properties of Generators of Generalized Translation

Operators 330 1.5. Algebraic Approach to the Infinitesimal Theory of Formal Generalized

Translation Operators 334 1.6. Some Facts from the Theory of Topological Vector Spaces 338

2. Analog of Lie Theory for Some Classes of Generalized Translation Operators

2.1. Infinitesimal Object For the Delsarte Generalized Translation Operators

341

341

viii Contents

2.2. Delsarte-Type Generalized Translation Operators and Generalized Lie Algebras 359

2.3. Infinitesimal Algebra of the Hypercomplex System L j (G, H) 367

3. Duality of Generators of One-Dimensional Compact and Discrete Hypercomplex Systems 372

3.1. Generators of One-Dimensional Compact and Discrete Hypercomplex Systems 372

3.2. General Case of the Construction of Generalized Translation Operators from a Generator 389

3.3. Analog of the Canonical Commutation Relations for the Delsarte Generalized Translation Operators 396

Supplement. Hypercomplex Systems and Hypergroups: Connections and Distinctions

1. Hypercomplex Systems with Locally Compact Basis. Definition and Properties

2. Examples of Hypercomplex Systems 3. Harmonic Analysis in the Locally Compact Case 4. Hypergroups and Hypercomplex Systems 5. Generalizations 6. Remarks on Terminology

Bibliographical Notes

References

Subject Index

405

405 415 419 424 426 428

431

439

481

PREFACE TO THE EnGLISH EDITIon

First works related to the topics covered in this book belong to J. Delsarte and B. M. Levitan and appeared since 1938. In these works, the families of operators that generalize usual translation operators were investigated and the corresponding harmonic analysis was constructed. Later, starting from 1950, it was noticed that, in such constructions, an important role is played by the fact that the kernels of the corresponding convolutions of functions are nonnegative and by the properties of the normed algebras generated by these convolutions. That was the way the notion of hypercomplex system with continuous basis appeared. A hypercomplex system is a normed algebra of functions on a

locally compact space Q-the "basis" of this hypercomplex system. Later, similar

objects, hypergroups, were introduced, which have complex-valued measures on Q as elements and convolution defined to be essentially the convolution of functionals and dual to the original convolution (if measures are regarded as functionals on the space of

continuous functions on Q).

However, until 1991, the time when this book was written in Russian, there were no monographs containing fundamentals of the theory (with an exception of a short section in the book by Yu. M. Berezansky and Yu. G. Kondratiev [BeKo]). The authors wanted to give an introduction to the theory and cover the most important subsequent results and examples. We now came to realizing that the first chapters of the book should have been written in more detail, but such a change into the English edition of the book would entail its complete revision, and so we only made small additions and clarifications.

But we still think that this book can serve as an introduction to the theory. We should also mention a recently published book by W. R. Bloom and H. Heyer "Harmonic Analysis of Probability Measures on Hypergroups," which contains interesting and important applications of the theory of hypergroups to problems in probability theory. It also contains an introduction to the theory of hypergroups and a large number of examples.

There are many mathematicians who currently work in this area of functional analysis. In 1993, a representative conference devoted to this range of problems was held in Seattle. The proceedings of this conference (Contemporary Mathematics, 183, 1995) give a good idea about the scope and depth of recently obtained results in this area.

Of course, we were not able to describe in this English edition the works published after 1991; we only confined ourselves to supplementing the bibliography with works in neighboring areas written by the authors or their students and colleagues. We also

IX

x Preface

include a supplement written on the basis of our joined communication at the mentioned conference. In particular, it may be considered as a survey of principal results presented in the first two chapters of the present book. Moreover, it contains a detailed exposition of the cases of compact and discrete bases, a brief description of the basic examples of hypercomplex systems, and the theory of almost periodic functions.

In this communication, we also suggested to introduce a certain uniformity into the terminology and to replace the term "hypercomplex system with locally compact basis" by the term "Ll-hypergroup". From this point of view, it would probably be more appropriate to call this translation of the book "Harmonic Analysis in Hypergroups."

The authors express their deep gratitude to the Kluwer Academic Publishers for publishing this monograph and to P. V. Malyshev, D. V. Malyshev, and Yu. A. Chapovsky who translated it and performed much editorial work that improved the clarity of the exposition.

April 1997 Yu. M. Berezansky, A. A. Kalyuzhnyi

In TRODUCTIon

In a series of works originated as early as in 1938, J. Delsarte [Dell]-[DeI4] and then Levitan [Lev 1 ]-[Lev 1 0] noticed that some facts of classical harmonic analysis can be

generalized by replacing exponential functions e iAq (q, A E 1R i) by some family of

complex -valued functions X (q, A) which inherit the following property of the indicated exponential functions: The exponential functions are connected with the family of

ordinary translation operators R p (p E 1R i) acting upon complex-valued functions

f(q) (qE 1Ri) according to the rule (RP/")(q) =f(q+p) ,l.e.,

(1)

for any A. A collection of functions X (q, A), where q varies within a set Q and A takes val

ues from another set Q, must also be associated with a family of linear operators Rp

(p E Q) of "generalized translation" acting upon functions of a point q E Q and satisfying an equality similar to (1) ("multiplication formula"), namely,

It is natural that the collection of operators R p must also have some additional properties imitating ordinary translations.

As became clear from the first works of J. Delsarte and Levitan, an important role is played not only by the translation itself but also by the convolution of functions generated by this translation. Thus, by analogy with ordinary convolution

(f*g)(q) = Jf(q-p)g(p)dp Rl

= f f(P)g(q-p)dp = f f(P)(R_pg)(q)dp (qE 1R i ), (2) Rl Rl

1

2 Introduction

it is possible to introduce generalized convolution * associated with the generalized translation operator in exactly the same way as in (2), i.e.,

(f*g)(q) = f!(P)(Rp*g)(q)dm(p) (qE Q), (3)

Q

where Q 3 P ~ p* E Q is an involution given in Q which replaces the transition to the

inverse element in 1R 1 and m is a fixed measure in Q inheriting some properties of the Lebesgue measure.

In 1950, Gelfand suggested Berezansky and S. Krein to investigate the properties of a priori given convolutions (3) of functions defined on Q which enable one to construct meaningful harmonic analysis. As a result, Berezansky and S. Krein introduced the concept of hypercomplex systems with continual basis (see [BKrl] and [BKr2]).

Recall that an ordinary hypercomplex system is defined as a d -dimensional (d < 00 )

associative algebra (i.e., a d-dimensional complex vector space with operation of multiplication! * g of its vectors f, g,... with standard properties) with a fixed basis Q consisting of d points p, q, .... By identifying a vector! with its coordinates, we can

understand this system as the space of functions Q 3 q ~ ! (q) E ([ 1 with the operations of addition and multiplication of functions by a scalar and with multiplication

(f*g)(r) = L !(P)g(q)c(p,q,r) (rE Q), p,qEQ

where c(p, q, r) is a function ("cubic matrix of structural constants") which determines multiplication and possesses certain properties guaranteeing the assosiativity (and, if necessary, the commutativity) of multiplication. It is clear that

c(p,q,r) = (p*q)(r).

A generalization of such hypercomplex systems proposed by Berezansky and S. Krein is connected with the transition from a finite basis Q to a certain locally compact space Q. According to already constructed examples, it seems reasonable to replace c with a "structure measure" c(A, B, r) (A, Be Q; r E Q) rather than by a function defined on Q x Q x Q. The most complete results of harmonic analysis are obtained in the case of commutative hypercomplex systems with nonnegative measure c whose properties are similar to the properties of the group algebra of a locally compact group (the so-called normal hypercomplex system with basis unity).

In 1950-1953, the theory of hypercomplex systems with continual basis was extensively developed by Berezansky and S. Krein [BKrl]-[BKr4] and Berezansky [Berl][Ber9]. They mainly considered the cases of compact and discrete (countable) bases. The detailed presentation of these results can be found in the Candidate's Degree Thesis of Berezansky [Ber9] which, unfortunately, was not published as a book. The develop-

Introduction 3

ment of this theory was significantly influenced by the Gelfand theory of commutative normed algebras (see [GRS]) and the works of M. Krein [Krel], [Kre2] and Raikov [Rai] on harmonic analysis on groups and homogeneous spaces. It is now reasonable to emphasize that the group algebra of a locally compact commutative group G is one of the most simple examples of commutative hypercomplex systems with locally compact basis

Q = G. The construction of general hypercomplex systems was mainly performed as a generalization of this classical object of harmonic analysis.

Since 1953, the interest of Berezansky and S. Krein to this and related subjects became much weaker, basically due to the small number of mathematicians working in the field and the lack of new examples. The situation changed only after the appearance, since 1973, of numerous works devoted to the theory of hypergroups which can be regarded as objects close to hypercomplex systems (Dunkl, Jewett, Spector, Ross, Bloom, Heyer, etc.). It became clear that the axiomatics of hypercomplex systems is broader than the axiomatics of hypergroups and, therefore, some results obtained in the theory of hypergroups were actually contained in older works devoted to the theory of hypercomplex systems. Since this time, the theory of hypercomplex system acquired renewed life. Since 1982, many works by Berezansky, Kalyuzhnyi, Vainerman, Podkolzin, and other authors have been devoted to the investigation of hypercomplex systems.

Note that, up to now, despite a vast amount of data accumulated in the theory of hypercomplex systems, hypergroups, and related subjects, there is no book with systematic presentation of all basic results obtained in this branch of functional analysis. In the present monograph, we try to fill this gap. This book contains the description of basic constructions and detailed analysis of applications. The monograph includes practically the entire thesis [Ber9] mentioned above.

At present, a similar book by Bloom and Heyer is prepared for publication. In this book, main applications deal with the problems of probability theory and random processes (we do not consider applications of this sort). One should also mention two wellknown monographs by Levitan [Lev9] and [LevlO] devoted to the Lie theory of generalized translation operators. A detailed presentation of the fundamentals of the theory of hypercomplex systems with locally compact basis, their representations by commuting operators, and the theory of generalized functions on the basis of a hypercomplex systems can be found in the book by Berezansky and Kondrat'ev [BeKo]. Principal results in the theory of hypergroups are presented in the monographic paper by Jewett [Jew].

Definitions and constructions of hypercomplex systems (both commutative and noncommutative) with locally compact basis are given in Chapter 1. In Chapter 1, we also present elements of harmonic analysis in the commutative case, i.e., generalizations of the Bochner and Plancherel theorems and duality theory. We do not dwell upon the theory of almost periodic functions to make the book more concise. In Section 5, we consider the theory of quantization of hypercomplex systems related to the concept of Kac algebra and harmonic analysis in the noncommutative case. In Section 2, we establish relationships between hypercomplex systems and the theory of generalized translation operators and hypergroups.

In Chapter 2, we present basic examples of hypercomplex systems with locally compact (in particular, compact and discrete) bases: the center of the group algebra of a com-

4 Introduction

pact group, Gelfand pairs, systems associated with J. Delsarte translations, with orthogonal polynomials, and with the Sturm-Liouville operator. It is shown that general facts presented in Chapter 1 imply principal results of harmonic analysis for these objects. In particular, in this chapter, we present the theory of representations of compact groups, including the Tannaka-Krein duality theory.

Chapter 3 contains principal notions and facts of the Lie theory of hypercomplex systems which deals with the investigation of infinitesimal objects associated with such systems. The idea of relevant constructions, clearly, goes back to the construction and investigation of the Lie algebra for a given Lie group G. However, unlike the case of

Lie groups, an infinitesimal object given at a single point of the basis Q of a hypercompie x system generally speaking, does not reconstruct the entire system because homogeneity is absent in this case (for groups, the character of the group operation in a neighborhood of a single point, e.g., in a neighborhood of the identity element, determines (after translations) the character of this operation in neighborhoods of all other points). In Chapter 3, we also present the theory of duality of generators of one-dimensional compact and discrete systems.

Let us now dwell upon some notions whose definitions available in the literature are

ambiguous and introduce necessary notation. We denote the boundary of a set A by

dA. The notions of compactness and bicompactness are equivalent. For convenience, we assume that all locally compact spaces used in what follows are a-compact, i.e., they can be represented as countable unions of compact sets (although all results remain true in the general case). We write

ff(r)drc(EpB,s)

Q

to denote the integral of a function fer) with respect to a measure !l (A) = c(A, B, s)

(B C Q and SEQ are fixed). The indicator of a set A is defined as

= {I, 0,

rEA

reA

The superscript IR (+) in the designation of a class of functions or measures means that this is the collection of all real-valued (positive) functions or measures from the indicated

class. The linear and closed linear spans of a subset M of a linear topological space are

denoted by I.s. (M) and c.1.s. (M), respectively. The space X' dual to a space X is defined as the collection of all linear continuous functionals. Unless otherwise stated,

(I, x) denotes the action of a functional 1 E X' upon a vector x EX. The topology of simple convergence on a set M C X in X' is defined as the weakest topology in which

each function E' 3 1 H (I, x) E a: (x E M) is continuous. The domain of definition of an operator A in a Hilbert space H is denoted by C2lJ (A). An operator A with dense

domain of definition C!.JJ (A) in the Hilbert space H is called Hermitian if

Introduction 5

(AX'Y)H = (x, AY)H (X,YE 0J(A),

self-adjoint if A * = A, and essentially self-adjoint if the closure of the operator A is

self-adjoint. We say that A is a *-algebra if it is an algebra over the field cr with involution, i.e., with an antilinear mapping A 3 X H x* E A such that (1) (x*)* = x and

(2) (xy)* = x* y* (x, YEA). A Banach algebra A with involution x H x * such that

II x* II = II x II (x E A) is called a Banach *-algebra. A weakly closed *-subalgebra of

the algebra of linear continuous operators in a Hilbert space is called a W* -algebra (or a

von Neumann algebra). A c* -algebra is defined as a Banach *-algebra A such that

IIxIl2=lIx*xll for any xEA. References are given both directly in the text and in Bibliographical Notes at the end

of the book. The list of references should in no case be regarded as complete. At the same time, it contains principal works even in those branches of the theory of hypergroups, generalized translations, convolution algebras, etc. which are not included in the book. It has already been noted that we do not dwell upon applications to probability theory, the theory of functions almost periodic with respect to generalized translations, problems of spectral analysis and synthesis, the theory of generalized functions on the basis of a hypercomplex system, algebraic combinatorics, and the theory of multi-valued groups and related problems in algebraic topology.

The authors express their deep gratitude to B. M. Levitan, S. G. Krein and L. 1. Wainerman for fruitful discussions and to G. L. Litvinov, D.1. Gurevich, B. P. Osilenker, and Yu. S. Samoilenko who read some parts of this manuscript and made useful critical remarks.

November 1991 Yu. M. Berezansky, A. A. Kalyuzhnyi

1. GEnERRL THEORY OF HYPERCOmPLEX SYSTEmS

In 1950, when extending the Delsarte-Levitan theory of generalized translation operators, Yu. M. Berezanskii and S. G. Krein introduced the concept of (commutative) hypercomplex system with continuous basis and developed harmonic analysis for such systems. Each hypercomplex system is a Banach *-algebra of functions on a locally compact space (the basis of a hypercomplex system). It generalizes the concept of hypercomplex system with finite basis and the concept of locally compact group algebra. The role of the group translation operators is played by generalized translation operators, which are naturally associated with a hypercomplex system. Note that it is possible to completely characterize hypercomplex systems in terms of such generalized translation operators (Theorem 2.1) and, hence, a hypercomplex system with continuous basis can be considered as a class of generalized translation operators that admit the construction of rich harmonic analysis and duality theory.

In Section 1, we introduce the required class of hypercomplex systems and establish their properties. In Section 2, we generalize the concept of hypercomplex system to the case where the "structure measure" c is real and establish the relationship between hypercomplex systems and related objects such as generalized translation operators, hypergroups, and convolution algebras. Section 3 is devoted to the construction of harmonic analysis on hypercomplex systems. We study representations of hypercomplex systems and prove analogs of the approximation theorem, the Plancherel theorem, the inversion formula, and the Bochner theorem. We also establish an analog of the Pontryagin duality theory for locally compact commutative groups. In Section 4, we study the properties of hypercomplex subsystems generalizing the concept of subgroups to the case of hypercomplex systems and introduce the concept of homomorphism for hypercomplex systems.

To construct duality theory for locally compact groups, Kac [Kac] introduced the concept of Hilbert bialgebras. Thus, functions defined on a group form a bialgebra with respect to the ordinary pointwise multiplication (commutative operation) and convolution (generally speaking, noncommutative operation). One can understand the transition from this special bialgebra to the general bialgebra with two noncommutative operations as a quantization of the group, moreover, the "conformity principle" is true, i.e., under certain conditions, the commutativity of one operation allows the bialgebra to be realized as a bialgebra of functions on a group. The development of this approach and its application to Lie groups leads to the quantization of these groups in terms of deforma-

7

8 General Theory of Hypercomplex Systems Chapter 1

tions of algebras (Drinfeld [DriD. In Section 5, we realize this construction for hypercomplex systems To do this, we introduce the concept of quantized hypercomplex system whose axiomatics is simpler and less restrictive than that of Kac bialgebras. If one operation is commutative, a quantized hypercomplex system can be realized as an object that generalizes a hypercomplex system with locally compact basis to the case where the measure c is real. Such hypercomplex systems also admit the construction of harmonic analysis and duality theory. At the end of Section 5, we present a generalization of the concept of hypercomplex systems to the case where the measure c is, generally speaking, not positive.

1. Fundamental Concepts of the Theory of Hypercomplex Systems with Locally Compact Basis

A hypercomplex system with locally compact basis Q is a set L 1 (Q, m) with generalized convolution, which can be defined in terms of a structure measure c(A, B, r) (A,

B c Q, r E Q) and generalizes the concept of locally compact group algebra. The structure measure c(A, B, r) is equal to the convolution ofthe characteristic functions of the sets A and B, whereas the positive measure m is an analog of the invariant measure. In this section, we prove the existence of this measure if the convolution is commutative. We extract a subclass of hypercomplex systems (the so-called normal hypercomplex systems with basis unity) which possesses many good properties which enable one to construct substantive harmonic analysis. In particular, it is established that every hypercomplex system of this sort is semisimple. In the last subsection, we present some technical facts about integration with respect to a measure depending on a parameter frequently used in this section. Examples of hypercomplex systems presented in Section 1 are pure illustrative. More important examples of hypercomplex systems are studied in Chapter 2.

1.1. Definition of Hypercomplex Systems. Characters. Assume that Q is a com

plete separable locally compact space, 13(Q) is the a-algebra generated by its Borel

sets, 130 (Q) is the subring of 13 (Q) consisting of sets with compact closure. In what follows, we consider regular Borel measures, i.e., measures nonnegative on Q and finite

on compact sets; a measure p of this sort is regular in a sense that p (A) = sup p(~)

for all A E 13(Q), where supremum is taken over all compact sets ~ cA. A function

Q 3 r ~ f C r) E a: is called finite if it is equal to zero outside a certain compact set. The support of a function fer) is denoted by suppj. The spaces of continuous functions, of finite continuous functions, of c.ontinuous functions vanishing at infinity, and of bounded continuous functions are denoted by C (Q), Co ( Q), Coo (Q), and C b( Q), re

spectively. The spaces Coo(Q) and CbCQ) are Banach spaces with norm

Section 1 Fundamental Concepts of the Theory of Hypercomplex Systems 9

II ·11 00 = sup I ( . ) (r) I, rEQ

Co (Q) is endowed with the topology of inductive limit

Co(Q) = limind CO(Qn), n

where Qn is an increasing sequence of compact sets whose union is equal to Q, and

CO(Qn) is the space with norm 11·1100 of functions from C(Q) with support in Qn.

Thus, convergence in Co (Q) is actually the uniform convergence of functions whose

supports belong to one of Q n' Recall that any continuous linear functional defined on

the space Co(Q) with inductive topology is called a (complex) Radon measure. We denote the space of Radon measures by M(Q). Let Mb(Q) = (Coo(Q))' be the Banach

space of bounded Radon measures with norm

1I1l1100 = suP{IIl(f)llfE Coo(Q), IfL::; 1},

and let Me (Q) be the 'space of Radon measures with compact support. The Riesz-Markov theorem establishes a one-to-one correspondence between positive Radon measures

M+(Q) (bounded positive Radon measures Mt(Q)) and regular Borel (finite regular

Borel) measures on Q. By M 1 (Q) c Mt (Q) we denote the set of Radon probability measures. The topology of simple convergence on functions from Co(Q) in the space of Radon measures M (Q) is called vague topology or v -topology. In the space M b( Q) of bounded Radon measures, one can also introduce the topology of simple con

vergence on functions from Cb(Q). This topology is called weak; we write Ila ~ Il

to denote that the net J-la E Mb(Q) weakly converges to the measure Il E Mb(Q).

A hypercomplex system with basis Q is determined by its structure measure c (A,

B, r) (A, B E '.B(Q); r E Q). The structure measure c(A, B, r) is a nonnegative regular Borel measure with respect to A(B) with fixed BE '.B(Q), r E Q(A, r), and satisfies the following requirements:

(HI) for each A, BE '.Bo(Q), we have c(A, B, r) E Co(Q);

(H2) for each A, B E' '.Bo(Q), C E '.B(Q) and SEQ, the following associativity relation holds:

f c(A, B, r)drc(Er. C, s) = f c(B, C, r)drc(A, En s) (1.1)

(if the domain of integration is not specified explicitly, this always means that integration is carried out over the entire Q).


A structure measure is called commutative whenever

(H3) c(A, B, r) = c(B, A, r) (A, B E ~o(Q); r E Q).

A regular Borel measure m positive on open sets

J c(A, B, r)dm(r) = m(A)m(B) (A, B E ~o(Q» (1.2)

is called multiplicative.

In what follows:

(H4) we assume that at least one multiplicative measure exists.

In the next section, we prove the existence theorem for a multiplicative measure under certain restrictions imposed on the commutative structure measure. The problem of whether the multiplicative measure exists for a noncommutative structure measure remains open.

We fix a mUltiplicative measure and denote the integration with respect to it by d r (the notions such as almost everywhere, orthogonality, etc. are regarded with respect to this measure). We also denote

J x(r)dr = x(A), A

c(A, B, C) = J c(A, B, r) dr (A, B, C E ~(Q». c

It is sometimes convenient to consider, together with the measure c(A, B, r), its ex

tension to the sets from Q x Q. For this purpose, we fix r and, for any (A, BE 'B(Q», put mr(A xB) = c(A, B, r). By additivity, the function of sets mr can be uniquely ex

tended to the algebra m spanned by the rectangles A x B (A, B E ~(Q». The regularity of structure measure implies the regularity of m r. Indeed, it suffices to check the

indicated property for a rectangle E = A ] x B]. Since ~(Q).3 A H c (A, B ], r) is reg

ular for fixed B], there exists, for any E > 0, a compact set F] e A I such that c (A], BJ,r)-c(FJ,B],r) < E. Similarly, we find a compact set G] eB] such that c(F],

BJ,r)-c(F],G],r) < E. This relation implies that mr(A]xBl)-mr(F]xGl) <2E,

whence it follows that m r is regular which means, as is known, that m r is a-additive on m. If we extend this'measure by using the well-known procedure to all Borel sets

from Q x Q, then we obtain a nonnegative regular Borel measure mr on Q x Q. For

fixed %( E ~(Q x Q), the function of r E Q m r(21) is, in any case, measurable be

cause it is obtained from continuous functions of the form c(A, B, r) by finite summations and limit transitions.

Consider the space L 1 (Q, m) = L 1 of functions on Q integrable with respect to the multiplicative measure m.

This space with the convolution


(x * y)(r) = f f x(p)y(q)dmr(p, q), (1.3)

is called a hypercomplex system with basis Q. The following theorem shows that this definition is correct.

Theorem 1.1. The space LI (Q, m) with the norm

IIxlll = J Ix(r)ldr

and multiplication (1.3) (the integral in (1.3) exists for almost all r) is a Banach alg

ebra and II x * y III ~ II x 11111 Y III. If, moreover, the structure measure is commu

tative, then the Banach algebra L I (Q, m) is also commutative.

Proof. First, we note that, by the Fubini theorem, the integral in (1.3) can be rewritten in the form

f J x(P)y(q)dmr(p, q) = f x(p)dp(f y(q)dqc(Ep, E q, r) ). (1.4)

Let us show that, for any x, yELl (Q, m), integral (1.4) exists for almost all r,

belongs to L I (Q, m), and

f If x(p)dpf y(q)dqc(Ep,Eq, r)ldr ~ f Ix(P)ldPJ ly(q)ldq. (1.5)

To do this, we note that the set Q contains a set Q 0 of full measure for every point r of which, the integral

f y(q)dqc(A, E q, r) (A E ~o(Q»

exists and is a measure with respect to A. Indeed, let Q I C Q 2 C ... be an increasing sequence of compact sets such that

By virtue of Lemma 1.5 presented below, for each Q n' we have

= f ly(q)ldqm(Qn) < 00.


This implies the existence of a set of full measure Q 0 c Q such that

for every n = 1,2, ... and r E Qo. Applying Lemma 1.7 presented below, we obtain the required result.

We can assume that the outer integration in (l.5) is carried out over Qo. By using Lemmas 1.4 and 1.5, we get

J I J x(p)dp(J y(q)dqc(Ep, E q, r) )1 dr

Qo

~ J (J Ix(P)ldp(J ly(q)ldqc(Ep,Eq,r») )dr Qo

= f Ix(P)ldPf ly(q)ldq.

Hence, convolution (1.3) exists for almost all r, belongs to L 1, and

i.e., multiplication is continuous and, in addition, clearly distributive. Therefore, it suffices to prove the relation of associativity (and commutativity relation if the structure measure is commutative) for a total set of functions from L 1, e.g., for the indicators of

sets from 'Bo(Q). The associativity and commutativity relations are true for these func

tions by virtue of (H2) and (H3) because (KA * KB) = c (A, B, r).

• We say that a hypercomplex system L 1 (Q, m) is commutative if its structure mea

sure is commutative. Unless otherwise stated, we consider general noncommutative hypercomplex systems. In what follows, the results related to the commutative case will be specially indicated. If a hypercomplex system has no unity, then we formally add the

unity element e and consider the algebra :c 1 (Q, m) of expressions of the form A, e + x, where A, is a scalar and x ELI with multiplication


An essentially bounded measurable complex-valued function x(r) (r E Q) not identically equal to zero on a set of positive measure is called a character of a hypercomplex system if the equality

f c(A,B, r)x(r)dx = X(A)X(B) (1.6)

holds for any A, B E ~o(Q). Every hypercomplex system has at least one character,

namely, the function identically equal to 1. We note that if X (r) (r E Q) is a character,

then x(r) is also a character. The following definition proves to be very helpful: a measurable complex-valued

function X (r) (r E Q) not identically equal to zero on a set of positive measure, is called a generalized character of a hypercomplex system if (1.6) holds for any A, B E

~o(Q)·

Theorem 1.2. The set of characters of a commutative hypercomplex system is in

one-to-one correspondence with the set of maximal ideals of the Banach algebra L 1 (Q,

m) (or of the algebra ~ 1 (Q, m) if L 1 (Q, m) has no unity; in this case, one must

consider only the ideals other than L 1)' The maximal ideal M and the character X

are connected by the relation

x(M) = f x(r)x(r)dr (XE Lt). (1.7)

Proof. Indeed, consider a character X (r). We associate each A e + x E ~ I (Q, m )

(if L 1 (Q, m) has unity, the proof is similar) with a number

A + f x(r)x(r)dr.

Since X (r) is bounded almost everywhere, this correspondence generates a linear multiplicative functional in L 1:

f (x * y)(r)x(r)dr = f (J x(p)dp(J y(q)dqc(Ep, E q, r) )x(r)dr)

= f x (p ) d p f y (q ) d q(J c (E p, E q, r) X (r) dr )

= f x(p)X(p)dp f y(q)X(q)dq.


Clearly, this mUltiplicative functional generates a maximal ideal M associated with the character X (r) by relation (1.7). The constructed ideal M"* L 1 since, otherwise, X (r) = 0 almost everywhere.

Conversely, assume that a maximal ideal M *" L 1 is given. The correspondence x H x (M) is a linear functional in L 1. By using the theorem on the form of linear functionals in L 1, we obtain

x(M) = f x(r)x(r)dr (x E L))

where X (r) is a function essentially bounded by one (because the norm of the functional is equal to one). The function x(r) satisfies (1.6) since

f c(A, B, r)x(r)dr = (KA * KB)(M) = KA(M)KB(M) = X(A)X(B).

In view of the fact that· M *" L 1, the function X (r) cannot be zero almost everywhere, and, hence, X is a character.

• It follows from the proof of Theorem 1.2 that every character of a commutative hy

percomplex system is bounded almost everywhere by 1. We note that the measure c (A, B, r) is absolutely continuous with respect to the

measure m for any fixed BE 11o(Q) and r E Q. Indeed, if A is a set of measure zero, then it follows from (1.2) that c(A, B, r) = 0 almost everywhere. Since c(A, B, r)

is continuous in r and the measure m is positive on open sets, we have c(A, B, r) for all r E Q.

• In contrast to the measure c (A, B, r), the measure m r (~) is, in general, not abso

lutely continuous with respect to the square of the measure m. Let us present several simple examples.

Example 1. Let Q'= G be a locally compact group and let 'A be its left Haar mea

sure. We set c(A, B, r) = 'A(B-1rnA) and m(A)= 'A(A). One can easily see that c(A, B, r) satisfies all axioms of structure measure and m is its multiplicative measure. Convolution (1.4) becomes an ordinary convolution of functions on the group

(x * y)(r) = f x(p)y(P-J r)dp

and the hypercomplex system L J ( G, 'A) is a group algebra. If G is abelian, then (1.6) implies that the characters of the hypercomplex system L 1 (G, 'A) satisfy the standard equality for characters of the group G, namely, X(pq) = X(p)X(q) almost everywhere.


In particular, if G = IR 1 is an additive group of all real numbers, then we arrive at the ordinary convolution on the real axis

(x*y)(r) = f x(p)y(r-p)dp (X,YE L1(IR1,dt»

and the functions X (t) = e its (s E IR 1) are characters.

Example 2. Let Q.= [0,00) and let A be the Lebesgue measure. We set

1 c(A,B,r) = -[A«A+r)nB) + A(IA-rlnB)] (A,BE 11o([0,00»);r20),

2

where IA-rl = {plp=ls-rl. sEA}. It is easy to check that c(A,B,r) is a commutative structure measure and the Lebesgue measure is its multiplicative measure. This structure measure generates the convolution

1 = (x * y)(r) = - f [x(p + r) + x(lp - r I)]y(p )dp (r 20),

2 0

and relation (1.6) implies that the characters of the related hypercomplex system satisfy the functional equation

1 X(p)X(q) = 2"[X(p+q) + x(lp-ql)] (P,qE [0,00)).

This yields X (p ) = cos s p (s 2 0) almost everywhere.

Example 3. Let Q = [0, 00 ),

c(A,B,r)=A«r-A)nB), and m(A)=A(A) (A,BE 11o([0,00));r20).

Then L 1([0, 00)), m) with the Duhamel convolution

r

(x*y)(r) = f x(r-p)y(p)dp

o

is a commutative hypercomplex system The functions X (p) = e ZP (z E a:: , Re z ::; 0; p::; 0) are characters of this hypercomplex system.

1.2. Theorem of Existence of a Multiplicative Measure. In this subsection, we present the proof of the existence of a multiplicative measure mentioned in Subsection 1.1. Most of the conditions.of Theorem 1.3 are trivial for the compact case.


Theorem 1.3. Assume that the structure measure c(A, B, r) is commutative and

satisfies the following conditions:

(i) c(Q,A,r)E Cb(Q) for all AE 'Eo(Qo);

(ii) for all A E 'Eo (Qo), there exists the limit

lim c(Q A, r) = c(Q, A, 00) r~oo

where the function 'Eo(Q) 3 A ~ c(Q, A, 00) is a regular Borel premeasure;

(iii) forallopen OE 'Eo(Q), the following inequalities hold: c(Q,O,r»O (rE

Q) and c (Q, 0, 00 ) > 0. Then there exists at least one multiplicative measure.

Proof. First of all, we note that, for any A E 'Eo(Q), any B E 'E(Q), and any compact set K, there exists a sequence of open sets On ~ A such that

c(On,B,r) ) c(A,B,r) n~oo

uniformly on K. Indeed, by virtue of the regularity of the structure measure for fixed r,

one can find an open set On,r~A such that C(On,r,B,r)-c(A,B,r) < lin. Sincethe structure measure is continuous in r, there exists a neighborhood V (r) such that,

C(On,r, B,p)- c(A, B,p) < lin for any p E U(r). We cover K by the neighborhoods

U(r) and take the finite covering V(rl)' U(r2)' ... , U(rk)' Then

C(On,r, B, r) - c(A, B, r) < 1 In

for the open set On = n:=1 On, r i and all r E K. This yields the required assertion.

Note that the regularity of the structure measure enables us to choose this sequence so

that all On lie in a compact neighborhood of the set A. If B E 'Eo ( Q), then we take

suppc(B, 01, r) as K and conclude that c(B, On, r) converges to c(B, A, r) uni

formly in r.

Let us show that, for any A E 'Eo (Q) and £ > 0, one can find an open set 0 E

'Eo(Q), 0 ~ A, such that II c(Q, 0, r) - c(Q, A, r) 1100 < 00. In fact, by virtue of the

regularity of the premeasure c(Q, A, 00), one can indicate an open set 000 ~ A such

that Ic(Q,Ooo,oo)-c(Q,A,oo)1 < £/5.

Let K E 'Eo (Q) be a compact set such that, for all r E Q \K,

and Ic(Q,A,r) _ c(Q,A,oo)1 < E 5


For every open ° C 0 00 that contains A and every r E Q\K, we have

c(Q, A, r) - c(Q, 000,00) :::; c(Q, 0, r) - c(Q, 0,00) :::; c(Q, 000, r) - c(Q, A, r).

This implies that I c(Q, 0, r) - c(Q, 0, 00) I < 2£/5 for all r E Q\K. Finally, by using the assertion established at the beginning of the proof, we find an open set OK:::::> A

such that

SUpIC(Q,OK,r)-c(Q,A,r)l:::; £/5. reK

By combining the arguments presented above, for the open set ° = OK n 0 00, we obtain

sup I c(Q, 0, r) - c(Q,A, r) I reQ

:::; suplc(Q,O,r)-c(Q,A,r)1 + sup Ic(Q,O,r)-c(Q,A,r)1 reK reQ\K

:::; ~ + ·sup Ic(Q,O,r)-c(Q,o,oo)1 5 reQ \ K

+ sup I c(Q, A, r) - c(Q, A, 00) I + I c(Q, A, 00) - c(Q, 0,00) I :::; £. reQ \ K

Hence, for any A E $o(Q), one can find a sequence of open sets On E $o(Q), On:::::>

A, such that II c(Q, On, r) - c(Q, A, r) 1100 ---7 0 as n ---7 00.

Consider the space C~ (Q) of continuous real bounded functions with uniform

norm. An operator TB (B E $o(Q» is defined by the formula

(TBf)(r) = f f(p)dpc(Ep, B, r)

Let us show that the operator TB maps the space C~(Q) into itself. For any r E Q,

we consider a sequence of points rn E Q such that rn ---7 r as n ---7 00. Since c(Q, B,

r) E Cb (Q), by virtue of (H I), we conclude that

f f(p)dpc(Ep, B, rn) ---7 f f(P)dpc(Ep, B, r) (fE Co(Q»

and c(Q, B, rn) ---7 c(Q, B, r) as n ---7 00. This means that the sequence of measures

/.in = c( ., B, rn) weakly converges to the measure /.i = c(·, B, r). Hence, for any func

tion fE C~(Q), we have


-n-~-oo~) J f(P)dpc(Ep, B, r)

and TB: C:(Q) ---., C(Q). The function (TBf)(r) is bounded. Indeed,

sup I (TBj)(r) I ~ sup J If(P)ldpc(Ep,B,r) ~ Ilflloosupc(Q,B,r). rEQ rEQ rEQ

The operator T B is clearly linear and continuous. If the function f (r) belongs to the

interior of the cone of nonnegative functions ct,JR (Q) in C~, i.e., there is p > 0 such

that infrE Qf(r) ~ p, then (TofXr) ~ pc(Q, 0, r) for any open set ° E tBo(Q).

The function c(Q, 0, r) is separated from zero, i.e., c(Q, 0, r) ~ c > O. In fact, one

can find a compact set K such that

1 sup I c(Q, 0, r) - c(Q, 0,00) I ~ -c(Q, 0, 00)

reQ \ K 2

(c(Q, 0, 00) > 0 by condition 3». Let CI = minc(Q, 0, r). Then the required rEK

This implies that, for any open set ° E $o(Q), the operators To map the interior of

the cone of nonnegative functions into itself.

The operators TB (B E 'Bo(Q» are commuting. Indeed, by virtue of (H2), (H3),

and Lemma 1.5, we get

= 5 f(p)dp(f c(Ep, B, q)dqc(Eq,A, r) )

=. f f(p)dp(f c(B, Ep, q)dqc(Eq, A, r) )

= f f(p)dp(f c(Ep, A, q)dqc(B, E q, r) )

= f f(p)dp(f c(Ep,A, q)dqc(Eq, B, r) ) = TB(TAf)(r).

Thus, we have the collection { To I ° E 'Eo ( Q) is an open set} of linear continuous

commuting operators mapping the interior of the cone of all nonnegative continuous

functions bounded on Q into itself. By the Krein theorem (M. Krein and Rutman


[KrR]), the adjoint operators T~ have a common eigenvector IE (C~(Q))', i.e.,

T~/='Aol ('Ao>O), where the functional I is positive (note that the space (C~(Q»' is isometrically isomorphic to the space of regular finitely additive measures on the algebra induced by closed sets; therefore we may regard I as a regular positive finitely additive measure). Thus, we have

(I, To!> = 'AoO,f) (1.8)

for all fE C~(Q) and 0 E ~o(Q). By setting fer) == 1 in (1.8), we obtain 'Ao = (I,

c(Q, 0, ·»/(/,1). For any A E ~o(Q), we define a finitely additive measure m' as follows:

(I, c(Q, A, .» m'(A) =

(/,1)

Since, for any A E ~o(Q), there is a sequence of open sets ~o(Q) 3 On::::l A such

that c(Q, On, r) E C~(Q) converges uniformly to c(Q, A, r) E C~(Q) and the func

tional I is continuous, the pre-measure m' is regular. By the well-known theorem, there exists a unique regular Borel measure m such that m(A) = m'(A) for all

A E 'Bo(Q). Let us show that this is the required measure. We fix A E ~o(Q) and an

arbitrary open set 0 E 'Bo(Q). As known, for the continuous finite function c(A, 0, r),

there exists a sequence of uniformly finite simple functions f nCp) that converges uni

formly to c (A, 0, p). Let S be a compact set that contains supp f n (n = 1, 2, ... ) and

suppc(A,O,r). Wehave

If c(A, O,p)dpc(Q, Ep, r) - f fn(p)dpc(Q, Ep, r)1

:::; IIc(A,O,p) -fn(p)lIooc(Q,S,r)

:::; Ilc(A,O,P)-fn(P)IIoollc(Q,S,r)lIoo. (1.9)

It follows from (1.9) that

f fnCp)dpcCQ, Ep, r)

converges to

f c(A, 0, p) dpc(Q, Ep, r)

uniformly in r. This fact, the Lebesgue theorem, the continuity of the functional I,


axiom (H2), and (1.8) yield

J c(A, 0, r)dm(r) = lim J fn(r)dm(r) n~oo

= (I, (To c)(Q,A,») = (1,1)

= m(O)m(A).

(I, c(Q,O,r») (I, c(Q,A,r»)

(I, 1) (I, 1)

( 1.10)

Assume that B is an arbitrary open set in ~o(Q). We choose a sequence of open sets On::> B so that c(A, On, r) ) c(A, B, r) uniformly in r from QN::>

n~oo

supp c(A, On, r) ::> supp c(A, B, r). Since the measure m is regular, there is a sequence

of open sets Un::> B such that m( Un) ) m(B). We set V n = On nUn. n~OQ

Obviously, m(Vn) ) m(B) and c(A, Vn, r) ) c(A, B, r) uniformly in n~~ n~~

r from QN' By setting 0 = Vn in (LlO) and passing to the limit, we get

J c(A, B, r)dm(r) = m(A)m(B).

• 1.3. Normal Hypercomplex Systems. Below, we consider only hypercomplex sys

tems with a certain symmetry property possessed by many important examples. For a hypercomplex system, this condition is an analog of the transition to an inverse element in a group. For simplicity, we consider the "unimodular" case; in the general case, the normality condition must be rewritten so that, in Theorem 1.8 presented below, modular bialgebras are used instead of Hilbert algebras (Vainerman and Kac [YaK]).

(H5) A hypercomplex system is called normal if there is an involutive homeomor

phism Q 3 P H p* E Q such that m (E*) = m(E), c(A, B, C) = c(A *, C, B) = c(C,B*,A) (EE ~(Q), A,B,CE ~o(Q».

If r* = r for all r E Q, a normal hypercomplex system is called Hermitian. Every


Hermitian hypercomplex system is commutative. In fact, for any A, B, C E 'Bo(Q), we

have c(A, B, C) = c(A, C, B) = c(C, B, A) = c(B, A, C), whence c(A, B, r) = c(B, A, r)

almost everywhere by virtue of the arbitrariness of C E '13o(Q). According to (Ht), both functions in this equality are continuous.

• In a normal hypercomplex system, the correspondence L I 3 X = x(p) ~ x(p*) =

x * ELI is an involution. Indeed, all properties of involution are evident except the

equality (x * y)* = y* * x* (x, y E Lr). It suffices to prove this equality for the indi

cators of sets from '13o(Q); here, this is equivalent to the relation c(A, B, C*) = c(B*,

A *, C) (A, B, C E 'Bo(Q», which can be established by using (H5) as follows:

c(A, B, C*) = c(A *, C*, B) = c(B, C, A *) = c(B*, A *, C). (1.11)

• If a hypercomplex system does not contain the unity, then one can extend the involu

tion into the algebra X; I (Q, m) by setting (Ae + x) * = Xe + x*.

A normal hypercomplex system has a unique multiplicative measure. Indeed, let

r E Q be an arbitrary fixed point and let VI => V2 => ... be a sequence of bounded

neighborhoods of r that contract to this point. We fix an arbitrary set A E '130 ( Q) and

take an increasing sequence of compact sets QI => Q2 => ... such that QI;:) supp c(VJ,

A, r) and Q = U;=l Qn' Then

c(Q,A *, r) = lim c(Qn, A *, r) = lim lim _1_ f c(Qn,A *, s)ds n-too n-too k-too m(Vk ) vk

= lim lim _1_ f C(Vk,A, s)ds n-too k-t oo m(""') Q n

= lim lim -1-f c(Vk,A, s)ds n-t oo k-too m(""')

= lim _1-m(Vk)m(A) = meA). k-t oo m(Vk )

In particular, this implies that c(Q, A, r) = const < 00 for any A E 'Bo(Q).

(1.12)

•


For a normal hypercomplex system, the following inequality is true:

mr(AxB) = c(A,B,r) ~ meA) (A,BE 13(Q);rE Q). (1.13)

Indeed, it suffices to prove (1.13) for A, B E 13o(Q). By virtue of (1.12), we have

c(A,B,r) = c(B*,A*:r*) ~ c(Q,A*,r) = meA).

• Denote La(Q, m) = La, II·IIL" = 1I·lIa, (".) L2 = (',' )2' and L~ = La', where

a-I + a' - I = 1 ( a E [1, 00 ]).

Lemma 1.1. For f E La and g E La' (a E [1, 00]), the convolution f * g

exists, is a bounded function, and satisfies the inequality Ilf* g II~ ~ IIfllallflla,. This

function belongs to the space C~(Q) for a E (1,00). If a = 1, then, for the con

volution to be continuous, it suffices to require, in addition, that g E L~ for some ~ E [1,00).

Proof Let a E (1, 00) and let a and b be step functions (i.e., linear combina

tions of the indicators of sets from 13o(Q»). Relation (1.3), the HOlder inequality, and (1.13) yield

= lIaliallblla' (rE Q), (1.14)

where the function (a * b) is continuous by virtue of (HI). Approximating the func

tions f and g in the metrics of La and La' by the functions a and b and using

estimate (l.4), we conclude that (a * b)(r) uniformly converges to (j* g)(r). Hence,

(j* g)(r) is a continuous function satisfying estimate 0.4). Lemma 1.1 is proved in the

case a E (1, 00 ).

Now assume that a= 1, IE Lt, and gE Loo. By analogy with (1.14), we obtain

1 (f *' g) ( r) 1 = If S f(P) g ( q ) d m r (P, q) I


which implies that f* g(r) exists and is bounded.

Assume, in addition, that gEL ~ for some ~ E [1, 00). It suffices to show that one

can find sequences a nand b n of finite step functions such that II an * b n - f * g IL ~ 0

as n ~ O. We choose sequences an and bn such that

By using the Holder inequality and (1.13), we get

I (j* g)(r) - (an * bn)(r) I

~ I (j* g)(r) - (an * g)(r) I + I (an * g)(r) - (an * bn)(r) I

--~) O.

• It follows from Lemma 1.1 that if a normal hypercomplex system contains the left

unity, then its basis Q consists of at most countably many points with discrete topology. Indeed, let eEL I be a unity. By virtue of Lemma 1.1, for ex, = 1 and any g E

LIn L 00, the function e * g is continuous and coincides almost everywhere with g. If

an arbitrary gEL] n Loo coincides almost everywhere with a continuous function and

the measure is positive on open sets, then the topology of Q must be discrete. It re

mains to take into account the separability of Q.

• A similar assertion is true ifthe left unity is replaced by the right unity.

Lemma 1.2. For XE L] and fE L2, the convolutions (x*fXr) and (j*xXr)

exist for almost all r E Q and belong to L2. Furthermore,

II x * fib ~ II x 11]lIfIl2,

(1.15)

Proof. It suffices to prove the first inequality in (1.15); the second inequality can be

proved by analogy. Let A, B, C E 'l3o(Q). Then, by virtue of (H5), we have


For step functions a, b, and c, this yields

(a * b, c) = (a, c * b*) = (b, a* * c). (1.16)

By using the estimate in Lemma 1.1 with a = 2, we obtain

and, therefore, II a * b 112 s; II a 11111 b 112 in view of the arbitrariness of c. Approximat

ing x and f by the functions a and b in the metrics of LI and L2, respectively, we complete the proof of the lemma.

• It follows from Lemma 1.2 that, for fixed x ELI, one can define the operators of

left and right convolution in the space L2 as follows: L2 3fH x * f=L(x)fE L2 and

L2 3fH f* x=R(x)fEL2. These operators are bounded and satisfy the inequalities

IIL(x)IIS;llxll l and IIR(x)IIS;lIxlll. It is clear that L(AX+IlY) = AL(x)+IlL(y)

and L(x*y)=L(x)L(y) (X,YELI, A,IlE CC). The adjoint operator satisfies the

equality L * (x) = L (x *) (x E L d. This is equivalent to the relation

obtained by passing to the limit in (1.16). The right convolution R(x) possesses similar properties. If a hypercomplex system LdQ, m) is commutative, then L(x) = R(x) and

L(x) is a normal operator:

L*(x)L(x) = L(x* * x) = L(x * x*) = L(x)L(x*) = L(x)L*(x).

In the space L2, in addition to the convolutions L(x) and R (x) (x E L d, we define one more operator family associated with hypercomplex systems, namely, left

(right) generalized translation operators Lp (Rp), p E Q. The operator Lp is given in

L2 by the bilinear form (Lpj,gh = U*g*)(p) (f,gE L2; pE Q). The operator Rp

is defined similarly: (Rpj,gh = (g* * f)(p), f, g E L2, P E Q (recall that we consider the case of unimodular hypercomplex systems). It follows from Lemma 1.1 with a = 2

that these forms are continuous and, hence, the operators L p and R p are well defined

and satisfy the inequalities II Lpll S; 1 and II Rpll S; 1. Since


we have L~ = L p* (p E Q). Similarly, R; = Rp •. In the case where Q is a locally

compact group and the hypercomplex system L, is its group algebra, we have Lpf(q) =

f(pq) and Rpf(q)=f(qp) (p,qE Q). The operator families Lp and Rp (pE Q) possess all properties of the operator family of Delsarte-Levitan generalized translations (Levitan [Lev9]); in Section 1.2, the relationship between hypercomplex systems and generalized translation operators is discussed in detail.

Obviously,

L(x) = f x(p )Lp• dp (XE Ld (1.17)

i.e., the operator of left convolution is the adjoint operator of left translation "averaged with function x". Let us show that Lemma 1.1 with a = 2 implies the weak continuity

of the function Q3p ~LpE 1;(L2,L2), where 1; (E"E2) is the space of linear continuous operators from E, to E2. Note that the integral in (1.17) should be understood in the weak sense.

Let us prove (1.17). Assume that f, g E L2. Then

((f x(p)Lp*dp )t: g)2 = f x(p)(Lp·f,g)zdp

= f x(p)(f*g*)(p*)dp = (x,g*f*). ( 1.18)

Passing to the limit in (1.16) as a tends to x in L, and band c tend to f and g, res

pectively,in L2, we get (x*j.g)z= (x,g*f*). Therefore, (1.18) can be continued as follows:

• Let L, (Q, m) be a commutative normal hypercomplex system. A character X (p)

is called Hermitian if X (p*) = X(p). Hermitian characters of an Hermitian hyper

complex system are real-valued: X(p) = X (p * ) = X (p) (p E Q).

Generally speaking, the Banach algebra L 1 (Q, m) is not symmetric, i.e., the equal

ity M* = M holds not for all maximal ideals of it (or, in view of (1.7), not all characters of it are Hermitian). Denote the set of all characters (generalized characters) and the set

of all Hermitian characters (generalized Hermitian characters) by X (Xg) and X h

(Xg,h), respectively.

Theorem 1.4. The characters of a commutative hypercomplex system are continu

ous functions (more precisely, coincide with such functions almost everywhere). If Q


is compact, then these characters are Hermitian; the set of such characters is at most countable, they are mutually orthogonal and form, up to constant factors, the system of idempotents of the algebra L 1, i.e.,

Proof. Let A, B E 'So (Qo), with X (B) *- O. We construct a compact set F so that c(A,B,r)=O for re F and set xF(r)=x(r) for rE F and xF(r)=O at all other

points. If E c A and ex = 1, according to (1.6), (HS), and Lemma 1.1, we obtain

X(E)X(B) = f c(E, B, r)x(r)dr = f c(E, B, r)xF(r)dr

= J (X p * KB*)(p)dp. E

Since E is arbitrary, we have

( 1.19)

for almost all rEA. The arbitrariness of A, this equality, and Lemma 1.1 imply the

continuity of X. Let Q be compact. The Hermitian property for X follows from the relation

x(A)llxll~ = J x(p)dp(f c(Ep,A, r)x(r)dr)

= j(x(p)dp(f x(r)drc(Ep,A,Er))

= x(A*)llxll~ (AE'S(Q)). (1.20)

To prove the orthogonality of two characters X' and X" (X' *- X"), we take B E


'B(Q) such that X'(B)-:t:X"(B)= X"(B*). Inviewof(1.19),wehave

(X'(B)- X"(B*) )f X'(p)X"(p*)dp

= (X' * KB*' x"h - (X', X" * KBh = 0, ( 1.21)

whence (X', x"h = O.

The separability of L 2 and the orthogonality of characters imply that there are at

most countably many characters and we can set X = (Xj )~:1' The last assertion of the

theorem easily follows from the relation (see (1.19))

• We introduce a topology in X by using the topology of the space of maximal ideals

of the commutative Banach algebra L 1 (Q, m). It is easy to check that the topology in X coincides with the topology of uniform convergence on compact sets. The set X h of

Hermitian characters is clearly closed in X. Consider the general case of noncommutative hypercomplex systems An element a

of the algebra K is called an annihilator if a * x = 0 for all x E K.

Theorem 1.5. The radical of a normal hypercomplex system coincides with the col

lection of its annihilators.

Proof If x is an annihilator, then it belongs the radical, because x * x = O. We show the inverse inclusion. Assume that u ELI is a fixed function such that u * x =

u * * x for any x ELI. In the Hilbert space L 2, we consider the convolution operator Lu with the function u, i.e., L2 E f H Luf = u * fE L2.

Let us show that the operator Luis selfadjoint. In fact, L: = L u*' But L u*f=

u* * f = u * f = L uf for an arbitrary function f ELI n LOQ' Therefore, the selfadjointness

of L u follows from the denseness of LIn L OQ in L2 and the fact that the operator L u

is bounded.

N ow let u (r) belong, in addition, to the radical ~ of the algebra L 1 (Q, m). Then,

clearly, the element 0 e + u belongs to the radical of the algebra 1; 1 (Q, m). We show

that the operator L u = O. Indeed, let R z be the resolvent of the operator L u' It is easy

to see that R zf = (u - Z e) -I * f (f E L 2). Since u (r) belongs to the radical, the func

tion (u-ze)-I is defined for all z-:t:O andbelongsto .n1(Q,m). Let (u-ze)-1 =


Ae+y (YE L1(Q,m)). By virtue of Lemma 1.2., for fE L2, we have (u-zer 1 * f = Af + y * f E L2· Hence, the resolvent R z is defined for all z '* O. Therefore, the

spectrum of the self-adjoint operator L consists of zero and, consequently, L u = O. In

particular, L uf= u * f = 0 for every function f ELI n Loo. In view of the continuity of

multiplication in the algebra L 1 (Q, m) and the denseness of LIn L 00 in Loa, we

conclude that u * f = 0 for all f ELI. The theorem is proved for the special case under

consideration (u * x = u* * x, X ELI)'

Let v E m be an arbitrary element of the radical. Since the radical of a *-algebra is

a symmetric two-sided ideal, v* also belongs to the radical 9t of the algebra L1 (Q,

m). In this case, u = v + v* E m. Therefore, u * x = 0 (x E L)) according to the al

ready proved result, i.e., v * x = -v* * x (f EL 1)- Consequently, the element iv satisfies the condition of the already proved part of the theorem and, thus, it is an annihilator.

together with v.

• Corollary 1.1. For a normal hypercomplex system to be semisimpZe, it is necessary

and sufficient that it do not contain annihilators other than zero.

We illustrate the ideas discussed in this section by several examples presented at the end of Section 1.1. It is easy to see that the hypercomplex system in Example 1 is nor

mal if we set r* = r- 1 and the group G is unimodular. The hypercomplex system in

Example 2 is Hermitian for r* = r. The hypercomplex system in Example 3 is not normal. The generalized translation operators associated with the hypercomplex system in Example 2 are given by the formula

1 (Lpx)(q) = 2"[x(P+q) + x(lp-ql)] (P,qE [0,00)).

1.4. Normal Hypercomplex Systems with Basis Unity. In this subsection, we discuss generalizations of finite-dimensional normal hypercomplex systems with unity element e in its basis to the continual case. We say that a normal hypercomplex system

possesses a basis unity if

(H6) there exists a point e in Q such that e* = e and c (A, B, E) = m (A * n B)

(A, B E 'B(Q».

The basis unity of a normal hypercomplex system is called strong if

(H7) for each neighborhood U of an arbitrary set FE 'Bo(Q), there is a neighborhood

V of the basis unity e such that suppc(F, V, r) cU.

A sequence of functions en ELI is called an approximative unit of a hypercomplex


e n* x --~) x weakly in LI. n~oo

If a normal hypercomplex system possesses the approximative unit {en};=1' then

en*f~f strongly in L2 for all fE L2 as n~oo.

In fact, assume that g E L2 and a and b are step functions. Then

by virtue of (1.15). Since (en);=1 is an approximative unit, this implies that en * f ~ f weakly in L2. Strong convergence now follows from the relation

--~) f, n~oo

(we have used (1.15) and weak convergence just proved).

• Lemma 1.3. A normal hypercomplex system with basis unity possesses an approxi

mative unit.

Proof. Assume that (On);=1' On~ On+l, is a sequence of bounded neighbor

hoods of the basis unity e contracting to this point. We show that the sequence of func

tions enE LI such that en(P)~O (pE Q), suppencOn, and Ile nlll=1 isanap

proximative unit of the hypercomplex system (specifically, one can take e n(P) =

(m( On»-l KOn (jJ) (p E Q; n = 1, 2, ... ). If necessary, one may take e n(P) E CO (Q)

(the existence of a function of this sort for all s is guaranteed by the Uryson lemma). Let us show that x * en ) x weakly in L I. Let IE L{ = Loo be a linear func-

n~oo

tional in L l. We must prove that I (e n * x) ~ I (x) as n ~ 00. It suffices to consider

the case x = KA (A, B E 'B(Q)). Then the required assertion is also true for step func

tions which are dense in L I and the proof is completed by using the inequality

Il(en*x)-I(x)1 ~ II(en*a)-I(a)1 + 211L1Llix-alll

(we have used the fact that IIen*(x-a)!ll ~ lIenlllllx-alil = IIx-alll)' Thus,let

x = KA. By F c Q we denote a sufficiently large compact set such that F ~ On (n = 1,

2, ... ). By virtue of (HI), there is a compact set G ~ A such that c(A, B, E)= 0 on the


complement of G. Let la be a function equal to I (r) for rEG and equal to zero at the other points. By using (1.16), we obtain

The function I a * KA * is continuous for a = 1 by virtue of Lemma 1.1. Therefore,

by applying the standard argument and (H6) to the last inequality, we obtain

• Corollary 1.2. A normal hypercomplex system with approximative unit is semi

simple. In particular, a normal hypercomplex system with basis unity is semisimple.

Indeed, in view of Theorem 1.5, it is necessary to show the absence of nonzero anni

hilators in the hypercomplex system. Let x be an annihilator. Then

x = lim x * en = 0 n~~

in a sense of weak convergence in L 1, i.e., x(r) almost everywhere.

• Clearly, the approximative unit en(p) constructed in proving Lemma 1.3 has the

following property:

f en(p)f(p)dp -7 fee) as n -7 00 (fE Cb(Q»· (1.22)

We now establish several properties of the collection of characters of a commutative normal hypercomplex system with basis unity. First, we recall the following definition:

the system ~ of functions on Q is called complete in a class ~ of functions on Q if

the fact that the equality

f x(q)q>(q)dq = 0

is true for all q> E ~ and some x E ~ (all x(q)q>(q) are supposed to be summable)

implies that x(q) = 0 almost everywhere.


Theorem 1.6. In order for a point e E Q to be the basis unity of a commutative

normal hypercomplex system, it is necessary and sufficient that 1) the system of all characters be complete in L I ;

2) for each character, X ( e ) = 1.

Proof. Necessity. The fact that a normal hypercomplex system with basis unity is semisimple yields the completeness of the system of all characters. We show that

X (e) = 1 for every character. We take a compact set A such that X (A) ;t; 0 and denote by UI::> U2::> ... a sequence of balls contracting to e. It follows from Lemma 1.3 that

Hence,

X (A) lim X(Un )

n~oo m(Un) = lim _1_ f c(A, U:, r)x(r)dr

n~oo m(Un)

= lim f (KA * en)(r)x(r)dr = X(A). n~oo

lim X(Un ) = 1. n~oo m(Un)

Together with the continuity of the character X, this yields the equality X (e) = 1.

Sufficiency. Let B be a compact set, and let U be a precompact neighborhood of

the point e. Consider a compact set F such that c ( U, B *, r) = 0 for reF and define

X F(r) as in the proof of Theorem 1.4. For E c U, X E X, we have

f (X F * KB)(r)dr = f XF(P)dpc(Ep, B, E) E

= f XF(P)dpc(E, B*, Ep) = f c(E, B*,p)X(p)dp

Since E c U is arbitrarily and the functions (X F * KB)( r) and X (r) are continuous,

've obtain

(1.23)

Consider the measure c(A, B, e) (A E ~o(Q». For A c Q\F, we have c(A, B,

e) = 0 since


c(A,B, U) = c(U,B*,A) = f c(U,B*,r)dr = O. A

Moreover, it follows from (1.13) that the magnitude of the derivative of the measure

c(A, B, e) with respect to meA) is bounded by the unit element. This implies that there exists a bounded function c (r) vanishing outside F such that

c(A,B, e) = f c(r)KB(r)dr (A E 'Eo(Q».

For any character, by using (1.23), we obtain

f [c(r) - KB*(r)]x(r)dr = f x(r)c(r)dr-x(B*)

F

Since c (r) - K B' (r) is bounded and finite, we have c (r) = K B *( r) almost everywhere

because the system of characters is complete in L 1. Hence, for all A E 'Eo ( Q),

c(A, B, e) = f c(r)dr = f KB*(r)dr = f KB(r)d = meA * n B). A A A

• As follows from the proof, for a point e E Q to be the basis unity of a commutative

normal hypercomplex system, it suffices to check the equality X (e) = 1 and the fact that the system of characters is complete in the class of essentially bounded finite functions.

It is clear that the hypercomplex systems considered in Examples 1 and 2 (Subsection

1.1) are systems with basis unity: In Example 1, the basis unity e coincides with the

unit element of the group, whereas in Example 2, e = O.

1.5. Even Subsystem of a Normal Hypercomplex System. In this subsection, we construct an Hermitian hypercomplex system which is a subalgebra of the initial normal hypercomplex system.

Let x(r) be a function on Q. We set x'(r) = x(r*) and call the function x(r)

even (odd) if x'=x (x'=-x). Clearly, every function on Q can be uniquely represented as the sum of its even and odd components, namely,

x = .!.(x+x') + .!.(x-x'). 2 2


Consider the mapping x ~ x' in the space L 1. This mapping is linear, involutive,

and (x * y)' = y' * x'. Therefore, the set of all even functions from L 1 forms a commu

tative sub algebra of L 1 (Q, m). We denote this sub algebra by HI (Q, m) and call it the

even subsystem. It can be regarded as an Hermitian hypercomplex system whose basis is a locally compact space Q obtained from the initial basis Q by uniting the points

rand r* into the class r.

In fact, we denote the preimages ofthe sets A, B, E ... E 'B(Q) under the mapping

r ~ r (r E r) by A, B, E, ... E 'B(Q). These preimages are invariant under *. We in

troduce a measure m on Q by setting m(E) = m(E) (E E 'B(Q)). Denote dm(r) =

dr. Every function x(r)E LI(Q,m) induces an even function x(r)E LI(Q,m), and vice versa. In addition,

J x(r)dr = J x(r)dr.

E E

This readily implies that the convolution of two functions from HI (Q, m) can be represented in the form

where the measure c is defined by the equality c (A, B, r) = c (A, B, r). Thus, H tC Q, m) is a hypercomple~ system for which the Hermitian property follows from the rela

tions c(A,B,C)=c(A,B, C) and B=B*.

• One can easily show that if X (r) is an Hermitian character of a commutative

normal hypercomplex system, then the function

1 vCr) = "2[x(r) + x(r*)] = Rex(r) (r E r) (1.24)

is an Hermitian character of its even subsystem.

Indeed, let M be a maximal ideal of the algebra X 1 (Q, m) associated with the

character X (for definiteness, we assume that L 1 (Q, m) does not contain the unit ele

ment). The mapping x ~ x(M) (x E HI (Q, m» is a homeomorphism of the algebra

HI (Q, m) in the field of complex numbers. This homeomorphism does not map all

HI (Q, m) into zero because if we assume the opposite, then it possible to find an odd

function x E LI such that x(M) '# O. But then (x* * x)(M) = Ix(M) 12 '# 0 in contra

diction with our hypothesis because x* * x is even. Thus, the homeomorphism x ~ x(M) (x E HI (Q, m» generates a character of the even subsystem. We denote it by vCr). Forevery XE H 1(Q,m), we have


f x(r)v(r)dr = f x(r)x(r)dr,

which gives the required equality.

• If the even subsystem of a commutative hypercomplex system is a symmetric alge

bra, then every its character admits representation (1.24).

Indeed, let v (r) be a character of the hypercomplex system H 1 (Q, m) and let M' be the corresponding maximal ideal. By virtue of the Shilov theorem on the extensions

of maximal ideals [Shi], the maximal ideal M can be extended as a maximal ideal of the

symmetric sub algebra to the maximal ideal M of the entire algebra. Since the mapping

x --? x(M) (x ELI) clearly does not map all L 1 (Q, m) into zero, M is generated by a

character X (r) of the hypercomplex system L 1 (Q, m). This is just the required character.

• One can show that ifthe hypercomp1ex system Ll (Q, m) is semisimple (with basis

unity), then the hypercomplex system HI (Q, m) is also semisimp1e (with basis unity).

The converse assertion"is weaker: If the even subsystem HI (Q, m) of a commutative

hypercomplex system L I (Q, m) possesses a basis unity and the hypercomplex system

L 1 (Q, m) is semisimple, then L 1 (Q, m) possesses a basis unity The hypercomplex system in Example 2 (Section 1.1) is the even subsystem of the

hypercomplex system L I (IR, d) with ordinary convolution. In this case, equality

(1.24) takes the form cos Ax = Re ei'Ax.

1.6. The Case of a Discrete Basis. In this subsection, we study basic notions of the theory of hypercomplex systems in the case where the basis of a hypercomplex system is a countable set Q with discrete topology The structure measure c(A, B, r) transforms into the collection of structure constants c (p, q, r) (p, q, r E Q) with the following

properties: (a) c(p,q,r)"C.O, (~) for any fixed p,qE Q, the sequence (c(P,q,

r )) r E Q is finite, and (y) the associativity relation

LC(p,q,r)c(r,l,s) = Lc(q,l,r)c(p,r,s) (1.25) r r

holds. The multiplicative measure turns into the multiplicative weight, i.e., the sequence of

positive numbers m (p) (p E Q) such that

L c(P, q, r)m(r) = m(p)m(q) (p, q E Q) (1.26) r


The hypercomplex system with basis Q is the space II (m) of sequences of complex numbers summable with respect to the multiplicative weight with convolution

(x*y)(r) = Lx(P)y(q)c(p,q,r) (X,YE II(m». p,q

Any bounded sequence X (p ) E a:: I (p E Q) not identically equal to zero and such that

L c(p, q, r)x(r)m(r) = x(p)m(p)x(q)m(q) (1.27) r

is a character.

This hypercomp1ex system is normal if there exists an involutive mapping Q 3

P H p* E Q such that m(p) = m(p*) and

c(p, q, r)m(r) = c(r, q*,p)m(p) = c(p*, r, q)m(q).

The involution in the hypercomplex system II (m) is given in the standard way:

The point e E Q is the basis unity of the hypercomplex system II (m) provided that

e* = e and c(p, q, e)= 8 pq* (8 pq is the Kronecker symbol and p, q E Q).

If the hypercomplex system II (m) possesses a basis unity, then there is a two

sided identity element with respect to the convolution in this hypercomp7ex system. The

sequence e (P), equal to 1/ m (e) for p = e and to zero for p *' e, is the required

element.

In fact,

1 1 (e * x)(r) = - L x(q)c(e, q, r) = -- L x(q)c(r, q*, e)

m(e) mer) q q

. 1 " = - £..J x(q)8 r,qm(q) = x(r) mer)

q

(p, q, rE Q).

The generalized translation operator in a normal hypercomplex system has the form

1 Lpx(q) = L x(r)c(p, q, r)m(r).

m(p)m(q) q

(1.28)

We give a simple example of a discrete hypercomplex system. Let Q = {O, I, ... }.


We set m (0) = 1/ 1t and m (p ) = 2/ 1t (p:to 0) and define the structure measure as follows:

m(p)m(q) c(p,q,r) = 2m(r) [6Ip-ql,r+6Ip+QI,r]'

The obtained hypercomplex system is Hermitian and possesses the basis unity e = 0. The functions X (p) = cos P t (t E [0, 1t]) are characters of this hypercomplex system. One can easily show that the hypercomplex system II (m) is isometrically isomorphic to the algebra of continuous functions on [0, 1t] with ordinary algebraic operations and absolutely convergent Fourier cosine expansions, i.e., it is isomorphic to the algebra of functions

1 2 00

x(t) = -x(o) + - L x(p)cospt 1t 1t p=1

such that

00

L Ix(P)1 < 00.

n=O

Theorem 1.7. The following conditions are equivalent:

(i) the basis Q of a normal hypercomplex system with basis unity is discrete;

(ii) the point e is isolated;

(iii) m(e) > 0.

Proof. The implication (i) => (ii) is evident. The implication (ii) => (iii) holds due to the fact that the multiplicative measure is positive on open sets. We now show that (iii) implies (i). First, we prove that the function

is the unit element in the algebra L 1 (D, m). By virtue of Lemma 1.1, (g * f)( r) E Coo(Q) forallfE Ll. We assume that f(r)E Cb(Q)nLl and En is a sequence of

neighborhoods of the point r E Q contracting to this point. Then

f* g(r) = lim _1_ J (j* g )(r)dr n~oo m(En)

En

Section 1 Fundamental Concepts of the Theory of Hypercomplex Systems

= lim 1 1 J f(p)dpc(Ep, {e}, En) n~oo m({e}) m(En)

= lim _1_ J f(p)dp = fer). n~oo m(En)

En

37

by virtue of Lemma 1.1 (see below). By approximating an arbitrary function f E L j by

functions from Lj n Cb(Q) in the metric of L j and applying Lemma 1.1, we conclude

that (j* g)(r) = (g * f)(r) = fer) almost everywhere.

• 1. 7. Hilbert Algebras. Let us show that a normal hypercomplex system with basis

unity is closely related to the theory of double Hilbert algebras (Hilbert bialgebras).

Recall some definitions (for simplicity, we consider the unimodular case). The lineal fj)

dense in a Hilbert space H is called a Hilbert algebra if fj) is a *-algebra with the following properties:

(i) (a,b)H=(b*,a*)H (a,bE fj);

(iii) for each a E fj), the operators L (a)b = a * b (b E fj) can be extended to con

tinuous operators in the entire space H;

(iv) fj) * fj) is dense in H.

Suppose that the following two pairs of operations are defined on a lineal fj) dense

in the Hilbert space H: the multiplication aU b and involution aU (the first pair) and

the multiplication an b and involution an (the second pair). In this case, the lineal fj)

is called a Hilbert bialgebra if

(i) for each pair of operations, fj) is a Hilbert algebra in H;

(ii) the operator W uniquely defined in H ® H by the equality


is continuous (W is called a binding operator).

Theorem 1.8. Let ~) (Q, m) be a normal hypercomplex system with basis unity.

Then the lineal 'lJ = L) n Loo dense in the Hilbert space H = L2 is a Hilbert bialgebra with respect to the operations f U g = f * g and f n g = f· g and the involu-

tions fer) H f(r*) and fer) H fer), respectively.

Proof. By virtue of Theorem 1.1, Lemma 1.1, and the normality condition, 'lJ is a

*-algebra with respect to convolution and the involution t. It is clear that 'lJ is a *-algebra with respect to pointwise multiplication of functions and complex conjugation.

The fact that ('lJ,.) is a Hilbert algebra is also obvious.

Let us show that (f]), *) is a Hilbert algebra. The first condition is satisfied by vir

tue of the fact that m(A)= meA *) (A E iJ3(Q», the second one follows from (1.16), the third condition is a consequence of Lemma 1.2, and, finally, the fourth condition follows from Lemma 1.3 and the fact that the existence of an approximative unit e nC r) in a

hypercomplex system allows one to approximate an arbitrary function f E 'lJ in L 2 by convolutions of the form en * f (it should be noted that the functions e n( r) constructed

in Lemma 1.3 lie in tJJ). Therefore, it remains to show that the binding operator is con

tinuous. Indeed, for all f),h, g), g2 E 'lJ, by virtue of Lemma 1.2 and the Schwartz inequality, we have

and, thus, II W II ~ 1.

• The fact that normal hypercomplex systems with basis unity have the structure of

Hilbert bialgebra implies that these systems possess many important properties (see Vainerman and Litvinov [VaL]). However, the presence of such a structure does not give a complete description of a hypercomplex system. In Section 5, we formulate axioms that distinguish hypercomplex systems from the class of Hilbert bialgebras.

Section I Fundamental Concepts oj the Theory oj Hypercomplex Systems 39

1.8. Parameter Depending Measures. In the construction of hypercomplex systems with locally compact basis, we have used several auxiliary lemmas concerning the integration over measures depending on a parameter. Though these assertions are rather simple, in this subsection, we give their proofs for completeness of presentation.

Let Q be a locally compact space. Recall that a nonnegative measure Pm (E) (E E

'B(Q)) is called a majorant of a measure p (E) (generally speaking, complex-valued) if

I p (E) I ~ Pm (E) (E E 'B(Q)). The obvious relation I Pm I(E) = (Var P )(E) ~ Pm (E)

(E E 'B(Q» implies the following statement:

Lemma 1.4. The inequality

If J(r)dp(r) I ~ f IJ(r) Idpm(r) (p E M(Q))

holds Jor any measurable Junction J( r) (r E Q).

Lemma 1.5. Let a(E, r) and p (E) (rE Q) be nonnegative measures with

respect to E E 'B(Q) and let a (E, r) and J(r) be nonnegative measurable Junc

tions oj r E Q such that

f a (E, r) d p ( r) < 00

Jor any E E 'Bo(Q). Then the existence oj at least one oJthe integrals

(1.29)

or

f J(r)dr(j a(Er. s)dp(s) ) (1.30)

implies that the other integral exists and they are equal. (The integral in (1.30) makes sense because one can easily verify that

f a(E, r)dp(r) (E E 'Bo(Q»

is a nonnegative measure}.

Proof. For a characteristic function J(t) (and, hence, for a step function), the statement is obvious.

Let J(t) be an arbitrary nonnegative measurable function for which integral (1.29) exists. Let us show that (1.30) also exists and is equal to (1.29). Indeed, it follows from the existence of (1.29) that J(t) is almost everywhere finite with respect to the measure


",(E) = f a(E, r)dp(r).

Then one can construct a sequence of step functions 0:::;, fl (r) :::;, fz(r) :::;, ... :::;,

fn(r):::;' ... :::;'f(r) that converges to fer) almost everywhere with respect to 'II (E). This

implies that this sequence converges to f(t) almost everywhere with respect to a(E, r)

for almost all r with respect to p (E). For such r, by virtue of the Lebesgue theorem on majorized convergence, we have

---7) ff(t)dta(E t, r). n~oo

(1.31)

By using the Lebesgue theorem on the integration of monotone sequences and the finiteness of integral (1.29), we get

f f(t)d",(t) = lim f fn(t)d",(t) n~oo

= lim f fn(t)dt(f a(Et, r)dp(r) ) n~oo

Let f(t) be a nonnegative measurable function for which integral (1.30) exists.

Then one can construct a sequence of step functions 0:::;, f I (t) :::;, fz (t) :::;, '" :::;, f( r) that

converges to f(t) almost everywhere with respect to ",(E). But then this sequence converges to f(t) a(E, r)-almost everywhere for p(E)-almost every r. For such r,

relation (1.31) is valid. Therefore, by virtue of the Lebesgue theorem on the integration of monotone sequences, we obtain

f (f f(t)dfa(E f, r) )dp(r) = lim f (f fn(t)dta(E t, r) )dp(r) n~oo

= lim f fn(f)dt(f a(Et, s)dp(s) ) n~oo

= f f(f)dt(f a(E" s)dp(s) ).

• Lemmas 1.4 and 1.§ readily yield the following statement:


Lemma 1.6. Let 0' (E, t) and p (E) (t E Q) be measures with respect to E E

1:l(Q) and let aCE, r) be measurable with respect to r E Q. Suppose that the measures aCE, r) and p·(E) have majorants a (E, r) and p (E), the first of which is measurable with respect to rand

J cr (E, r)ip (r) < 00 (E E ~(Q)).

Then the existence of at least one of the integrals

f(J 1i(t)ldtcr(E,r) )dp(r)

or

J If(t) I dt(J cr(E,r)dp(r)),

where f(t) is a measurable function, implies that integrals (1.29) and (1.30) exist and coincide.

Lemma 1.7. If \jI(A, B) (A, B E ~(Q)) is a nonnegative regular measure with

respect to A (B) with fixed B E ~o(Q) (A E ~o(Q)) and f(t) (t E Q) is a

measurable function summable with respect to the measure d r \jI (E n A) for all

A E 1:lo(Q), then the function

P (A) = J f(t)d\jl(E t, A)

can be extended to a unique regular a-additive measure on ~(Q).

Proof. By virtue of Theorem 503 in [Schw, p. 584], it suffices to consider a nonnegative function f(t). The function p (A) is finitely additive, p(0) = 0, and, evidently, p (A 1) ~ p (A 2) if Ale A 2. Therefore, p (A) is a compact volume. It follows from the regularity of the measure \jI(A, B) in B that the compact volume is regular. This

implies that p (A) can be uniquely continued to a regular a-additive measure on ~(Q) .

•


2. Hypercomplex Systems and Related Objects

Generalized translation operators were first introduced by Delsarte [Dell]-[De13] as an object that generalizes the idea of translation on a group. Later, they were systematically studied by Levitan [Levl]-[LevlO]; for some classes of generalized translation operators, he obtained generalizations of harmonic analysis, the Lie theory, the theory of almost periodic functions, the theory of group representations, etc. The fact that generalized translation operators arise in various problems of analysis is explained by the following result of Vainerman and Litvinov [VaL]: Transformations of Fourier type for which the Plancherel theorem and the inversion formula hold, as a rule, are closely related to families of generalized translation operators. As shown in Section 1, each hypercomplex system can be associated with a family of generalized translation operators. In Subsection 2.1, we present conditions that distinguish the class of hypercomplex systems from the class of generalized translation operators. Then we clarify the relationship between hypercomplex systems and hypergroups, which were independently introduced (with slight differences in axiomatics) by Dunkl [Dun2], [Dun3], Spector [Spel]-[Spe3], and Jewett [Jew] and are extensively studied. A hypergroup is a locally compact space Q on which the Banach * -algebra M b( Q) of bounded Radon measures

on Q with the identity element be (e E Q) is defined; the operation of multiplication

in M b( Q) (in what follows, called "convolution") satisfies certain conditions, the most important of which is that the convolution of probability measures is also a probability measure. Actually, hypercomplex systems and hypergroups describe the same algebraic object. The only difference is that operation of convolution in a hypergroup satisfies stronger topological conditions than in a hypercomplex system Note that the concepts of discrete hypercomplex system and discrete hypergroup are equivalent. It seems that the language of hypercomplex systems is more suitable for the construction of duality theory, whereas the language of hypergroups is preferable for studying the structure of an object itself (see, e.g., Section 4). At the end of this section, we present one of possible extensions of hypercomplex systems to the case of complex structure measure and establish the relationship between such hypercomplex systems and the generalized convolution algebras introduced by Ionescu Tulcea and Simon [loS].

2.1. Generalized Translation Operators and Hypercomplex Systems. Let us establish the relationship between the hypercomplex systems and Delsarte-Levitan generalized translation operators outlined in Subsection 1.3. Below, for convenience, we present some necessary facts from the theory of generalized translation operators (see Levitan [Lev9], [Lev 10]).

Let Q be an arbitrary set and let . be a space of complex-valued functions on Q. Assume that an operator-valued function Q 3 P 1---7 Rp: is given such that the

Section 2 Hypercomplex Systems and Related Objects 43

function g(P) = (Rpf)(q) belongs to for any fE and any fixed qE Q. Denote by L q the operator 3 f(P) H (L q J) (p) = (R pf)( q ) E . The operators R pare called generalized translation operators, provided that the following axioms are satisfied:

(Tl) Associativity axiom. The equality (RfiCRqf))Cr) = (R;(Rpf))(r) holds for

any elements p, q E Q (the notation (R$CRqf))Cr) means that the operator Rp

acts on the function (Rqf)(r) of two variables q and r in the variable q).

(T2) There exists an element e E Q such that R e = I, where I is the identity oper

ator in .

The associativity axiom can be rewritten in a more transparent form (Delsarte [Dell]-[DeI4]):

(TI') LqRp = RpLq (p, q E Q).

Indeed,

i.e., (TI) implies (TI'). Properly rewriting this chain of equalities, one can obtain the converse statement.

The associativity axiom (Tl) also holds for the operators Lp.

Indeed,

= ( R( C Lpf) ) (q) = (Eq (Lpf) ) ( r) (p, q, r E Q) .

• Furthermore, the operators Rp and Lq commute:

The operator Leis a projector, i.e., L~ = L e (the subspace = L e is invariant

under the action of Le). Indeed,


= (Re(Rpf»(e) = (Rpf)(e) = (Le!)(p) (p E Q).

The operator Leis not necessarily identical. Therefore, generally speaking, the oper

ators L p are not generalized translqtion operators. To avoid this discrepancy be-

tween Rp and L p, one should replace cI> by = LecI> and consider, instead of Q,

the set Q obtained from Q by identifying all points p' and p II such that f(P') = f(p ") for all f E into the class p. Since

(Rp'f)(q) = (Lqf)(p') = (Lqf)(p") = (Rp"f)(q) (p',p" E p E Q; q E Q),

for all f E <b, the operators (Rpf) ( ij) = (Rpf) (q) (p E p, q E ij) are well defined

on Q. Clearly, they form the family of generalized translation operators. The operators

Lp = Lp (p E P E Q) associated with Rp are, in fact, generalized translation oper

ators because 4 cI> = Lecl> = cI>. The described process is called reduction and

 = LecI> is called the principal space. As a result of reduction, a smooth manifold may be transformed into a manifold with edge. For example, for any value of a parameter q, the operator

1 (Lqf)(p) = (Rpf)(q) = "2[j(p+q) +f(q-p)] (p,qE JRl;fE C(JR 1))

transforms an arbitrary function into an even function and, therefore, as a result of reduc

tion, the manifold JR I turns into the half line [0, 00). In the general case, Q may

have no smooth structure even if Q is a manifold. Below, we consider only reduced generalized translation operators, i.e., we assume

that reduction is already performed and = cI> or, which is the same, Le = Re = 1.

,Generalized translation operators are called commutative if, for any p, q E Q, we

have (R;(Rqf») (r) = (R;(RpJ») (r). For commutative generalized translation oper

ators R P' the equality

(RpJ)(q) = (RqJ)(q) (p, q E Q), (2.1)

holds and, hence, the operators R p and L p coincide. Indeed, if the operators R p

commute, the associativity axiom implies that


By setting r = e and using the fact that generalized translation operators are reduced, we obtain (Rei) (p) = (Rpf)(q)·

• The equality R p' = R Ii implies that p' = p". Indeed,

(Lqf}(p') = (Rpf)(q) = (RpI)(q) = (Lqf}(p") (fE cD)

for all q E Q. This, in particular, implies that f(P') = (Lef)(p') = (Lef)(P") = f(p")

(f E cD) and p' = P /I because the generalized translation operators are reduced. We are interested in the case where Q is a locally compact space with regular Borel

measure m positive on open sets and bounded generalized translation operators R pact

in the space of functions = L2(Q, m) = L2. Note that this choice of disagrees with the formal definition of generalized trans

lation operators (because functions from the space L2 (Q, m) have no values at points). We modify this definition by assuming that the relation of associativity for generalized

translation operators Rp holds for almost all p, q, and r with respect to the measure

m X m X m. The case under consideration is not exhaustive. Thus, Vainerman [Vai6]

studied unbounded generalized translation operators in L 2 ([ 0, 00 ), d t ]). As usual, we

denote integration with respect to the measure m by dp.

Consider an involutive homeomorphism Q 3 P ~ p* E Q. The generalized translation operators Rp are involutive if the equalities

and e * = e hold for almost all p, q E Q with respect to the measure m X m. The generalized translation operators Rp are real if they commute with complex conjugation,

i.e., (RpJ)(q) = (Rpf)(q) for almost all p, q E Q. Generalized translation operators

Rp preserve positivity if (Rpf)(q) ~ 0 almost everywhere in m whenever f(q) ~ O.

Clearly, generalized translation operators are real if they preserve positivity. The family of operators R p is called weakly continuous if the operator-valued function

Q 3 P ~ Lp is weakly continuous. We say that the measure m unimodular if m (A) =

m (A *) for all A E '13(Q).

By virtue of the fact that the functions from L2(Q, m) do not have values at points,

the left generalized translation operators L p cannot be defined by using the equality

(Lpf)(r) = (Rrf)(p) (p, r E Q). However, one can define the family of operators L p

(p E Q) for a weakly continuous family of generalized translation operators R p

(p E Q) in the space L 2 (Q, m) with unimodular measure m by the bilinear form


Since R p is involutive and m is unimodular, we have

whence Lpf(r) = Rr/(P) for almost all p and q with respect to the measure m X m,

i.e., the operators L p (p E Q) form a family of left generalized translation operators

(it follows from the equality e* = e that L p are indeed generalized translation oper

ators). It is easy to understand that left generalized translation operators are involutive

(real, weakly continuous, and preserving positivity) if and only if the relevant right generalized translation operators have the same properties.

Let R; be the operator adjoint to R p The operators R; form a family of adjoint

generalized translation operators. The left adjoint generalized translation operators

L~ are defined similarly. The measure m is called strongly right-invariant (strongly

left-invariant) if R; = Rp' (L~ = L p.) for all p E Q.

If weakly continuous generalized translation operators L p are commutative, then

relation (2.1) holds for almost all p and q. Indeed, for any f, g, h E L2, we have the following chain of equalities true for

almost all r:

= J J (Lf(Rqf))(p)g(p)h(q)dpdq. (2.2)

Since Lr are weakly continuous, the first and the last expressions are continuous in r.

Hence, they coincide for all r E Q. By setting r = e, we complete the proof.

• Assume that the generalized translation operators L p satisfy the finiteness condi

tion:

(F): For any A, BE tBo(Q), there is a compact set F so large that (Lpf)(q) = 0

for almost all pEA and q E B provided that suppfn F = 0.

It is obvious that if L p satisfy (F), then right generalized translations also satisfy (F).

By using condition (F), one can define the action of generalized translation operators L p


upon all functions f from C(Q). Namely, for any sufficiently small neighborhoods V,

WE '.8oCQ) of arbitrary points p, q E Q, we take a set F according to condition (F)

andput (LpJ)(q) = (LpfFXq), where the function fF(r) is equal to fCr) for rE F

and equal to zero, otherwise (here, fer) is an arbitrary function from C(Q». We say

that generalized translation operators L p preserve the function identically equal to one

if their extensions to C C Q) satisfy the equality C L pi) C q ) = 1 for all P E Q and al

most all q E Q.

If the measure m is strongly left-invariant and the finiteness condition (F) holds,

then the subspace L2.0 of finite functions from L2 is invariant with respect to the

action of generalized translation operators L p (p E Q).

In fact, we fix some p E Q and a function f E L 2. 0, supp f = A E '.80 (Q). Let B

be a compact set which contains p and p *. For given sets A and B, we choose the set

F from condition (F). By virtue of (F), we have

J Lpf(q)g(q)dq = J f(q)(Lp.g)(q)dq = J f(q)(Lp.g)Cq)dq = 0 A

for any function g E L2 whose support does not contain points from F. Hence,

suppLpf(q) c F.

• We now can explain the definition of strongly left-invariant measure.

Let m be a measure strongly left-invariant with respect to the generalized transla

tion operators L p (p E Q) which preserve the unit element and satisfy the finiteness

condition ( F). Then

J (Lpf)(q)dq = J f(q)dq (p E Q). (2.3)

for any fE L2,0·

Indeed, assume that p is a fixed point from Q, A = suppf, B is a compact set con

taining p and p *, and F is a compact set constructed from the sets A and B ac

cording to condition (F). Since supp(LpJ)(q) c F and (LpKF)(q) = (Lp l)(q) for all

p E Band q E A, we have

J (Lpf)(q)dq = J (LpJ)(q)KF(q)dq

= J f(q)(L p' KF)(q)dq = J f(q)(L p' KF)(q)dq = J f(q)dq. A

•


A similar assertion is true for the strongly right-invariant measure.

We say that a measure m E M+ (Q) satisfying (2.3) is left-invariant. The right

invariant measure is defined similarly. In the next subsection, we present conditions for an invariant measure to be strongly invariant. If a left-invariant (strongly left-invariant) measure is, at the same time, right-invariant (strongly right-invariant), then it is called invariant (strongly invariant).

Theorem 2.1. There exists a one-to-one correspondence between normal hyper

complex systems L 1 (Q, m) with basis unity e and weakly continuous families of

bounded involutive generalized translation operators L p satisfying the finiteness con

dition (F), preserving positivity in the space L 2 (Q, m) with unimodular strongly

invariant measure m, and preserving the unit element. Convolution in the hypercom

plex system L 1 (Q, m) and the corresponding family of generalized translation oper

ators L p satisfy the relation

(j* g)(p) = f (Lpf)(q)g(q*)dq = (Lpf, g*)2 (f, g E L2) (2.4)

Moreover, the hypercomplex system L 1 (Q, m) is commutative if and only if the gen

eralized translation operators L p (p E Q) are commutative.

Proof. Let Lp be a family of generalized translation operators satisfying the condi

tions of the theorem. By using (2.4), we define the function c(A, B, r) = KA * KB(r) (A,

BE 'Bo(Q» and show that c(A, B, r) is a structure measure. The fact that c(A, B, r) is nonnegative follows from (2.4) and the fact that positivity is preserved by the generalized translation operators. The equalities

B* A

imply that the function c(A, B, r) can be uniquely extended to a regular Borel measure

with respect to B(A) with fixed A E 'Bo(Q), r E Q (B E 'Bo(Q». The weak continuity of the family of generalized translation operators Lr implies that c(A, B, r) is con

tinuous in r.

Let us show that c(A, B, r) is finite in r for all fixed A, BE 'Bo(Q). For this pur

pose, we take a compact set F from condition (F) such that L pf(q) = 0 for almost all

pEA and q E B provided that suppfn F = 0 and show that c(A, B, r) = 0 for any r fl. F. Indeed, consider a point r fl. F such that c(A, B, r) > o. Since c(A, B, p) is continuous, one can indicate a neighborhood V of the point r such that V n F = 0 and c(A, B, p) > 0 foraH p E V. Then

Section 2 Hypercomplex Systems and Related Objects

o < f c(A, B,p)dq = f f (LpKA)(q)KB*(q)KV(q)dqdp V

By virtue of (F), the last integral is equal to zero.

49

Let us now prove the associativity relation (H2). Since the axiom of associativity

holds for generalized translation operators and the measure m is invariant, we obtain

f c(A, B,p)dpc(Ep, C, s)

= f c(B, C, q)dqc(A, E q, s).

Hence, the function c(A, B, r) is a structure measure. It follows from the Fubini the

orem, the invariance of m, and (2.4) that

f c(A,B,p)dp = f f (LpKA)(q)KB(q*)dpdq

= f KB*(q) f (RqKA)(p)dpdq = m(A)m(B*) = m(A)m(B),

i.e., the measure m is multiplicative. To show that c(A, B, r) is normal, we use the

50 General Theory oj Hypercomplex Systems Chapter 1

following equalities, which are true for all A, B, C E $o(Q):

= f f KA(P)(LpKc)(q)K8'(q*)dpdq = c(C,B*,A).

The equality c(A, B, C) = c(A *, C, B) is proved similarly.

Now let the generalized translation operators Lp (p E Q) be commutative. To

check the commutativity relation (H3), we show that (f * g )(p) = (g * f)(p) (j, g E

L2)' In fact, since the generalized translation operators are involutive, the measure m is unimodular and invariant, and relations (2.1) and (2.4) are true, we have

(f* g)(p) = f (Lpj)(q)g*(q)dq

= f j(q)(Lp.g*)(q)dq = f j(q)(L q• g)(p)dq

= f j(q)(Lpg)(q*)dq = f j(q*)(Lpg)(q)dq = (g * f)(p),

Let us prove the converse assertion. Assume that L 1 (Q, m) is a normal hypercomplex system with basis unity e and structure measure c(A, B, r). We define a family of

operators Lp (p E Q) by using the bilinear form (2.4). In Subsection 1.3, it was shown that this definition is correct, the multiplicative measure is invariant, Lp are weakly

continuous, and II Lpll :::;; 1. We now define the operators Rp (p E Q) by the equality

(Rpf, g) = (g* * f)(p) (J, g E L2) and show that

(Lpj)(q) = (Rqj)(p) (fE L2) (2.5)

for almost all p and q. Indeed, if we approximate arbitrary functions J, g E L2 in L2

and h E Co (Q) in L 1 by step functions, then, in view of (1.16), we obtain

f f (Rpj)(q)g(q)h(p)dpdq = (g**J,hh = (J,g*hh

= (f* h*, gh = J J (Lqj)(p)h(p)g(q)dpdq


but this yields (2.5). In its turn, relation (2.5) implies that the function <p(p) = (Lpj)(q) E L2 for almost all q E Q. It follows from the strong invariance of the multiplicative measure and the associativity and involutivity of convolution in the hypercomplex system that

f f L~(Lqj)(r)g(r)h(q)drdq = f f (Lqj)(r)g(r) dr(Lp.h)(q) dq

= f f (Lpj)Cr)(Lq.g)Cr)h(q)drdq

(2.6)

for all jE L2, g, h E CoCQ), and p E Q. This means that (L~(Lqj»)(r) = ( ~ C Lpj») (r) for almost all p, q, and r, i.e., the operators L p are generalized trans

lation operators. The generalized translation operators Lp are involutive because the

hypercomplex system L I (Q, m) is normal, i.e.,

f f (Lpj)(q)g(q)h(p)dpdq = (j* g*, hh

= f f (Lqh*)(p)j*cp)g(q)dpdq

The fact that the generalized translation operators L p are finite follows from the relation

f (LKAKB)(r)dr = c(A,B, C) = c(C,B*,A) = 0 c

which is true for all A, B, C E r.Bo(Q), whenever A is located far enough from B


and C.

Positivity is preserved because c(A, B, r) is nonnegative and the following chain of

equalities holds for any A E ~o(Q), r E Q, and an arbitrary sequence of neighborhoods Vn of the point r contracting to this point:

J (Lrf)(q)dq = lim _1-J J (Lpf)(q)KA(q)Kv (P)dpdq A n-700 m(Vn) n

= lim _1_ J J f(q)(RqKA)(P*)Kv (P)dpdq n-700 m(Vn) n

= lim-1-Jf(q)c(Vm A,q)dq ~ 0 (fE L2,f(q)~O). n-700m(~)

Let Lp be a family of generalized translation operators associated with a hyper

complex system L I (Q, m). The inequality

(2.7)

holds for almost all q E Q.

Indeed, we fix p E Q, choose a decreasing sequence of neighborhoods Vn of the

point p contracting to this point, and set

Let gEL 1 (Q, m) be an arbitrary finite function and let A = supp g. Then (Lpf)(q) = (LpfF)(q) for all p E VI and q E A, where F is taken from axiom (F). We have

J (LpJ)(q)g(q)dq = (IF *g*)(p)

lim f (g~ *fF)(q)g(q)dq. n-700


Since IIg~lIl = II gnlll = 1, we have

If (Lpl)(q)g(q)dql = lim If (g~ *IF)(q)g(q)dql n---7 00

~ lim h~ *fF/UlgII 1 ~ II11LlIg1I1' n---7 00

which implies the required inequality.

We now show that the generalized translation operators L p preserve the function

identically equal to one. Let Q n be an increasing sequence of compact sets which

covers all Q. By virtue of the definition of Lpl (q) and (1.12), we have

f (Lpl )(q)dq = !~ f (LpKQ)(q)KA (q)dq A

= lim c(Qn,A*,p) = c(Q,A*,p) = meA) n---7 OO

for any A E ~o(Q) and p E Q. Since positivity is preserved, it follows from (2.7) that

O~Lpl(q)~ 1, whence (Lpl)(q) == 1.

Assume that the hypercomplex system L 1 (Q, m) is commutative. Let us prove

equality (2.1). Since the generalized translation operators Lp are involutive for all I, g E L2 and p E Q, we obtain

f (Lpf)(q)g(q)dq = (j* g*)(p)

* f * -*-= (g * f)(p) = (Lpg )(q)1 (q) dq

which means that relation (2.1) is true for almost all p and q. The commutativity of L p

now follows from (2.1) and the commutativity of the hypercomplex system L 1 (Q, m):

f f ~Lpl(r)g(r*)h(q*)drdq = f f L~Lql(r)g(r*)h(q*)drdq


= f f ~Lrf(q)g(r*)h(q*)drdq

= f f ~Lqf(r)g(r*)h(q*)drdq (p E Q).

Since Lq is weakly cOl'l:tinuous,

J ~Lqf(r)g(r* )dr = (Lqf, Lp.g*)

and

J ~Lpf(r)g(r*) dr = (Lpf, L q* g*)

are continuous in q for every p. This implies that ~Lqf(r) = ~Lpf(r) for almost

all r, p, q

• According to Theorem 2.1 and the results obtained in Subsection 1.3, every family of

generalized translation operators L p satisfying the conditions of Theorem 2.1 is uni

formly bounded in L2 (Q, m): II Lp II ~ 1 (p E Q).

Inequality (2.7) allows one to extend the generalized translation operators L p

(p E Q) to operators continuous in L 00 (Q, m). This result, together with Theorem 2.1,

can be regarded as a formalization (for the case of the duality (Loo (Q, m ), L I (Q, m ») of Levitan's heuristic ideas (see [8, Section 2, Subsection 2.1]). Levitan conjectured that a linear topological space of functions (measures, functionals, germs, etc.) on a locally compact space Q is an associative topological algebra with right identity element if and only if the action of the right generalized translation operators R s (s E Q)

is defined on the dual space <1>' and, moreover, the operation * in and the corresponding generalized translation operators Rs in <1>' satisfy the relation

(F * G,f) = (F, (G s, RsI» (fE '; F, G E <1»,

where the symbol GS denotes the action of a functional G in a variable s). A formalization of these ideas was proposed by Litvinov [Litl]-[Lit3] (see Chapter 3, Section 1)

for the cases (Mc(Q), C(Q», (Mb(Q), Coo(Q» (the hypergroup case), ('D(Q),

COO(Q» (in this case, Q is a smooth manifold, COO(Q) is the space of infinitely differentiable functions on Q equipped with the topology of uniform (including all

derivatives) convergence on compact sets, and 'D( Q) is the space of generalized

functions with compact support), (Jl(Q), j{(Q» (here, Q is a complex analytic

manifold, j{( Q) is the space of functions holomorphic on Q with the topology of

Section 2 . Hypercomplex Systems and Related Objects 55

uniform convergence on compact sets, and JI.( Q) is the dual space which consists of analytic functionals), and for the case of the duality "real analytic functions-hyperfunctions with compact support".

In order that a function X (r) E C (Q) be a generalized character of a normal

commutative hypercomplex system L 1 (Q, m) with basis unity e, it is necessary and

sufficient that the equality

(2.8)

hold for almost all p, q E Q.

Indeed, assume that a function X (p) E X g is a generalized character, i.e., equality

(1.6) holds. Then

X(A)X(B) = f c(A, B, r)x(r)dr

= f f (LrKA)(s)dsx(r)dr = f f (Ls*KA)(r)x(r)drds B* B

= f f (Lsx)(r)drds BA

for any A, BE 1Jo(Q), which yields (2.8). The converse statement can be proved by analogy.

• Note that if a hypercomplex system L I (Q, m) satisfies the condition of separate

continuity (see Subsection 2.2 below), then (2.8) holds for all p, q E Q. Note that all results obtained in this section can easily be extended to the case of non

unimodular hypercomplex systems (with nonunimodular multiplicative measure).

2.2. Strong Invariance of Invariant Measures. Below, we present a condition that guarantees the strong invariance of an invariant measure.

Let L s (s E Q) be generalized translation operators acting in the space of bounded

functions on a locally compact space Q equipped with an involutive homeomorphism

Q 3 P H P * E Q. We say that the generalized translation operators L s satisfy the con

dition of separate continuity if the function (Lsf)(t) is continuous in each variable for

any fE Co(Q)·

Lemma 2.1. Let L s (s E Q) be a family of involutive generalized translation op-


erators in Cb(Q) that preserve finiteness and satisfy inequality (2.7) and the condi

tion of separate continuity. and let Il E Mt CQ). Then the equalities

Ls f (Ld)(r)dll(r) = f (~(Ltf)) (r)dll(r) (2.9)

and

(2.10)

holdfor all t, SEQ and f E Co(Q).

Proof. Equality (2.9) obviously holds for any Dirac measure. Note that the cone 'l(

generated in M b( Q) by Dirac measures is dense in Mt (Q) in the v -topology (the

topology of simple convergence on functions from Co (Q)). Let the net of Il a E 'l( be

such that v - lim Ila = Il. Then

lim f (~(Ltf)) (r)dlla(r) = lim f (L;(Lsf))(r)dlla(r) a a

= f (L; (Lsf)) (t)dll(r) = f (~(~f)) (r) dll(r)

for f E Co (Q) and t, SEQ by virtue of the fact that the generalized translation operators Ls (s E Q) are separately continuous and preserve finiteness. Let us show that

lim f Ls f (Ltf)(r)dlla(r) = Ls f (Ltf)(r)dll(r). a

Denote

Obviously, for any t, we have

lim h a ( t) = h (t ), a

where

h(t) = f (Ltf)(r)dll(r).

Since generalized translation operators are separately continuous, the functions h a (t)

and h (t) are continuous. Therefore, h a (t) uniformly converges to h (t) on every


compact set. We fix t, SEQ. By virtue of the finiteness condition (F), there exists a

compact set Ft,s such that (Lsf)(t) = 0 if suppfn Ft,s =0. It is clear that, for all a,

the function (h u - h) I F (t) can be extended to the function (hu - h) - (t) so that the 1$

net (hu - hf (t) uniformly converges to zero. Then, by using (2.7) and the finiteness condition, we get

lim I (Ls(hu-h»(t)1 = liml(Ls(hu-hf)(t)I:::; lim II (hu-hfIL = O. u u u

Let us prove (2.10): Assume that 11 = 3 r (r E Q). Then

because

The proof of relation (2.10) can now be completed by analogy with the proof of (2.9) .

• Consider a family of generalized translation operators L t (t E Q) in L 2 (Q, m )

with respect to a left-invariant measure m. Let On. On => On+ J, be a sequence of bounded neighborhoods of the identity element contracting to this element. Let us con

struct a sequence of functions e n(P) E Co (Q) such that e n(P) ;:: 0, supp en C On.

e~ = en' and

J en(p)dm(p) = 1.

Lemma 2.2. Let L t be a family of involutive generalized translation operators in

the space of functions square integrable with respect to a left-invariant measure;

furthermore, we assume that these operators preserve finiteness and positivity and satisfy inequality (2.7), the condition of separate continuity, and the following con

dition:

(C) For any neighborhood W of a point SEQ, one can find a neighborhood U


of the identity element e such that suppLs*<p(-) C W for every <p E Co(Q)

with supp <p C U.

Then every function f(s) E Co (Q) can be uniformly approximated by the sequence

Proof. Note that, by virtue of inequality (2:7), the continuity of functions f n (s)

follows from the Lebesgue theorem and the fact that generalized translation operators are separately continuous. For any t > 0, one can find a neighborhood Wt of an arbitrary

point t E supp f such that I f( t) - f( s) I < t for all s E Wt- We cover supp f by these

neighborhoods Wt and take a finite subcovering Wtl , ••• ,Wtn • Denote by U1, ••• , Un

the neighborhoods of the identity element e that correspond to Wt I' ... , Wr n according

to condition (C). Denote

00

U = n Ukk=1

Then, for any no. By using (2.3),

we obtain

for all s E supp f Since the generalized translation operators are involutive, we have

and, obviously, fn(s)=O for

Hence, Ilfn- flloo ~ 0 as n ~ 00.

Theorem 2.2. Let L s (s E Q) be an involutive family of generalized translation


operators in the space L2 (Q, m) with left-invariant measure m such that

(i) these operators are separately continuous;

(ii) inequality (2.7) holds;

(iii) the operators Ls preserve positivity;

(iv) the finiteness condition (F) is satisfied;

(v) condition (C) in Lemma 2.2 is satisfied, i.e., for any neighborhood W of a

point SEQ, one can find a neighborhood U of the identity element e such that supp L s *<p (. ) c W for every <p E Co (Q) with supp <p c U.

Then L: = Ls* (s E Q), i.e., the measure m is strongly left-invariant.

Proof. Since the generalized translation operators are continuous and preserve

positivity, it suffices to show that CLsf,g)Z=(J,Ls*f)2 for any f,gE ctCQ), SEQ. By virtue of (2.7) and the condition of separate continuity, the generalized translation

operators Ls act continuously in CbCQ). Therefore, by virtue of Lemmas 2.1 and 2.2, we obtain

f (Lsf)(t)g(t) dt = lim f Z; f (Lt* e nHr)f (r) dr g(t) dt a

for any J, g E ctCQ) and SEQ.

•

60 General Theory oj Hypercomplex Systems Chapter 1

2.3. Hypergroups f!,nd Hypercomplex Systems. In this subsection, we establish the relationship between hypercomplex systems and hypergroups.

Let Q be a locally compact Hausdorff space. The space Q is called a hypergroup if the following conditions are satisfied:

(Gl) There is a separately continuous mapping QXQ03 (p,q) ~ Op*OqE Ml(Q),

which is called convolution, where M 1 (Q) is equipped with weak topology and Op is the Dirac measure. This mapping can be extended to M b(Q) according to the formula

(Jl * v,f) = f f (Op * Oq,f)dJl(p)dv(q). (2.11)

The operation of convolution is assumed to be associative, i.e., 0 p * (0 q * 0 r ) =

(Op * Oq) * or.

(G2) supp (Op * Oq) is compact.

(G3) There is an involutive homeomorphism Q 3 P ~ p* E Q such that (Op * Oq) * =

o q* * 0 p*; here, Jl * denotes the measure

(Jl*,f) = f J(p*)dJl(p).

(G4) There exists a point e E Q, called the identity element, such that 0 e * ()p=

= Op * oe = Op (p E Q).

(G5) e E supp Op * Oq* if and only if p = q.

(G6) The mapping (p, q) ~ (supp 0 p * 0 q) continuously maps Q x Q into the

space of compact sets Sl'(Q) equipped with the Michael topology [Mic]; this

topology is generated by the pre-base {K E S'r(Q) I K n u,* 0, K ~ V}, where U and V are open sets in Q.

A locally compact space Q that satisfies (Gl)-(G4) is called a weak hypergroup.

If Op* Oq= Oq* Op for any p, q E Q, the hypergroup Q is called commutative. We

define the action of generalized translation operators R q (q E Q) upon an arbitrary Bo

rel function J on Q by the formula (R q.f)(p ) = (0 p * 0 q,f) if the last integral exists

(but is not necessarily finite). In any case, the action of Rq is defined on bounded func

tions on Q. By virtue of (Gl) and (G4), the operators R q form a family of generalized translation operators and are reduced. Obviously, the generalized translation operators Rp and Lp (p E Q) are involutive, preserve positivity and identity element, and satisfy


the conditions of separate continuity and finiteness and inequality (2.7).

A regular Borel measure m positive on open sets is called the left Haar measure of

a hypergroup Q if it is weakly invariant with respect to the family of generalized translation operators Lp (p E Q). The right Haar measure is defined in a similar way. If a Haar measure exists, then it is unique up to multiplication by a constant ([Jew], see also Theorem 4.1) Spector [Spe4] showed that any commutative hypergroup possesses the Haar measure. Jewett [Jew] proved the existence of the Haar measure for noncommutative hypergroups in compact and discrete cases. At present, the problem of existence of the Haar measure for general locally compact hypergroups is still open and the

presence of this measure is postulated. We fix a left Haar measure m. As usual, we de

note integration with respect to the measure dm (p) by dp.

Let us introduce the convolution of sets A, BE tB(Q):

A * B = U supp (Op * Oq). pEA qEB

Clearly, this operation of convolution is associative, i.e., A * (B * C) = (A * B) * C,

and involutive, i.e., (A * B)*= B** A * (A, B, C E tB(Q)).

We now show that (A * B) n C is nonempty if and only if B n (A ** C) is non

empty.

Indeed, let (A * B) n C be nonempty. Then, by virtue of (G5), C * * (A * B) con

tains the identity element e. At the same time, C * * (A * B) = (A * * C) * * B. Since the

last set contains e, we have (A * * C) n B,* 0 by virtue of (G5). Note that we have, in fact, also proved the converse statement (it suffices to the rename sets in a proper way) .

• Let us show that the generalized translation operators L p associated with the hy

pergroup Q satisfy condition (C) in Lemma 2.2.

By virtue of (G6), for any neighborhood W of an arbitrary point SEQ, one can

find a neighborhood of the unit element U 3 e such that {s} * U c W. Condition (C)

means that supp ( 0 s. * 0 t) n U '* 0 for any t ~ W. Assume the contrary: ( {s*} * (Q \ W)) n U *- 0. Then we have (Q \ W) n ({ s} * u) '* 0 by virtue of the assertion proved above. The last inequality contradicts the choice of U.

• Thus, the generalized translation operators L p (p E Q) satisfy the conditions of

Theorem 2.2 and, hence, the left Haar measure of the hypergroup Q is strongly leftinvariant with respect to the family of generalized translation operators L p associated


with the hypergroup Q, i.e., L~ = Lp. in the space L2(Q, m) (by virtue of (2.7), the

generalized translation-operators Lp are bounded in L2(Q, m»). By applying the Le

besgue theorem and the fact that the generalized translation operators L p are separately

continuous, we show that the generalized translation operators L p are weakly continu

ous in L 2 (Q, m). Thus, all conditions of Theorem 2.1 hold for the generalized translation operators Lp- By this theorem, the space L 1 (Q, m) is a normal hypercomplex sys

tem with basis unity e with respect to the convolution

f* g(P) = J (Lpf)(q)g(q*)dq (f, g E L2(Q, m», (2.12)

and involution p*.

We have shown that it is possible to construct a normal hypercomplex system with

basis unity e for any unimodular hypergroup. Let us now analyze the possibility of the reverse procedure. Let Ll (Q, m) be a normal hypercomplex system with basis unity In order to define the convolution of measures, we assume that the generalized translation operators Lp (p E Q) defined in Subsection 1.3 and acting in the space L2

can be extended to a family of separately continuous operators in Coo(Q). One can now define the convolution of measures as (op * Oq,f) = (Lpf)(q) (fE Coo(Q» and extend

it to M b(Q) by using (2.11). The associativity (and, if necessary, commutativity) of this convolution follows from the associativity of the generalized translation operators Lp and relation (2.11). Indeed,

Since Lp preserve the function identically equal to one, op * Oq is a probability mea

sure: (op* Oq, 1) = Lp1(q) = 1. Condition (G2) readily follows from the finiteness condition (F) for generalized translation operators, (G4) follows from (T2), and (G3) results from the fact that the generalized translation operators are involutive. Thus, if the operators Lp can be ex~ended to continuous operators from Coo(Q) into Coo(Q), a weak

hypergroup can easily be associated with any hypercomplex system. It is a priori clear that condition (G6) is, generally speaking, not satisfied. It is also impossible to check the validity of (G5). Nevertheless, in the case of a discrete base Q, the concepts of nor

mal hypercomplex system with basis unity and hypergroup are equivalent because there is no need to check (G6) and (G5) is equivalent to the equality c (p, q, e) = 0 p, q* m,

where p, q E Q and Op,q* is the Kronecker symbol.


Let us now present a formula which establishes the relationship between the structure

measure c (p, q, r) of a hypercomplex system with discrete base Q and the coefficients

g (p, q, r) of the expansion of the measure l> p * l> q in the basis l> n i.e., l> p * l> q =

Lrg(p,q,r)l>r (L r g(p,q,r)=1 because l>p*l>q is a probability measure):

c(p, q, r)m(r) g(p,q,r) =

m(p)m(q) (2.13)

2.4. Hypercomplex Systems Whose Structure Measure Is Not Necessarily Nonnegative. In this subsection, we present a generalization of the concept of hypercomplex systems to the case of, generally speaking, complex-valued structure measures and reestablish the results of Sections 1 and 2 for these hypercomplex systems.

First, we introduce the following definition:

A complex-valued function c(A, B, r) (A, B E ~(Q» is called a structure measure

if it is a bounded Radon measure in A (B) for fixed B and r (A and r) and satisfies the conditions

(H 1)' the function c (A, B, r) is continuous in r for fixed A, B E ~o (Q) and for all

E I C A and E2 c B, there exists a compact set Fe Q such that supp c(E I,

E I , r) c F (A, B E ~o(Q»;

(H2)' the relation of associativity

J c(A, B, r)drc(Er' C, s) = J c(B, C, r)drc(A, E r, s)

holds for all A, B, C E ~o(Q) and SEQ;

(H3)' the commutativity relation c(A, B, r) = c(B, A, r) holds for all A, BE 1Jo(Q)

and rE Q.

As in Subsection 1.1, we define a measure mr~) (~E ~(Qx Q» by setting

mr(Qx Q) = c(A, B, r) (A, B E ~o(Q» and extending the measure mr to all Borel

subsets of Qx Q (in view of its regularity, this extension is always possible).

We say that a regular Borel measure dm(r) = dr defined on Q and positive on open sets is <: multiplicative measure if

J V(A,B, r)dm(r) S; m(A)m(B) (2.14)

where V(A, B, r) is a variation of the measure mr~:{) on the set A x B for any A, BE

~o(Q).


If the base Q is discrete, then, clearly, inequality (2.14) has the form

L I c (p, q, r) I m (r) ::; m (p) m (q) (p, q E Q). (2.15) r

(H 4 )' We assume that at least one multiplicative measure exists.

However, in the case of a complex-valued structure measure, it is impossible to prove the existence theorem similar to Theorem 1.3 for multiplicative measure and the problem of existence of multiplicative measure remains open.

Consider the space· L 1 (Q, m) = L 1 of functions on Q summable with respect to a

multiplicative measure m. This space equipped with convolution

is called a hypercomplex system with complex-valued structure measure. This definition is meaningful because even in the case of a complex-valued structure measure one can

show that this hypercomplex system is also a Banach algebra with II x * y 111 ::;

II x 11111 Y 111· The definition of a character remains unchanged. Note that, for a complexvalued structure measure, the function identical to one is not, generally speaking, a character of a hypercomplex system. As for the positive structure measure, one can show that, in the commutative case, the set of characters is in one-to-one correspondence with

the set of maximal ideals of the Banach algebra L 1 (Q, m) (or :c 1 (Q, m) if L 1 has no unit element; in this case, one must consider ideals other than LI). A maximal ideal and the corresponding character satisfy relation (1.7).

If (H5) is satisfied, a hypercomplex system with complex-valued structure measure is called normal and if (H 6) is satisfied, then the hypercomplex system possesses a basis unity

Lemma 2.3. In a normal hypercomplex system,

V(A, B, r) ::; meA) (2.16)

for all A, B E tB(Q) and r E Q.

Proof. In fact, it suffices to prove (2.16) for A, BE tBo(Q). We fix r E Q, A, B E

'130 (Q). For any E > 0, one can find finitely many disjoint sets ~ j c Q x Q constructed from finitely many rectangles such that

n

V(A, B, r) - E < L I mr~j)I, j=1

n

U ~j = AxB. j=1


We decompose the sets A and B into small sets Au and B ~ (ex. = 1, 2, ... , k; ~ = 1,

2, ... ,1) such that each ~j can be represented as a sum of disjoint rectangles Au x B~.

Then

k I

V(A,B,r)-£ < L Llc(Au,B~,r)l. u=l ~=l

The lemma will be proved if the relationship

I

L Ic (A u, B~, r) I :::; m(Au) ~=l

is established. For any C E ~o(Q), we have by (2.14),

± I J C(Au'B~,r)drl = ± I c(A~, C, B~) I ~=l C ~=l

This and (HI) imply the required inequality.

• By using Lemma 2.3, one can show that all assertions in Section 1 and Subsection 3.1

remain true for normal hypercomplex systems with complex-valued structure measure and basis unity. The proofs are practically the same as in the case of nonnegative structure measure. It is easy to see that Theorem 3.4 is also true for such hypercomplex sys

tems, although the Plancherel measure dX is not, in general, a multiplicative measure of the dual hypercomplex system Indeed,

for any <1>, '¥ E ~o(Q). Then it is clear that inequality (2.14) is, generally speaking, not

true for the measure dX (the corresponding example was given by Gasper [Gas3]). Hence, in this axiomatics, it is impossible to construct an acceptable duality theory for hypercomplex systems with complex-valued structure measure. In Section 5, we present another approach (based on the ideas of Kac [Kac]) to the investigation of hypercomplex systems with complex-valued structure measure.

The results established in Subsection 3.3 also remain valid for hypercomplex systems

with complex-valued structure measure if the definition of A -systems introduced in the

next subsection is properly changed. An A -system is now defined as an orthogonal sys-


tern L = { q> it)} pE Q of continuous functions complete in L 2 (E, 0') and such that

(A 1)' the sequence {f q> p( t)q> i t)q> r (t) dO' (t)} r E Q is finite for all fixed p, q E Q;

(A2)' if q>it) E L, then q>p(t) E L; (A3)' there is a sequence {m(r)L E Q of positive numbers such that

r

for any fixed p, q E Q, and (A4)' there is a point e E Q such that q> e( t) == 1 (t E E).

2.5. Convolution Algebras and Hypercomplex Systems. In this subsection, we establish the relationship between hypercomplex systems and convolution algebras introduced for the first time by Ionescu Tulcea and Simon [loS] (see also [Mall] and [MaI2]).

Let Q be a locally compact space and let m r be a real Radon measure on Q X Q

for all r E Q. For any f, g E Co (Q), we define their convolution f * g as

(j* g)(r) = f f f(P )g(q)dmr(P' q).

Suppose that the following conditions are satisfied:

(Cl) for any A c Q, there exists a compact set KA c Q such that suppf* g c KA

whenever Stipp f c A and supp g c A;

(C2) Co (Q) is a commutative associative algebra with respect to convolution;

(C3) there exists a Radon measure m on Q strictly positive on nonempty open sets and such that

f 1(f*g)(r)ldm(r) ~ f If(P)ldm(p)f Ig(q)ldm(q)

for any f, g E Co(Q);

(C4) there is an involutive homeomorphism r --7 r* of the space Q such that

f f(r)dm(r) = f f(r)dm(r*)

and

f (g * h)(r)k(r)dm(r) = J g(r)(h* * k)(r)dm(r)


for any f,g,h,kE Co(Q) (here, as usual, j*(r)= j(r*»;

(C5) there is a point e E Q (the identity element) such that e*= e and

f j(P)g(q)dme(P, q) = f j(p)g*(p)dm(p)

forallf,gE Co(Q).

Clearly, the maps (f, g) -7 j* g and j -7 j*, extended by continuity to L1 (Q, m),

transform L 1 (Q, m) into a commutative Banach algebra with involution. By L we denote the set of all finite functions summable with respect to the measure m. We also impose the following condition:

(C6) for any functipn gEL and any compact set A c Q, the family of maps S-7

(k*f)(s), wherejE Co(Q), suppjcA, and IIjIL~l, isequicontinuous. Here,

g(s) = g(s*).

The space L1 (Q, m) satisfying (Cl)-(C6) is called a convolution algebra.

Let us show that any convolution algebra is a normal hypercomplex system with,

generally speaking, complex-valued structure measure and basis unity. Indeed, we set c(A, B, r) = mr(A xB) and show that c(A, B, r) is a structure mea-

sure. Since m r is a real Radon measure on Q x Q, c(A, B, r) is a real measure on Q

for fixed Band r. Further, since m r (A x B) = (KA * K8 )(r), by using the Lebesgue

theorem, we conclude from (C6) that c (A, B, r) is continuous in r for fixed A, B E

~o(Q). Moreover, it follows from (Cl) that there exists a compact set Fe Q such that supp C(E1, E2, r) c F if E1 c A and E2 c B. The relation of associativity now fol

lows from (C2) and the following chain of relations valid for all A, B, C E -Bo(Q) and rE Q:

The relation of commutativity for c(A, B, r) is a consequence of (C2). Condition (C3) implies that the measure m is a multiplicative measure. Thus, L 1 (Q, m) is a hypercomplex system This hypercomplex system is normal by virtue of (C4), (C6), and the


Lebesgue theorem. Condition (C5) implies that e is the basis unity of this hypercomplex system.

• On the other hand, (Cl)-(C5) are satisfied for all commutative normal hypercomplex

systems with basis unity as follows from Section 1 (as indicated in Subsection 2.4, all results established in Section 1 remain valid for hypercomplex systems with complex-valued structure measure) but condition (C6) is, in general, not true.

Section 3 Elements of Harmonic Analysisfor Normal Hypercomplex Systems

3. EI ements of Harmonic Analysis for Normal Hypercomplex Systems with Basis Unity

69

In this section, we establish some principal theorems of the harmonic analysis on locally compact groups for hypercomplex systems. In Subsection 3.1, we introduce an analog of the Fourier transformation, prove analogs of the Bochner and Plancherel theorems, and establish the inversion formula for commutative normal hypercomplex systems with basis unity (the case of noncommutative hypercomplex systems will be considered in Section 5). Then we generalize the Pontryagin duality theory to hypercomplex systems and describe an analog of the Fourier algebra. Further, we consider representations of normal, generally speaking, noncommutative hypercomplex systems with basis unity and prove the approximation theorem.

3.1. Fourier Transformation and the Plancherel Theorem. Let us show that analogs of the Bochner and Plancherel theorems and the inversion formula hold for commutative normal hypercomplex systems. The relevant results for noncommutative hypercomplex systems are presented in Section 5.

Let X j (Q, m) be a commutative normal hypercomplex system with basis unity,

i.e., axioms (Hl)-(H6) are satisfied; Let X and X h be the sets of characters and Hermitian characters, respectively, endowed with the topology of the space of maximal ideals. We denote the spaces of generalized and Hermitian generalized characters by X g

and Xg, h, respectively. The space X is compact (locally compact) if L1 contains the identity element (does not contain). We now consider the second more complicated case. At the end of the subsection, we describe the situation occurring in the first case.

An essentially bounded function q> (r) (r E Q) is called positive definite if

f q>(r)(x* * x)(r)dr ~ 0 (3.1)

for all x E L j •

We also present another definition of positive definiteness. A continuous bounded function q> (r) (r E Q) is called positive definite if the in

equality

n

L Ai~j(Rrtq»(rJ ~ 0 i,j=l

(3.1)'


lfthe generalized translation operators R t extended to Leo map Cb(Q) into

Cb(Q x Q), then the definitions of positive definiteness (3.1) and (3.1)' are equivalent

for the functions <per) E Cb(Q).

Indeed,

J <p(r)(x* * x)(t)dt = J <p(t) J (Lsx)(t)x(s)dsdt

= JJ (Ls·<p)(t)x(s)dsx(t)dt

= JJ (Rt<p)(s*)x(t)x(s)dtds ;;:: o.

By the condition, we have (Rt<p)(s*) E Cb(Q x Q), then the last inequality clearly im

plies (3.1)'. Let us prove the converse assertion. Let Qn be an increasing sequence of

compact sets covering the entire Q. We consider a function y(r) E Co(Q) and set

Ai=y(ri) in (3.1)'. This yields

n

L (R,;*<p)(rj)y(ri)y(rj) ;;:: O. i,j=i

By integrating this inequality with respect to each ri"'" rn over the set Qk (k EN)

and collecting similar terms, we conclude that

Further, we divide this inequality by n 2 and pass to the limit as n ~ 00. We get

J J (Rr·<p)(s)y(r)y(s)drds ;;:: 0

Qk Qk

for each kEN. By passing to the limit as k ~ 00 and applying Lebesgue theorem, we

see that (3.1) holds for all functions from Co(Q). Approximating an arbitrary function

from LI by finite continuous functions, we arrive at (3.1).

• In what follows, we use definition (3.1) which is formally more general. Clearly, the

set of functions positive definite in the sense of (3.1) is weakly closed in L eo(Q, m). By

P(Q) we denote a set of continuous positive definite functions and by ffi we denote the

Section 3 Elements of Harmonic Analysis for Normal Hypercomplex Systems 71

linear span of the functions from L} n P(Q).

Let L 2. 0 be the set of functions with compact support.

Lemma 3.1. The linear span of the set of functions from m n Co (Q) of the form

a * b (a, b E L 2,0) is ~ense in L2 and L1.

Proof. Let en be the approximative unit of a hypercomplex system constructed in

Lemma 1.3. The sequence a * en E Co(Q) converges to a in L2 for any a E L 2,0,

Thus, the functions of the form a * b* E Co(Q) are dense in L 2,0 (in the metric of the

space L 2). Hence, they are dense in L 2. On the other hand, a * b* is a linear combi

nation ofthe positive definite functions (a ± b) * (a ± b)* and (a ± ib) * (a ± ib)* and, hence, it belongs to m.

Sinc.e step functions are dense in L}, it suffices to prove the second part of the lem

ma for the indicators KA of sets A E 'Bo(Q). Let On be the sequence of neighbor

hoods constructed in Lemma 1.3. Denote F = supp c (A, 0 1, r). Taking into account that supp en cOn cOl and the structure measure is nonnegative, we establish that

supp KA * en c F. Thus,

IIKA*en-KAIiI = JI(KA*eJ(r)-KA(r)ldr

Q

• Lemma 3.2. Assume that the function Lsf(t) is separately continuous for each

fE Co(Q) (i.e., the corresponding hypercomplex system satisfies the condition of

separate continuity). Then the set of functions from m n Co (Q) of the form

N(<p)

<per) = L Ak(ak * b;)(r) (A k E a::; a", b k E Co(Q)) k=1

is dense in Coo (Q).

Proof. Since Co(Q) is dense in Coo(Q), it suffices to prove this statement for an

arbitrary function x E Co (Q). First, we show that x * e n( r) ~ x( r) as n ~ 00 for all

r E Q. Indeed, since e~ = en' we have


x* en(r) - x(r) = f «Lsx)(r)-x(r»)en(s)ds-70, n-7 00 ,

Qn

because (Lsx)(r) is continuous for fixed r E Q. Since II x * e niL,,, ~ II x 1100 II e nil 1 =

II X 1100, by virtue of the Lebesgue theorem, we have (~, x * en> -7 (~, x> as n -7 00 for

any Radon measure ~ E Mt(Q). This implies that x * en(r) weakly converges to

x (r) in C"", (Q). Then there exists a sequence of functions from the linear span of

{ x * en I n EN} that uniformly converges to x.

• Lemma 3.3. If fE L 2 (Q, m), then f* *fE P(Q).

Proof. By virtue of Lemma 1.1, we have f* *fE C"",(Q). We fix x E Ll (Q, m).

Passing to the limit in the equality (a* * a, b* * b h = II a * b II; (here, a and bare

step functions) as a -7 f in the space L 2 and b -7 x in L l' and using Theorem 1.1 and Lemmas 1.2 and 1: 1, we get

• For any function x ELI and any character X E X, we set

x(X) = f x(r)x(r)dr.

This integral exists and x is a continuous function on X. It is called a Fourier trans

form of the function x ELI' By Theorem 1.2, x(X) is simply x(M), where M is a

maximal ideal of the algebra ~ 1 (Q, m) associated with the character X· This implies that

for all X E X. For all X E X h , we have (x*)"(X) = x(X). In particular, (x** x)"(X) =

I x(X) 12. We also note that the set of Fourier transforms of functions from Ll (Q, m) IS

dense in the space C"", (X h) with uniform norm.

Theorem 3.1 is an analog of the Bochner theorem for hypercomplex systems.


Theorem 3.1. Every function <p E P(Q) admits a unique representation in the form of an integral

<per) = f x(r)d/l(X) (r E Q), (3.2)

Xh

where /l is a nonnegative finite regular measure on the space Xh. Conversely, each

function of the form (3.2) belongs to P(Q).

Proof. Let in L j (Q, m) defined as follows:

(x) = J <p(r)x(r)dr (x E L J ).

It is clear that this functional is positive. The functional can be extended to a positive

functional in .n j (Q, m). To do this, it suffices to show that the functional is

real (i.e., (x*) = (x) for all x E L j (Q, m» and satisfies the inequality 1 (x) 12 ~ C(x**x) for all XE Lj(Q,m), where C is a constant (see, e.g., [Nail,p.223]). Let

e n( r) be the approximative unit constructed in the proof of Lemma 1.3. Since is

positive, we have

(x*) = lim (e~ *x*) = lim (x * en) = (x) n~oo n~oo

for all x E L j (Q, m). Further, by using Lemma 1.3, we obtain

1(x)1 2 = lim 1(en*x)1 2 ~ lim (e~*en)(x* *x) ~ II II (x* *x). n-+oo n-+ oo

Consequently, it is possible to extend to a positive functional on .n j (Q, m ). By virtue of the theorem on representations of positive functionals on commutative

Banach *-algebras with identity element, the functional (and, hence, <I» can be uniquely represented in the form

(x) = J J x(r)x(r)drd/l(X), xh Q

where /l is a finite regular Borel measure on 1JO(Xh). Consequently,

<per) = J x(r)d/l(X) xh


almost everywhere on Q. Since the characters of normal hypercomplex systems are continuous, both functions in this equality are also continuous. This yields (3.2).

The second part of the theorem follows from the relation

= f Ix(x)12d/l(X) ;::: 0 xh

and the Lebesgue theorem on the limit transition.

Corollary 3.1. If the product of any two Hermitian characters is positive definite,

then the product of any two continuous positive definite functions is also positive definite.

Indeed, by virtue of Theorem 3.1,

f f(r)g(r)(x* * x)(r)dr = f f x(r)d/l(X) f 'V(r)dv('V)(x* * x)(r)dr

= J J J x(r)'V(r)(x* *x)(r)drd/l(x)dv('V) ;::: 0 xh xh Q

for all f, g E P(Q) and x ELl'

• Corollary 3.2. Assume that Ll (Q, m) is a commutative hypercomplex system with

basis unity. Then a continuous bounded function <p (r) is positive definite in the sense of(3.1) if and only if it is positive definite in the sense of (3.1)'. Moreover, it has the following properties:

( i) <p ( e) ;::: 0;

(ii) <p(r*) = <per);

( Ii i) I <p ( r ) I ~ <p ( e);

( v) I <p (s ) - <p ( t ) 12 ~ 2 <p ( e) [ <p ( e ) - Re (R s <p )( t *) ].


Proof. At the beginning of this subsection, we have shown that positive definiteness

in the sense of (3.1)' implies positive definiteness in the sense of (3.1). Let us show that

the converse assertion is also true (Le., the requirement of continuity of (R s <p)( r) (f E

Cb(Q») is unnecessary). Assume that <per) E Cb(Q) is positive definite in the sense of

(3.1) and let A1''''' An E CI: and r" ... , rn E Q. Relation (3.2) and the fact that the

generalized translation operators are continuous in Loo (Q, m) imply that

n n

L Ai~ARr;.<p)(rj) = L Ai~j J (Rr;*X)(rj)dJl(X) i,j=1 i,j=1

= J .t Ai~j X(lj)x(rj)dJl(X) = J I I Akx(rk) \2 dJl(X) ~ O. X h 1,)=1 X h k=1

It also follows from relation (3.2) that

lep(r)l::; J Ix(r)ldJl(X)::; Jl(X) = epee); Xh

I (Rsep)(t)12 = If X(s)X(t)dJl(X) 12 Xh

Finally,

lep(s) - ep(t)1 2 ::; If (X(S)-x(t»dJl(X)12 ::; Jl(X h ) JI(X(s)-x(r»1 2dJl(X) X h X h

= ep(e) J (lx(s)12 + Ix(t)12 - 2Re x(s)X(r) )dJl(X) Xh


::;; <p ( e) J 2 ( 1 - Re (Rr* X ) ( s ) ) d ~ (X ) xh

= 2 <p ( e ) [ <P ( e) - Re (Rr* <p )( s ) ].

• Note that properties (i )-(v ) hold for continuous functions positive definite in the

sense of (3. 1), on a noncommutative hypercomplex system. Inequality (3.1)' means that the kernel K(t, s) = (Rt.<p)(s) is positive definite.

Therefore, this kernel possesses the following properties: K (t, t) ~ 0, 1 K(t, s) 12 ::;; K(t, t)K(s, s), K(t, s) = K(s, t), and

1 K(t, r) - K(s, r) 12 ::;; K(r, r)(K(t, t) + 2ReK(t, s) + K(s, s))

(see, e.g., [BeKo, p. 469]). This immediately yields (ii). Indeed,

<p(r*) = (Re<P )(r*) = K(e, r*) = K(r*, e) = (Rr<p)(e) = (Le<p)(r) = <per).

Similarly, (Rr<P )(s) = (Rs.<p)(r*)(r*). This implies that

i.e., (iv). By setting s = e in (iv), we obtain 1 <per) 12::;; <p(e)(Rr.<p)(r). In view of

(2.7),wehave (Rr.<p)(r)::;; 11<pIL. Consequently, 1<p(r)1 2::;; <p(e)II<pIL whichim

plies (iii) and, hence, (i). Finally, (v) follows from the last inequality for K(t, s),

where s = e,

::;; 2 <p ( e ) [ <p ( e) - 2 Re (Rt* <p )( s ) ].

• Remark. For simplicity, we consider only bounded positive definite functions on

hypercomplex systems. But omitting the boundedness condition, one may study functions continuous and positive definite in the sense of (3.1)' on hypercomplex systems with the property of separate continuity. For example, the positive definite functions in Example 2 in Section 1 are even-positive definite functions. The reader can find an analog of the Bochner theorem for such functions, e.g., in [Ber1O]. An analog of the Bochner theorem for general unbounded positive definite functions associated with commutative normal hypercomplex systems with basis unity and the property of separate con-

Section 3 Elements of Hannonic Analysis for Nonnal Hypercomplex Systems 77

tinuity, can easily be obtained by using the theorem on spectral decomposition for representations of commutative hypercomplex systems (see Subsection 4 below). General unbounded positive definite functions and measures on hypergroups were studied by Bloom and Heyer [BIH3] and Voit [Voi7].

Theorem 3.2 (an analog of the inversion formula). There exists a unique regular A

Borel measure dm(x) = dX on the space Xh such that, for any function x E m, its Fourier transfonn x belongs to Ll and satisfies the equality

x(r) = J x(X)X(r)dx· xh

Proof. Denote by I the set of all functions x E Ll (Q, m) n Cb(Q). Lemma 1.1

implies that I is a *-ideal in L 1• In I, we define a linear functional F(x)=x(e)

(x E I), where e is the basis unity of the hypercomplex system l3 1 (Q, m). The func

tional F(x) is real (since e = e*) and positive on I because

For what follows, we need the Weil-M. Krein-Godeman theorem (see, e.g., [Nail, p. 336]): Let R be a commutative Banach *-algebra without identity element. Then,

for any real positive functional f defined on an ideal I which is dense in R, there exists a unique measure 11 on the space M h of symmetric maximal ideals of the al

gebra R such that

(i) forany <pE P, we have <p(t)E L 1(Mh,l1) and

f(<px) = J x(M)<p(M)dl1(M);

Mh

(U) the mapping <p H <p(M) can be extended to an isometric mapping of the Hil

bert space H; into the space L2 (M h, 11).

If I· I is dense in I in the sense of II· lin ' then <p H <p(M) can be extended 'I

to an isometric mappingfrom the entire Hf into L 2(Mh , 11).

Here, the symbol P denotes the collection of all elements positive with respect to f (i.e., elements <p E I such that the functional f!p(x) = f(<px) (x E R) is real and posi-


tive and satisfies the condition If cp (x) 12 ::;; Cf cp (x*x) for all x E R), H f is the Hilbert

space generated by elements x E I with scalar product (x, y)=f(xy*), and H; is the

subspace in H f generated by elements x E P.

By virtue of the denseness of the ideal I in L I' we can apply this theorem to the A

functional F. Consequently, there exists a unique regular Borel measure dm('1J = dX,

given on the space X h of Hermitian characters of a hypercomplex system such that the

Fourier image of any function (X)x(X)dX· (3.3)

xh

Let (x) = F( constructed in the proof of Theorem

2.1. The latter can be extended to a positive functional in :c 1 (Q, m) because L 1 con

tains an approximative unit. Hence, functions <p ELI n P(Q) are positive with respect

to the functional F. Relation (3.3) holds for any function <p E LI n P(Q) and, hence,

for any <p E ffi. In (3.3), we replace x by x*, substitute the corresponding expressions

for F((x)x(r)dx]dr

(here and below, we omit the indices Q and X h of the domain of integration if this

does not lead to misunderstanding). In view of the arbitrariness of x ELI' this com

pletes the proof of the theorem.

• The measure m in Theorem 3.2 is called the Plancherel measure. Theorem 3.2

implies, in particular, that, for any two functions f, gEL 1 n L 2, the convolution f * g

belongs to ffi and satisfies the equality

(3.4)

Denote by Q c X h the support of the Plancherel measure.


Corollary 3.3. If the Fourier transforms of functions fJ, hELl (Q, m) coin

cide on Q, then fl = h.

Indeed, by virtue of Lemma 3.1, the convolution en * f converges to f in L I and, according to (3.4),

e n*fl(r) = f (en*fd"(x)x(r)dx = f en(x)A(x)x(r)dx

Q Q

= f en(x)A(x)x(r)dx = en *h(r). Q

Passing here to the limit as n -7 00, we obtain fl = h.

• Corollary 3.4. A function <p ELI n C b( Q) is positive definite if and only if

q,(X) ~ 0 for all X E Q.

Indeed, let <pE LI n P(Q). By virtue of Theorem 3.2, q, E LI(Q, m) and

<per) = f q,(r)x(r)dx·

Therefore,

= If (x** x)(r)X(r)q,(x)dx dr = f <p(r)(x** x)(r)dr ~ 0

for any function x E LI (Q, m). Therefore, since all functions x (x ELI) form a

dense set in Coo (X h), we get

for any nonnegative function v E Coo(X h ). This is possible only if q,eX) ~ 0 (X E Q).

Conversely, let <p ELI n Cb(Q) and let q,(X) ~ 0 for all X E Q. By using The

orem 3.2 and the fact that g* * g E P(Q) for any function gEL I n Loo, we obtain

f <p(r)(g** g)(r)dr = f <per) f (g** g)"(x)x(r)dx dr


Approximating an arbitrary function y ELl by functions from gEL I n L co in the

metric of L I and using the boundedness of <per) and the continuity of convolution, we get

f <p(r)y** y(r)dr~ 0,

i.e., <per) E P(Q).

• Remark. It follows from Lemma 3.3 and Corollary 3.4 that if f* * f ELI for some

functionfE L 2 (Q,m), then (f**f)"(xJ ~ 0 (XE Q).

Applying (3.4) to the function c(A, B, r) = KA * KB(r), we obtain

c(A, B, r) =. f f c(A, B,p)X(p) dpx(r)dx = f X(A)X(B)x(r)dx· Q Q Q

Let us prove an analog of the Plancherel theorem for hypercomplex systems. The proof is based on a continual analog of the Schur lemma (see, [N ai 1, Section 26, Subsection 5]). For convenience, we present the assertions necessary for what follows.

Let T be a locally compact space, let !l be a regular Borel measure on T, and let R be a Banach algebra with involution.

Theorem (a continual analog of the Schur lemma). Suppose that a representation

x H A x of the Banach algebra R with involution is given by the direct integral of

representations of this algebra in a separable Hilbert space H, namely,

Furthermore, assume that the following conditions are satisfied:

(i) thefunction T3 t H II Ax(t) II E C",,(T) for any XE R;

(ii) the operator Ax(t) is compactfor all x E Rand t E T;

(iii) each representation x H Ax(t) is irreducible;

(iv) the representations x H AxCtl) and x H Ax(t2) are not equivalent for any


Then any bounded operator B acting in

EB

j{= J Hdll T

and commuting with all operators Ax (x E R) has the form B = {~(t)ll}, where

~ (t) E Loo(T, 11). Furthermore, any subspace we of j{ invariant under the action

of all operators A x is a collection of vectors ~ = {~( t)} E j{ satisfying the con

dition ~ (t) = 0 for almost all t E .1., where .1. is a fixed Il-measurable set in T.

Theorem 3.3 (an analog of the Plancherel theorem). The Fourier transformation

x 1--7 X is an isometric mapping from a set dense in L 2 (Q, m) onto a set dense in

L 2 ( Q, m) and, hence, it can be uniquely extended to an isometric operator :J that

maps L2 (Q, m) onto L2 (Q, m).

Proof. Let I, F, and P(Q) be the same as in Theorem 3.2. By setting x = '1'* in (3.3), where 'I' E ffi, we get

Therefore, the Fourier transformation <p H <p is an isometric mapping from the sub

space ffi of L2 (Q, m) into L2 (Q, m). Since ffi is dense in L2 (Q, m), this map

ping can be uniquely extended to an isometric operator :J that maps L2 (Q, m) into

L 2 (Q, m). It remains to show that :J maps L 2 (Q, m) onto L 2 ( Q, m). It suffices to show

that the image of the space L 2 (Q, m) under the mapping :J is dense in L 2 ( Q, m). For this purpose, we use the continual analog of the Schur lemma presented above. If we

A 1 A

take L 1(Q,m) as R and set T= Q, dll=dX, and H=IR, then j{=L 2(Q, m).

Let Ax be the operator of multiplication by the function x(X) (X E Q) in L2 (Q, m);

then A x< t) is the operator of multiplication by the number x (X) in IR 1. Clearly, the

operators A x thus chosen satisfy all conditions of the theorem. Let us show that the

image we of the space L2 (Q, m) under the mapping :J is invariant under the action of

the operators A x (x ELI (Q, m». Consider a function v (X) E :J(L I (Q, m) n L 2 )·

Then v=:Jy, whereYEL I(Q,m)nL 2(Q,m) and Axv=x(X)v(X)=(x*y)''(X).

By virtue of Theorem 1.1, x * y (r) ELI (Q, m). According to Lemma 1.2, we have

(x * y)(r) E L 2(Q, m). Therefore, the manifold :J(L I (Q, m) n L 2 (Q, m» (and,


hence, we) isinvariantundertheactionoftheoperators Ax (XE LI(Q,m»). By using the continual analog of the Schur lemma, we find that we consists of functions equal to

zero almost everywhere on some measurable set ~ in Q. Let us show that m (~) = O. Denote by U I :::::l U 2 :::::l ••• a sequence of balls that contracts to e and consider the se

quenceoffunctions en(r)=Kunr)/m(Un). Clearly, en(r)E LI n L2 and en(x)=

X( Un) / m(Un)· As shown in Theorem 1.6,

which implies that m (~) = 0 by virtue ofthe continuity of the functions en (X).

• Denote by S the Fourier image of the set m n Co (Q).

Lemma 3.4. The set S is dense in L I (Q, m) and L2 (Q, m).

Proof. Consider the set SIc L 2 ( Q, m) of Fourier transforms of functions from

Co(Q). Since Co(Q) is dense in L 2(Q,m), the set SI is dense in L 2(Q, m). Let

XE LI(Q, m). Then x=y·z, where y,z E L 2 (Q, m). Let us find YI,ZI E Co(Q)

such that

Theorem 3.2 implies thatthe function Yl * Z 1 (r) belongs to m n Co(Q)· Hence, the

function (YI * Z 1 )A(X) = YI 21 (X) belongs to S and, therefore,

Finally, by virtue of Theorem 3.3 and Lemma 3.1, we conclude that S is dense in

L 2 (Q, m).

By virtue of Theorem 3.2, the relation

fer) = S !(x)x(r)dx

Q

•

determines the inverse Fourier transformation of the set S dense in L2 (Q, m) onto the


set ffi n Co (Q) dense in L 2 (Q, m). Consequently, the operator :r* = r I is given by the equality

v ] f fer) = (r f)(r) = f(X)X(r)dx (f(x) E S). (3.5)

Q

Relation (3.5) can be extended to the entire space L I ( Q, m) by passing to the limit as

f(x) tends to x(X) E Ll (Q, m) in the same way as in the proof of Lemma 3.4. By v

Theorem 3.3 and Lemma 1.1, the functions f (r) = (y] * Z 1 )(r) E Co(Q) n ffi (we

use the notation of Lemma 3.4) uniformly converge to some function <per) E C",,(Q).

Hence, rl maps L](Q, m) into C",,(Q).

3.2. Duality of Commutative Hypercomplex Systems. Recall the Pontryagin duality principle for commutative locally compact groups: The group of characters constructed for the group of characters of some group is isomorphic to the original group. If the groups in this chain are regarded as bases of the corresponding group algebras regarded, in tum, as hypercomplex systems, then the Pontryagin duality principle can be reformulated as follows: On a given group, we construct a hypercomplex system that coincides with its group algebra. Then, by using the set of characters of this hypercomplex system as a basis, one can construct a hypercomplex system whose characters coincide with the elements of the original group. In this subsection, we show that this construction can be realized in the case of general commutative hypercomplex systems. This duality can be explained by the fact that, along with associative convolution (or, which is the same, generalized translation), there is another operation -ordinary multiplication of functions on Q. Thus, we have a double topological algebra (see {Lit2J and [Lit3]); in the case of hypercomplex systems, this algebra is a Hilbert bialgebra (see Subsection 1.7). As we

pass to the dual object Q (the space of multiplicative functionals), the operations in the double topological algebra interchange, namely, convolution (or generalized translation)

transforms into multiplication of functions defined on Q while multiplication of func

tions defined on Q turns into the dual operation (convolution of functions on Q).

Let ~ I (Q, m) be a commutative hypercomplex system satisfying conditions (H 1)

(H6). Denote sets from 1J 0 ( Q) by the letters <l>, '¥, e, ... and the characters of the

hypercomplex system ~ 1 (Q, m) by X, <p, 'II, e, .... For what follows, we need the functions

XcJ> = f KcJ>(x)x(r)dx· Q

By virtue of (3.5), xcJ>(r) E L2 (Q, m) and xcJ>(X) = KcJ>(X). Theorem 3.1 implies that


XcI>( r) E P (Q). Obviously, I xcI>( r) I ~ xcI>( e) = m (<I» ( E tEo ( Q». For all <1>, 'P E

tEo ( Q) and X E Q, we set

c(, 'P, X) = f x(r) f <p(r)d 'I'

Clearly, the integral on the right-hand side converges. We now show that, under certain

conditions, c( <1>, 'P, X) is a structure measure which defines convolution in the dual hypercomplex system.

We say that an Hermitian character X is essential if X E Q.

Theorem 3.4. In order that c (, 'P, X) be a structure measure, it is necessary

and sufficient that its support is compact for fixed <1>, 'P E '130 ( Q) and that the pro

duct of any two essential characters X, 'II E Q be positive definite.

Proof. Necessity. By virtue of Corollary 3.4 and the fact that 8(<1>, 'P, X) =

(XcI>X'l')"(X) ~ 0 for all X E Q, the function xcI>(r)x'I'(r) is positive definite. Let us

show that the product of any two essential characters n)~=\ and ('Pn)~=\ besequences

of neighborhoods in Q that contract to (r) I m(n) - ,

by passing to the limit as n --7 00 and using the fact that the function rex) = X (r) IS

continuous, we conclude that

for any function x E L J (Q, m). The nonnegativity of the limit follows from the positive

definiteness of the function xcI> (r) x'I' (r) for all n. n n

SUfficiency. Let <1>, 'P E '13 0 ( Q). Since xcI>(r)xlj/(r) E L \ (Q, m), we have

8(<1>, 'P, X)E Cb(X,J Moreover, xcI>(r)x'I'(r)E P(Q) by Corollary 3.1. Hence,

8(<1>, 'P, X) = (xcI>x'I')"(X) ~ 0 (X E Q, <1>, 'P E tEo ( Q» by Corollary 3.4. Clearly,

8( <1>, 'P, X) is a regular Borel measure with respect to 'I' for any fixed E tEo ( Q) and X E Q. Clearly, (H3) is also satisfied. It remains to check the associativity relation (H2). Indeed, Theorems 3.2 and 3.3 yield

Section 3 Elements of Harmonic Analysis for Normal Hypercomplex Systems

f c(<1>, 'P, X)dxc(Ex' n, q» = f f (x<pX'I')"(x)x(r)dXxn(r)q>(r)dr

Q Q

= f X ( r) x'I'(r) x Q( r) q>(r)d r

Q

Assume that the conditions of Theorem 3.4 are satisfied.

85

• The Plancherel measure can be regarded as the multiplicative measure for the

structure measure 2(<1>, 'P, X). In fact,

by virtue of Theorems 1.6 and 3.2.

• It is easy to see that the dual hypercomplex system is normal if the complex conju

gation x*(r) = x(r) is taken as the involutive homeomorphism. Every point r E Q

generates an Hermitian character r (X) = X (r) (X E Q) of the dual hypercomplex system.

Indeed,

2(<1>, 'P, n) = f f x(r)x'I'(r)x(r)drdx = f x(r)x'I'(r)xQ*(r)dr QQ

= f f x(r*)dx f 'I'(r*)d'P f ro(r*)drodr Q, 'P, n E 'Bo( Q), whence we conclude that the dual hypercomplex system is normal.


By using Theorem 3.2, we can show that the function rex) = X (r) (x E Q) is a character of the dual hypercomplex system for each r E Q:

= x(r)x'P(r) = J X(r)dx J 'I'(r) d'l' = r(<1»rCP). 'P

• Each character of a compact hypercomplex system is essential.

Indeed, by Theorem 1.4, every character Xn is an idempotent. Hence, Xn E ffi. The

characters are orthogonal and, therefore,

This and Theorem 2.2 imply that

00

Xn(r) = L Xn(p)Xp(r)m(p) = Xn(r) IIxn II; men). p=o

By setting r = e, we obtain

men) 1

= --2 (Xn E X = CX)~~)' IIXn Ib

whence it follows that men) > o.

(3.7)

• If a character identically equal to one is essential, then it is a basis unity of the dual

system.

Indeed, by Theorem 3.3, for all <P, 'P E r.Bo( Q), we have

2(<1>, 'P, 1) = f f <p(r)d 'P Q 'P*

= f x(r)x'P*(r)dr = J K(r)K'P*(x)dX = m(<1> n 'P*). Q Q

•


To prove duality, we suppose (throughout the remaining part of this subsection) that the identity character is essential and the conditions of Theorem 3.4 are satisfied. In addition, we require the following condition to be satisfied:

(P) If <p and \jf are continuous positive definite functions on Q which are the A

characteristic functions of measures concentrated on Q, then the measure associated

with the positive definite function <p. \jf is also concentrated on Q.

By Qh we denote the space of Hermitian characters of the dual hypercomplex sys-A

tern and let Q c Qh be the support of the Plancherel measure :it constructed accord-

ing to the dual hypercomplex system. We have shown that Q c Qh. It is therefore rea-

sonable to ask under what conditions we have Q c Q or Q = Q. In the second case,

we say that duality takes place. Let us describe the range of the Fourier transformation.

Lemma 3.5. The equality

(fg)A(X) = i * 8(X) (X E Q)

holds for all functions f, g E L2 (Q, m ).

Proof Let f, gEL 2 (Q, m) be arbitrary fixed functions. Consider sequences

fn E m and gn E m which converge in L2 (Q, m) to the indicated functions (the fact that such sequences exist follows from Lemma 3.1). By Theorem 3.2,

= J J in(<p)<p(r)dJ 8n(\jf)d'l'c(Eq>' E'I" X) = (in * 8n)(X) (X E Q). Q Q

On the other hand, by virtue of Theorem 3.3,

II (fngn)"(X) - (fg)"(X) ILo = su~ I f[fngn(P) - fg(p) ]x(p)dp I XEQ


--~) o.

• Corollary 3.5. The rank of the Fourier transformation :r(L1 (Q, m») is equal to

L2 (12, m) * L2 (12, m). In particular, L2 (12, m) * L2 (12, m) is a linear space.

Indeed, by representing a function fELl (Q, m) as f = y z (y, z E L2 (Q, m ) (e.g.,

one may set y = If I 1 /2, Z = Ifl- 1 /2 f if f::f. 0 and 0 if f= 0) and applying Lemma 3.5,

we conclude that :re L1 (Q, m ») c L2 ( 12, m) * L2 ( 12, m). To prove the inverse inclu

sion, for any two functions f, g E L2 (12, m), we find (according to Theorem 3.3) their v v v v Vv

Fourier preimages f, g E L2 (Q, m). Then f· gEL 1 (Q, m) and e f g) ('X,) =

(f * g )(x) by virtue of Lemma 3.5.

• Let P(Q, Q) be the set of continuous positive definite functions on Q which are

characteristic functions of measures concentrated on 12 and let ffi (Q, (2) be the linear

span of P(Q, (2) n L 1• Consider the manifold R= ffi(Q, (2) n Co(Q). If x(r)E R,

then I x(r) 12 E R because, by virtue of condition (P), the product of any two functions

from P (Q, (2) is also an element of this set.

Throughout the remaining part of this subsection, we suppose that the hypercomplex

system L1 (Q, m) satisfies the condition of separate continuity.

By virtue of Theorem 3.2, the linear span of the set {a * b I a, b E L 2,o} lies in R.

Consequently, R is dense in C,,(Q) according to Lemma 3.2.

Let us show that R majorize Co (Q), i.e., for any function x (r) E Co (Q), there

exists a function x'(r) E R such that 1 x(r) I ~ x'(r). In fact, consider the function

y(r) E Coo(Q) such that y(r) > max {I x(r) 1+ 1,2} for all r E supp x. We find a

functionz(r)ER such that Ily-zll <1/2. Then Iz(r)l>max{lx(r)l+l/2,l} 00

(rE suppx) and one can take Iz(r)1 2 = z(r)z(r) ERas x'(r).

• By using the property proved above, we may conclude that every positive linear

functional given on R can be extended to a positive linear functional given on the entire

space Co(Q) (see, e.g., [KVP]).

We define the convolution of a measure J.1 E Mt (Q) with a function f E Co (Q)

as follows:

(f*J.1)(r) = f (LrJ)(s*)dJ.1(s) = f (Lrf)(s)dJ.1*(s),

Section 3 Elements of Hannonic Analysis for Nonnal Hypercomplex Systems 89

where I-l *(A) = I-l(A *). By virtue of (2.7), we have

I (f* I-l)(r) I ::; IIflLo II 1-l1I·

The function (f * I-l) C r) is continuous. Indeed, consider a sequence r n E Q which

converges to r E Q. Since CLrf)(s) is separately continuous, we have (Lr f)(s) ---7 n

CLrf)(s) for each SEQ. The continuity of (f* I-l)(r) follows from the Lebesgue

theorem.

• Let us show that (f * I-l)(r) is integrable. In fact, since generalized translation

operators are linear, it suffices to check this assertion for nonnegative f E Co C Q). By

using the fact that Ls preserves positivity, the invariance of the multiplicative measure,

and relation (2.1), we obtain

= If fCr)drdI-lCs) = III-llillfll].

• Hence, (f * I-l)(r) E Cb(Q) n L] (JE CoCQ), I-l E MtCQ»).

We define the Fourier transfonn of a measure I-l E Mt (Q) by the relation

~(;O = f X(t)dl-l(t) (X EX).

Clearly, ~(X) E Co(X).

If the Fourier transforms of two measures from Mt (Q) coincide on Q, then

these measures coincide.

Indeed, it suffices to show that ~(X) =1= 0 (X E Q) for any nonnegative measure

I-l E Mt C Q). Since

(f* I-l)(e) = f f(s*)dI-lCs),

for each I-l E Mt C Q), one can find f E Co (Q) such that f * I-l =1= 0 (r E Q). Since

f * I-l E L], in view of Corollary 3.3, there exists a character X E Q such that

(f * I-l)" (X) :f:. O. At the same time,


(f*l-l)"(X) = fJ (LJ)(s*)d/l(s)x(r)dr = JJ (Ls·J)(r)x(r)drd/l(s)

but this implies that ~ (X) '# o.

• The characters from Q separate points of the set Q, i.e., for any points p', p" E

Q, one can find X E Q such that X (p') '# X (p"). Suppose the contrary. By virtue of Lemma 3.2, there exists a function fER such

that f(P') '# f(p"). Then Theorem 3.2 implies that

f(P') = f j(X)X(p')dX = f j(X)X(p")dX = f(P") Q Q

and we arrive at a contradiction.

• Theorem 3.5. Let L J (Q, m) be a commutative normal hypercomplex system with

basis unity which possesses the following properties:

(i) the identity character is essential;

(ii) the product of essential characters is a positive definite function and the measure that corresponds to this function according to Theorem 2.1 is concentrated on the set of essential characters;

(iii) the structure measure c (<p, '¥, X) of the dual hypercomplex system has com

pact supportfor fixed <1>, '¥ E 11o( Q);

(iv) the hypercomplex system L J (Q, m) satisfies the condition of separate con

tinuity.

Then the dual hypercomplex system also satisfies conditions (i)-(iv) and duality

takes place.

Proof. We have already shown that each point r E Q generates an Hermitian char

acter rex) = x(r) (X E Q) ofthe dual hypercomp1ex system. In view of the fact that

characters Q separate the points of Q, we have Q c Qh' where Qh is the set of Hermitian characters of the dual hypercomplex system.

Consider a family of functions R. Functions from Coo(Q) separate points in Q.

For any point r E Q, there exists fE Co(Q) such that fer) '# O. By Lemma 3.2, R is

Section 3 Elements of Hannonic Analysisfor Nonnal Hypercomplex Systems 91

dense in Coo(Q). Therefore, R also possesses the indicated properties. It is easy to see

that the weak topology in Q generated by the family of functions R coincides with the original topology in Q (see, e.g., [Nail]). By Theorem 3.2, this implies that the sets

(Xl, ••• ,XnE R, rE Q, £>0)

form a base of neighborhoods in Q. Since reX) = x(r), by the definition of the topology in the space of characters, U (r; £; xl' ... , X n ) are neighborhoods in the space Q regarded as a subspace of Qh' In Lemma 3.4, it was actually proved that the Fourier

image of the linear span of {y*ZIY,ZE Co(Q)} cR is dense in L1(Q, m). There

fore, U (r; £; x 1, ... , X n) form a base of neighborhoods in Q c Qh' A A

We show that Q c Q. Let 't E Q be an essential character of the hypercomplex

system J3 1 ( Q, m). Consider a linear functional

J(x(r») = J x(X)'t(X)dX (XE R).

By Lemma 3.5 and Theorem 3.2,

J(x(r)y(r») = J (xY)"(X)'t(X)dX = f (x * Y)(X)'t(X)dX

= J x(X)'t(X)dX J y<e)'t(e)de = J(x(r»)J(y(r») (x, y E R),

i.e., the functional is mUltiplicative on R. Let us now show that J is positive on R. Let

x(r):2:0 (XE R) and y(r)= ~x(r). Then y(r)E Co(Q) and,hence, yE L2 (Q,m). Consequently, by virtue of Lemma 3.5,

Since x(r) E R, in view of Theorem 3.2, we have x(X) = y* * Y<X) E L1 ({2, m). By

applying the remark to Theorem 3.2 to the function Y<X), we obtain

i.e., J(xCr»):2: O. Since R majorizes Co(Q), the functional J(x(r») can be extended

to a positive linear functional j on the entire Co (Q). By the Riesz-Markov theorem,


there exists a regular Borel measure 11 on Q such that

l(x(r») = f x(r)dll(r).

for all x E Co C Q). This and the multiplicativity of the functional/imply that

f xCr)dll(r) f y(r)dll(r) = f x(r)y(r)dll(r) (x, y E R). (3.8)

We show that the measure 11 is finite. Indeed, let Qn be a sequence of compact sets

such that U:=l Qn = Q and Qn C Qn+ 1· For each open A E 'Bo(Q), we consider a

sequence of functions

1 xnCr) = --cCQn,A, r).

meA)

Clearly, XnE R. By virtue of (1.13), we have O~xn(r)~ 1. In view of (1.12),

xn(r) ~ 1 as n ~ 00 for all r E Q. Since x~(r) ~ xn(r), we find

Hence,

By using the Fatou lemma, we obtain

Il(Q) = f lim xn(r)dll(r) ~ lim f xn(r)dll(r) ~ 1. n~oo n~oo

If we approximate arbitrary functions J, g E Coo(Q) by functions x, y ERin the

uniform norm, then it follows from (3.8) that

f f(r)dll(r) f g(r)dll(r) = f f(r)g(r)dll(r).

This relation immediately implies that there exists a unique point So E Q such that

/(x(s») = f x(s)dll(s) = x(so)·

A

Let us show that Q = Q. By Theorem 3.3, L 2 (Q, m) is isomorphic to L2 ( Q, m)


~

which is, in tum, isomorphic to L2 (Q, th). Consequently, L2 (Q, m) is isomorphic to

L2 ( Q, th). Denote this isomorphism by l' 0 :r. By Theorem 3.2, we have

fer) = f J(x)x(r)dx = Q

A

(rE Q)

for all fER, i.e., under the isomorphism 'J 0 'J, the function fER, is mapped into A A

fl Q. Since the topology in Q is induced by the topology from X, the space Q is closed, and R is uniformly dense in Coo(Q), we conclude that, for each x E Co(Q), its

A ~ A

image §: 0 'Jx = x I Q. Since Q is closed in Q, U = Q\ Q is an open subset of Q.

If U * 0, then. by virtue of the regularity of m, there exists a compact set A E 'Bo(Q) such that A c U and m (A) > 0. Let the function f E Co (Q) be such that f( r) = 1 on

~

A, 05J(r) 5, 1, andf(r)=O on Q. Since m(A»O, we have f*O in L2 (Q,m).

Since §: 0 'Jf = f I Q = 0, we arrive at a contradiction with the assertion that L2 (Q, ~ A

m) is isomorphic to L2 C Q, th). Thus, U = 0 and Q = Q. Consider functions of the form U*f')(x) (f(X)E CoCQ»). Clearly, f*f*E

ffi ( Q) n Co ( Q). By applying Lemma 3.5 and Theorem 3.2 to the dual hypercomplex system, we get

flf(r)1 2 x(r)dm(r) =f*f*(x) = fIJ(r)12 r(X)dth(r). Q Q

It follows from the relations !t 0 'Jx = x I Q and Q = Q that ](r) = fCr). There

fore, in view of the uniqueness of the Fourier transforms of measures from Mt ( Q), we obtain

1 ~ 2 A 12 ~ d~Cr) = fer) 1 dm(r) = itCr) dm(r).

Since the image of ffi ( Q) n Co ( Q) is dense in L2 ( Q, m), we have m = th.

Relations (3.4) and (3.6) imply that the structure measure 2(A, B, r) (A, B E 'Bo(Q»)

of the hypercomplex system dual to L1 ( Q, m) coincides with the initial structure mea

sure c(A, B, r).

By virtue of the fact that e (X) = X( e) = 1 (X E Q), the basis unity e E Q gener-

ates the identity character of the dual hypercomplex system. Since Q = Q, this character is essential.

Let us show that the hypercomplex system dual to L1 ( Q, m) satisfies condition (ii)


of the theorem. Since Q = Q, it suffices to show that the function P*(X)q*(X) = X (p) X (q) (X E Q) is positive definite for any p, q E Q and the support of the measure associated by Theorem 3.1 with X(p)X(q) lies in Q. By virtue of (1.19), for any

BE 'Bo(Q), one can find a compact set FE 'Bo(Q) such that

In this equality, we set B = Un! where Un is a sequence of neighborhoods of the point

q contracting to q. Clearly, we can take the same compact F for all n. Then

(3.9)

where

Clearly, Iln(E) :?: ° and Iln(Q) = 1. Let us show that the sequence of probability

measures Iln is uniformly dense, i.e., for any E> 0, one can indicate a compact set K

such that Iln(K) > 1 - E. In fact, let Fn be a sequence of neighborhoods of the point p

contracting to p. We set K = supp c (F I' U I' .). Then, in view of the continuity of the structure measure, we have

lim 1 J c(K, U~, p)dp i-too m(Fj)m(Un)

F;

. c(K, U~, Fj) = 11m = i-too m(Fj)m(Un)

i cCFj, Un' p)dp = lim ....!Q"-------

i-too m(Fj)m(Un ) = 1.

Hence, by virtue of the Prokhorov theorem, one can choose a subsequence Ilnk of the

sequence 11 n weakly convergent to a nonnegative regular measure d 11 (r) = d 11 p, qC r).

By passing in (3.9) to the limit in the subsequence Ilnk' we obtain

x(p)x(q) = J XF(r)dllp,qCr) = J x(r)dllp,q(r). (3.10)

F


It is easy to show that the function on the right-hand side of (3.10) is positive definite. Indeed,

f f xCr)d/lp.q(r)(x* * x)(X)dX Q F

for all x E L] U;2, m).

:::; f f X(r)(x** x)(X)dXdllp.q(r) FQ

:::; f 1 xCr) 12 d/lp,qCr) F

Finally, let us show that condition (iv) is satisfied for the hypercomplex system

Ll (Q, m). Let X, 'V E Q. By virtue of (ii), the function x(r)'V(r) is positive definite. By Theorem 3.1, this yields

X (r) 'VCr) :::; f 8Cr) dllx. '1'(8),

Q

where Ilx. '1" by virtue of (ii), is concentrated on Q. Let us show that

for all fE Co(Q).

(Lxf)('V):::; f f(8)dll x.'I'(8) Q

A

(X, 'V E Q) (3.11)

Let fER, then, clearly, JELl (Q, m) n eoo ( Q) and for arbitrary g E L2 (Q, m), we obtain

:::; I (f fer) I 8(r)dllx''I' C8)drJ g('V)d'V Q Q Q

:::; f f f(r)xCr)'VCr)drgC'V)d\jl. Q Q

On the other hand, by virtue of Theorem 3.2 and (3.6), we have

(j * g * )( x) = f f j( 8 )g('V)dc( E a, E'I" X) Q Q


= f f i(9)9(r)d9 f g(\jI)\jI(r)d\jlx(r)dr

Q Q Q

= f f f(r)x(r) \jI(r)drg(\jI)d\jl.

Q Q

A

Thus, relation (3.11) holds for all f(x) (fE R). By Lemma 3.1, R is dense in L 1•

Hence, for any x E L1, one can choose a sequence f n E R that converges to x in L 1 .

Then the relevant Fourier transforms in uniformly converge to x in C",,( Q). In view

of (2.7), the generalized translation operators L X are continuous in L oJ Q, m). By

passing to the limit on both sides of relation (3.11), we conclude that (3.11) holds for all

x(X) (XE L 1). Since the Fourier image of L1(Q,m) is dense in C",,(Q), we arrive at (3.11) by passing to the limit.

We fix \jI E Q, assume that X n ~ X in Q, and choose ~ E 130 (Q) such that

Xn E ~ for all n E N. The structure measure c(c:t>, '1', X) has a compact support with

respect to X by the condition of the theorem. Thus, by virtue of Theorem 2.1, the gen

eralized translation operators Lx (X E Q) associated with the dual hypercomplex system satisfy the finiteness condition (F). Recall that, according to condition (F), there

exists a compact set K c Q such that (Lxf)(\jI) = 0 for all X E~, provided that suppfn K= 0. By virtue of(3.11), this means that supp ).lx,'V c K for all X E ~. In

A

addition, it is clear that sup ).lXn,'V(Q) = 1. Thus, the sequence of measures ).lxn,'V is n

uniformly dense and, by the Prokhorov theorem, weakly compact, i.e., any its subse-quence ).lxn"'V contains a weakly convergent subsequence ).lxn"''V· Assume that ).lxn"''V

weakly converges to a measure V ().lXn"''V => v). This enables us to conclude that, for

all r E Q,

Xn,,(r)\jI(r) = f 9(r)d).lxn",'V(9) -n--7-""~) f 9(r)dv(9).

On the other hand, since Xn ~ X, we have

for all r E Q. This yields

f 9(r)d).lx,'V(9) = f O(r)dv(O).

Each integral in this equality is a continuous positive definite function on Q. By Theorem 3.1, the integral representation (3.2) is unique. By using this fact, we conclude that

v = ).lx.'V' whence ).lxn.'V => ).lx.'V· In view of (3.11), this immediately yields the con-


tinuity of (Lxf)('¥) in X for fixed '¥. The continuity of (Lxf)('¥) in '¥ can be proved analogously.

• Let us now describe an analog of the Fourier algebra. Let L 1 (Q, m) be a normal

hypercomplex system with basis unity satisfying the conditions of Theorem 3.5. By virtue of this theorem, all results established in 3.1 and 3.2 hold for the dual hypercomplex

system L1 (Q, m). Denote

A

By Lemma 1.1, A(Q) cCo(Q). Since L 1(Q, Jz)=L 1(Q,m), by applying Corol-

lary 3.5 to the hypercomplex system L 1 (Q, m), we can show that A (Q) is a linear

space. A norm in A (Q) is introduced by setting

By the same Corollary 3.5, A(Q) with this norm is complete.

Theorem 3.6. Assume that L 1 (Q, m) is a commutative normal hypercomplex

system with basis unity satisfying the conditions of Theorem 3.5, i.e.,

(i) the identity character is essential;

(ii) the product of essential characters is a positive definite function and the measure that corresponds to this function according to Theorem 3.1 is concentrated on the set of essential characters;

(iii) the structure measure c( <1>, \{1, X) of the dual hypercomplex system has com

pact supportfor fixed <1>, \{1 E 'Bo( Q);

(iv) the hypercomplex system L 1 (Q, m) satisfies the condition of separate con

tinuity.

Then A (Q) = L 2 (Q, m) * L2 (Q, m) is a Banach algebra with respect to the or

dinary multiplication of functions and the norm II·IIA (Q) The algebra A (Q) is

called the Fourier algebra of the hypercomplex system L 1 (Q, m).

Proof. First, we show that the set Co (Q) * Co (Q) = {u 1 * u 2 I u l' u 2 E Co (Q) } is dense in A(Q). We fix arbitrary f 1,12 E L2 and take u 1, u2 E Co(Q) such that


By applying Theorem 3.3, we obtain

Now let ul'u2,vl'v2E Co(Q). Denote <l>1=u1*v 1 and <l>2=u2 *v 2. Then

<1>1' <1>2' <1>1<1>2 E LI (Q, m) n Coo(Q). By virtue of Lemma 3.5, we have (12/(X) = <PI * <P2 (X)· Since i = U i * viE ffi (i = 1, 2), by virtue of Theorem 3.2, we conclude

that <Pi(X) ELI (Q, m) (i = 1,2). At the same time, by Theorem 3.4, L I (Q, m) is a

Banach algebra. Therefore, ( I 2)" (X) = (<PI * <P2)( X) ELI ( Q, m). Consequently,

one can find elements h l ,h2 E L2(Q, m) such that (l<1>2)"(X) = (<P1 * <P2)(X) = v v

hI (X) h 2 (X)· Denote by hi' h2 E L 2 (Q, m) the preimages of the functions h 1 and

h2 under the Fourier transformation. By applying Corollary 3.5 to the hypercomplex

system L I ( Q, m), we obtain

v v (u l *v 1 )(u 2*v 2)(r) = (I<1>2)(r) = (r 1(h 1h2»(r) = hi * h2(r)E A(Q).

Moreover, in view of the fact that L 1 ( Q, m) is a Banach algebra, we get

To complete the proof, it remains to use the fact that Co (Q) * Co (Q) is dense in A (Q) .

• In passing, we have proved the fact that the Fourier algebra A (Q) is isomorphic to

the algebra L1 (Q, m). By passing to the dual object, we conclude that the Fourier al

gebra A ( Q) of the dual hypercomplex system is isomorphic to L I (Q, m ).


3.3. The Case of Discrete Hypercomplex Systems. Let us study hypercomplex systems with discrete bases in more details. A bounded numerical sequence Tl (P) (p E Q) is called positive definite if

n

L (Rr;Tl)(rz )AkI [ ~ 0 (3.12) k,I==1

for any AI'"'' An E a:: and r l' ... , r n E Q. This definition coincides with the definition presented in Subsection 3.1.

The spaces of characters X and Hermitian characters X h are compact sets because

the hypercomplex system 11 (m) possesses an identity element. In view of relation (3.4), we have

c(p, q, r) = m(p)m(q) J p(x)q(x)r(x)dx, (3.13)

xh

where dX is the Plancherel measure and p(X) = X(p) (p E Q). Let us show that hypercomplex systems with compact and discrete bases are dual to

each other, namely, the following theorem is true:

Theorem 3.7. Let 11 (m) be a commutative normal hypercomplex system with

discrete basis and basis unity. In order that the function 2(<1>, '1', X) defined by relation (3.6) be a structure measure of the dual hypercomplex system with compact basis, it is necessary and sufficient that the product of any two essential characters be positive definite. If, in addition, the measure corresponding to this product according to Theorem 3.1 is concentrated on the set of essential characters and the identity character is essential, then duality takes place, the product of any two continuous positive definite functions f (X) and g (X) from the dual hypercomplex system is positive definite, and the dual hypercomplex system satisfies the condition of separate continuity.

Proof. The fact that the basis of the dual hypercomplex system is compact follows from the fact that the hypercomplex system 11 (m) contains the identity element. The space of characters of this compact hypercomplex system is discrete by virtue of Theorem 1.4 and each character in this space is essential. The proof is completed by applying Theorem 3.5.

• It is easy to see that a similar result is true for hypercomplex systems with compact

bases. We introduce an important concept of A -system of functions. Let E be a compact

space with nonnegative Borel measure dcr(t) such that supp cr = E, let Q be a count

able set, and let L = { <P pC t) } pE Q be a system of continuous functions orthogonal and

100 General Theory of Hypercomplex Systems Chapter I

complete in L2 (E, dcr). This system is called an A-system if

(AI) for all fixed p, q E Q, the sequence

is finite and consists of nonnegative members;

(A2) if <ppCt) E L, then <pp(t) E L;

(A3) there is a point e E Q such that <Pe(t) == I (t E Q);

(A4) there is a point a E E such that <pp(a) = I (p E Q).

For every normal hypercomplex system 1\ (m) with basis unity and discrete basis

Q whose identity character belongs to Q, it is possible to construct an A-system of

functions.

Namely, we set

cr = m, E = supp cr = Q, L = {P(X)}pE Q

and show that L is an A-system. First, we note that the function p(X) = X(p) is the

Fourier transform of the sequence K p (q) = () p, q / m (p) (q E Q and 0 p, q is the Krone

cker symbol). Therefore, the functions p(X) are continuous and, in view of Theorem

3.3, form an orthogonal system in L2 ( Q, m), i.e.,

J p(X)q(X) dX Q

(3.14)

The completeness of the system L follows from the fact that the set of sequences

(Kp(r))rE Q (p E Q) is total in 12, Condition (AI) follows from (3.13) and the fact

that c(p, q, r) is nonnegative and finite. Since p(X) = X(p) = X(p*) = p*(X), (A2) is

satisfied. Condition (A3) follows from the equality X (e) = 1 (X E Q), and condition (A4) is a consequence of the fact that the identity element is an essential character of the

hypercomplex system 11 (m ).

• Note that the normal hypercomplex system 11 (m) with basis unity is isometrically

isomorphic to the algebra of continuous functions on E = Q with ordinary algebraic


operations whose expansions in the complete orthogonal system of functions p (X)

(X E Q, p E Q) are absolutely convergent, i.e., to the algebra of functions

such that

x(X) = L x(p)p(x)m(p) p

L Ix(p)lm(p) = IIxll < 00

p

The validity of the converse statement is established by the following theorem:

Theorem 3.8. For any A -system of functions L :::; {<l' pC t)} pE Q' it is always

possible to construct a commutative normal hypercomplex system with discrete basis Q and basis unity. Each point tEE generates, generally speaking, a generalized

Hermitian character of the derived hypercomplex system X t (P) :::; <l' pC t) (p E Q). If

the sequence <l'p(t) (p E Q) is bounded, then the relevant Hermitian character is

ordinary.

Proof. We set

( 2 )-1 2 m(p) = JI<l'pct)1 dcr(t) :::; ll<l'p ll~ ,

c(p, q, r) = m(p)m(q) f <l'p(t)<l'q(t)<l'r(t)dcrCt). (3.15)

By virtue of (AI), c(p, q, r) is nonnegative and finite along r for fixed p,.q E Q. The commutativity relation is obvious, and the associativity relation (1.25) follows from the Parseval equality:

r r

r

= L c(q, l, r)c(p, r, s). r


Thus, c(p, q, r) is a structure measure. We expandthe continuous function <p/t)<pit)

into the Fourier series in the orthogonal system <Pr(t), i.e.,

r

By virtue of (AI), the sum in (3.16) is finite. Hence, both functions in this equality are continuous and (3.16) holds at every point tEE. By setting t = a in (3.16), we obtain

r

Multiplying the last equality by lI<pplI~211<pqIl;2, we get (1.26). Hence, "<pp,,~2 is a

multiplicative measure. We introduce involution in Q as follows: Let p* be the index

of the function <PpCt) E L. Then

and

c(p, q, r)m(r) = m(p)m(q)m(r) f <pp(t)<pqCt)<Pr(t)dcr(t)

= c(r, q*,p)m(p),

i.e., the hypercomplex system is normal; recall that, in view of (AI), we have

The point e E Q is the basis unity of the derived hypercomplex system. Indeed, e = e * and

c(p, q, e) = m(p)m(q) f <pp(t)<pq*(t)d1:(t) = m(p)bp,q*'

Multiplying equality (3.16) by m (p)m(q), we obtain (1.27), which proves the last assertion of the theorem.

•


Let 11 (m) be a normal hypercomplex system with discrete basis Q and basis unity

constructed from the A-system of functions <pp(t) (p E Q, tEE). Assume that the li-

near span of {<pit) I p E Q} is dense in C (E). We endow X g ,h with the topology of

uniform convergence on compact sets. Consider a continuous map a: E ~ X g,h de

fined as follows: a(t)(p) = <pit) (p E Q, tEE). Since the functions <pp(t) (p E Q)

separate the points of E, the map a is injective. Moreover, aCE) is homeomorphic to

E and is a compact subset of Xg,h (as the image of the compact set E under a continu

ous map). Thus, without loss of generality, we may consider the functions <pp(t) as

given on aCE). As already mentioned, under the condition that the identity character of

the hypercomplex system 11 (m) is essential, the set of functions <pix) = X (p) (X E Q, pE Q) is also an A-system such that the sequence (<PiX))pEQ is bounded for each

X E Q and <pp(X) = <pp(t) whenever X = aU) E Q n aCE) (note that Q n a(E):;i: 0

because a(a) E Q).

Thus, each function <P p( t) from an arbitrary A -system of functions on the compact

set E such that the linear span of these functions is dense in C (E) and the identity

character of the hypercomplex system 11 (m) constructed from this A -system is es

sential can be extended to a function <P~ on, generally speaking, another set E'

(E n E':;i: 0) so that the functions <P~ I E' also form an A-system generating the same

hypercomplex system II (m), and the sequence (<p~ (t')) pE Q is bounded for all t'E E'.

Hence, without loss of generality, we may consider A-systems of functions (<pp(t)) pE Q

(t E E) such that the sequence (<pit)) pE Q is bounded for all tEE. In this case,

aCE) c X h and the image a (cr)(dX) of the measure dcr(t) under the mapping a

is a Plancherel measure on X h . Hence, Q = aCE). Indeed, consider the functions

In view of orthogonality, we have

= f Xt(P)Xt(q)dcr(t) = f <pp(t)<pq(t)dcr(t) E

= _1_0 m(p) p,q

E

r

(3.17)

Since the functions Kp(r) (p E Q) form a total set in 12 (m), the Parseval equality


(3.17) holds for all fE 12(m). In view of the uniqueness of the Plancherel measure, we

get da(a)(x) = dm(x).

• The most difficult problem encountered in proving that a given orthogonal system of

functions {<Pp(t)}pE Q is an A-system is to show that the integrals in expansion (3.16)

are nonnegative. Below, we present a sufficient condition for the nonnegativity of these integrals which belongs to Koomiwinder [Ko02]).

Let E and F be compact Hausdorff spaces with nonnegative regular Borel measures

a and Il, respectively, where supp a = E and supp Il = F. Let {<PP(t)};=1 and

{rp ('t)} ;=1 be families of continuous functions orthogonal with respect to the measures

a and Il, respectively, and let ro ('t) == 1. Assume that the system {<ppCt)} ;=1 is com

plete in L2 (E, a) and only finitely many terms in expansion (3.16) differ from zero.

Theorem 3.9. Assume that there exists a continuous mapping A: E x Ex F ~ E

such that the functions <P pC t) satisfy the additional formula

(3.18)

where <P;(t) are functions continuous on E, <P~(t) = <P pet), c p ,k ~ 0, c p,o ~ 0,

and, for each n, only finitely many coefficients c n,k differ from zero. Then the inte

grals in expansion (3.16) are nonnegative.

Proof. Denote

a(p, q, /) = J <pp(t) <PqCt)<pz(t)da(t).

We have

) ( ) ~ <pz(A(t, s, 't») <piA(t,s,'t) <Pq A(t,s,'t) = 7 a(p,q, I) II<pIl@

Consequently,

a(p, q, I)

= (cl,oll<p/II~r J J J <pp(ACt, s, 't») <piA(t, s, 't») <p/(t)<p/(s) da(t)da(s) dll('t) E E F

Section 3 Elements of Hannonic Analysisfor Nonnal Hypercomplex Systems 105

• We note that additional formulas of the form (3.18) are true for some classes of or

thogonal polynomials (of single or several variables), in particular, for the Jacobi poly-

nomials p;a,~) (t) (a > ~ > -1/2) (for details, see [Ko02]). Relations of the form

(3.18) appear in studying spherical functions on the homogeneous space G/ H, where G is a compact group and H is a subgroup of G (see Section 2 in Chapter 2).

3.4. Representations of Hypercomplex Systems and Approximation Theorem. This subsection contains a brief investigation of representations of hypercomplex systems. The commutativity of hypercomplex systems is not supposed.

Let L} (Q, m) be a normal hypercomplex system with basis unity e. The family of

bounded operators U = (Up)pE Q in a separable Hilbert space J{ is called a represen

tation of a hypercomplex system if

(1) Ue=l;

(2) U; = Up' (p E Q);

(3) for each ~ E :J£, the vector function Q 3 P H Up~ E J{ is weakly continuous;

(4) for all A, BE 'BO(Q),

J c(A, B, r) Urdr = J Updp J Uqdq. (3.19)

A B

The integrals in (3.19) are understood in Bochner's sense with the a-algebra 'B(Q),

and the integrands are vector functions Q 3 P HAp E f.J... J{ -? J{) or their products with continuous functions. Their existence follows from (3) and general properties of the Bochner integrals; it should be noted that, in view of the Banach-Steinhaus theorem,

condition (3) implies that the function Q 3 pHil Up \I is locally bounded (see, e.g., [Yos]).

Let us show that the family of generalized translation operators L p' (p E Q) de

fines a representation (so-called left regular representation) of the hypercomplex sys-


tem L 1 (Q, m) in the Hilbert space L2 (Q, m). Indeed, the generalized translation operators Lp' obviously satisfy conditions (1)-(3). Let us prove (3.19). Denote

By virtue of (1.17), Lx is the operator of left convolution with x ELI (Q, m). Hence,

In particular, we have

which is obviously equivalent to (3.19).

• Similarly, one can show that the right generalized translation operators Rp (p E Q)

define a right regular representation.

In [BeK3] and [BeK4], it was shown that any representation Up of a commutative

normal hypercomplex system with strong basis unity (see Subsection 1.4) admits the following spectral decomposition:

Up = J X(p)dE(X) (p E Q). (3.20)

Xg,h

Here, E is a resolution of the identity on the a-algebra

and cru ( <t Q) is the a-algebra generated by cylindrical sets with finite-dimensional

Borel supports (see, e.g., [BeKo n. The resolution of the identity E is concentrated on

the set {XE Xg.hIIX(p)I~IIUpll, pE Q}. If the function Q03p~IIUpll is

bounded, then conditions (1)-(4) are necessary for the existence of representation

(3.20); in this case, E is concentrated on the set X h and given on the a-algebra

'B(X h )·

This statement remains true for any representation of the hypercomplex system

L1 (Q, m) by unbounded operators. Such a representation is defined as a function

Q 3 P ~ Up whose values are normal operators (generally speaking, unbounded) in a

separable Hilbert space J{ that satisfy the following conditions:


(i) the operators Up (p E Q) commute in the sense of commutation of their

spectral projections;

(ii) there exists a lineal 'lJ c npEQ 'lJ( Up) dense in :J{ and invariant under the

action of the operators Up;

(iii) the space :J{ admits a decomposition

(iv)

(v)

(vi)

such that 'lJ n :J{k is invariant under the action of Up (p E Q) for all k,

and each :J{k contains a vector ~k E 'lJ n:J{k such that the closure of the

linear span {Up~k I p E Q} coincides with :J{ k (this condition means that

the representation Q 3 P H Up can be expanded into a sum of cyclic repre-

sentations);

U; = Up.;

Ue=l;

the function Q 3 P H ( Up~, TJ. ) J{ is continuous for all ~ E 'lJ and 11 E :J{;

(vii) for all A, B E ~o(Q),

( f c(A, B*, r)Urdr~, TJ.)J{ = (f Urdr~, f UsdsTJ. J ; A B J{

here, the operator

is defined on 'lJ as follows:

Furthermore, if there exists at most a countable subset Q 0 c Q such that


then condition (iii) is automatically satisfied (this simple sufficient condition was established by Ionescu Tulcea). For a detailed exposition of results related to decomposition (3.20), see the monograph [BeKo].

We now return to the general case of noncommutative hypercomplex systems. Let

Q 3 P H Up be a representation of the hypercomplex system L I (Q, m). Below, we

consider representations that satisfy conditions (1)-(4) and the following additional condition:

(5) The function Q 3 pHil Up II is bounded.

Such representations are called bounded.

Let L\(Q,m)3xH Ux be a representation of the Banach algebra L 1(Q,m) ina

separable Hilbert space Jr. Denote S't' = {~ I Ux~ = 0 for all x ELI (Q, m)}. Ob

viously, ~ is an invariant subspace. Below, we consider the case Kt = 0 (i.e., the rep

resentation Ux is nondegenerate).

Theorem 3.10. Let L \ (Q, m) be a normal hypercomplex system with basis unity

that satisfies the condition of separate continuity. Then there is a bijection between the

set of non degenerate *-representations of the Banach *-algebra L\ (Q, m) and the

set of bounded representations Q 3 P H Up of the hypercomplex system. This bijec

tion is given by the relation

Ux = f x(p)Updp (XE Lj(Q,m». (3.21)

Q

Proof. Since any *-representation of a Banach *-algebra can be decomposed into a direct sum of cyclic representations (see, e.g., [Nai 1]), it suffices to prove the theorem

for cyclic representations. Let L \ (Q, m) 3 X H Ux be a cyclic representation of the

Banach *-algebra L\ (Q, m) in a separable Hilbert space Jr with cyclic vector ~o E Jf.

Let Jf = {~E Jrl ~ = Ux~o for all x E Co (Q)} be a dense lineal in Jr. The generalized translation operators Lp (p E Q) act continuously in Loo(Q, m).

Furthermore, by virtue of (2.7), we have II Lp II ::; 1. Hence, for any p E Q, we can de

fine an operator L~: L\ -7 (Loo/ by restriction ofthe adjoint operator to the space L\.

It is clear that L~x = Lp' x ELI n Co (Q) and II Lp' x III ::; II x 11\ for all x E Co (Q).

In Jf, we introduce an operator Up by setting Up~ = ULp'x~O for ~ = Ux~o, X E Co (Q). The operator Up thus defined does not depend on the choice of x. Indeed,

if ~= Ux~o= Uy~O (X,YE Co(Q», then Ux_y~o=O and,hence, Uz~o=O for all z

from the closed left ideal I in L \ (Q, m) generated by the element x - y. Let us show

that Lp' (x - y) E I. Indeed, let en be the approximative unit of the hypercomplex sys-


tern L 1 (Q, m) constructed in Lemma 1.3, and let Z E I n Co (Q). Then Lp* en * Z E

I n Co (Q). Since generalized translation operators are associative, we have

= J ~*Lren(q)x(q*)dq = Lp* (e n* x)(r).

Since en*z~z in Ll for all ZE InCo(Q) and en*ZE Co(Q) for all n, we have

Lp* en * Z = Lp* ( e n * z) ~ L pZ in L 1. Hence, L pZ E I n Co ( Q) because I is closed.

By setting z=x-y, we obtain Lp*x-Lp*YElnCo(Q). Consequently, ULp'X~O=

ULp*Y~O = Up~.

The operator Up maps J{' into itself and is bounded in Ji. Indeed, assume that

~= Ux~OE J{' (XE Co(Q». Since en*x~x in L 1, we conclude that Uen~ tends

to ~ in Ji. Note that

x * Lpy = J (Lrx)(s)(Lpy)(s*)ds = J (Lrx)(s)(Ls/)(p*)ds

Therefore,

But

Passing to the limit as n ~ 00, we obtain II Up~ II J{::;; II ~ II J{, i.e., Up is continuous in

Ji and II Up II ::;; 1. Since Ji' is dense in Ji, we can extend Up by continuity to the

entire space Ji. Let us show that the operator family Up satisfies conditions (1)-(5). The validity of

(1) and (5) is obvious. Condition (2) can easily be verified by direct calculation. Let us show that condition (3) is satisfied. Since the generalized translation operators

L p in L 1 (Q, m) are uniformly bounded with respect to p, by using the Lebesgue the

orem and the condition of separate continuity, we conclude that II LPn X - Lpx III ~ 0 as

n~oo for each XE Co(Q). We set ~=Ux~o and ll=Uy~O (X,YE Co(Q». Then


This relation immediately yields (3) because II Up II ~ 1.

Finally, let us verify that condition (4) is satisfied. For all z E Ll (Q, m), we introduce the operators

where the integral on the right-hand side is understood in Bochner's sense. For all ~ = Ux~o and 11 = Uy~o E J-t we have

Thus, Vz = Uz. (z ELI (Q, m). Hence, for any A, BE '.Eo(Q), we obtain

J Updp J Uqdq = U U = U = J c(A, B, r) Urdr, "A "B "AHB

A B

i.e., condition (4) is satisfied. Note that it follows from the equality Vz = Uz. (z ELI)

that the representation Up of the hypercomplex system associated with the cyclic repre

sentation Ux of the algebra L 1 (Q, m) is also cyclic.

Conversely, let Q 3 P ~ Up be a bounded representation of the hypercomplex sys

tem L 1 (Q, m) in a separable Hilbert space J{. For each x ELI (Q, m), we define a

bounded operator Ux according to (3.21). Clearly, U; = Ux*. By virtue of (3.19), the

equality Ux Uy = Ux*y holds for step functions. Taking into account that L 1 (Q, m) is

a Banach algebra and passing to the limit, we establish that L 1 3 x ~ Ux is a *-repre

sentation of the Banach *-algebra L I (Q, m). Let us show that this representation is

nondegenerate. Let ~ E J{ be an arbitrary nonzero vector. We choose a neighborhood

o E '.Eo ( Q) of the basis unity so that m ( 0) ~ 1 and

for all p E Q. Then


I(UKO~,~h{- (~,~h{1 ~ J I(Up~-~,~)j{ldp + (l-m(O»II~II~ o

~ ~ m ( 0) II ~ II~ + (1 - m ( 0) ) II ~ II~ < II ~ II~

and, hence, U KO ~ "* o.

• Corollary 3.6. For any bounded representation U r of a normal hypercomplex

system with basis unity that satisfies the condition of separate continuity the following

relations hold:

(3.22)

Proof. Let Ux be the representation of the algebra L I (Q, m) associated with a

representation Up of the hypercomplex system according to Theorem 3.lO. Without

loss of generality, we can assume that Ux (or Up) is cyclic. Let ~o be a cyclic vector.

According to Theorem 3.lO, the representations Ux and Up satisfy the relation Up~ =

ULp.x~O' where ~= Ux~OE JI. Then ULp'Y~= U(Lp.Y)*x~O= ULp*(Y*X)~O= UpUY~ for all ~ E JI by virtue of (2.lO).

Clearly, it suffices to prove relations (3.22) for ~,11 E JI. Let SEQ be fixed. By

virtue of the equalities (Rs'x)*(p) = (Rs'x)(p*) = (Rpx*)(s) = (Lsx*)(p), for any

x E LI' we get

Hence, R s (Ur 1;, 11) j{ = (Ur Us 1;, 11) j{ for almost all r. By virtue of the condition of

separate continuity, the left-hand side of this equality is continuous in r; the right-hand side is continuous by definition. Thus, we have proved the first equality in (3.22). The second equality can be proved similarly.

•


Note that, in the definition of representations of hypercomplex systems, one can replace condition (4) by (3.22). Relations (3.22) can be used for the construction of the

representation of the measure algebra M b(Q) associated with a representation Q:3

P H Up of a hypercomplex system L) (Q, m) (for the definition of the convolution of

measures on the basis of hypercomplex systems, see Subsection 2.3). For this purpose,

one should set

tend this operation to the entire algebra M b( Q).

In what follows, we assume that hypercomplex systems satisfy the condition of sepa

rate continuity. A representation of a hypercomplex system L) (Q, m) is called irre

ducible if any closed subspace invariant under the action of Up (p E Q) coincides

with either {O} or Jl. It is easy to see that a bounded representation of a hypercom

plex system is irreducible if and only if the corresponding representation of the algebra

L) (Q, m) is irreducible. It is also clear that two representations of a hypercomplex

system L) (Q, m) are unitarily equivalent if and only if the corresponding representa

tions of the algebra L) (Q, m) are equivalent. We say that a system of representations

of a hypercomplex system L) (Q, m) is complete if, for any p * q (p, q E Q), it con

tains a representation Q:3 r H Ur such that Up * Uq ..

Theorem 3.11. The system of all bounded irreducible representations of any nor

mal hypercomplex system with basis unity that satisfies the condition of separate continuity is complete.

Proof. First, recall some necessary results concerning representations of Banach al

gebras. A Banach *-algebra R is called reduced if, for any x E R, x * 0, there exists

a positive functional <1> such that <1> (x * * x) * O. The collection of all irreducible *

representations of a reduced Banach *-algebra forms a complete system. Let us show

that the hypercomplex system L 1 (Q, m) is reduced. Assume that x * 0 and L x is the

operator of convolution with x in L2 (Q, m). Also assume that <1> (x * * x) = 0 for any

positive functional <1>. Then, in particular, II Lxfll2 =0 for all fE L 2(Q,m), whence

x * f = 0 (f E L 2)' Let f = en' where en is the approximative unit constructed in Lem

ma 1.3. Then x * e n-'t x in L l' whence x = 0, which contradicts the assumption.

Hence, the algebra L) (Q, m) is reduced.

Since the generalized translation operators Lr satisfy the conditions of finiteness (by

virtue of Theorem 2.1) and separate continuity, they map Co (Q) into itself and, hence,

are reduced on Co (Q). Therefore, for p * q, one can find x E Co (Q) C Ll (Q, m )

such that Lp* x * Lq* x. Since the Banach algebra Ll (Q, m) is reduced, there exists

Section 3 Elements of Hannonic Analysis for Normal Hypercomplex Systems 113

an irreducible representation L] (Q, m) 3 X 1---7 Ux such that UL ox-L ox"* 0, i.e., p q

ULox~O"*ULox~O for some vector ~OEJ{ Let Ux~o=~· Then UL.x~O=Up~' p q p

U Lq* X ~ 0 = U q ~, and the last inequality takes the form Up ~ "* U q ~.

• The statement below describes the relationship between bounded representations of

the hypercomplex system L] (Q, m) and positive definite functions on Q.

Theorem 3.12. Let L] (Q, m) be a nonnal hypercomplex system with basis unity

satisfying the condition of separate continuity. Then there is a bijection between the

collection of continuous bounded functions on Q positive definite in the sense of (3.1) and the set of classes of unitarily equivalent bounded cyclic representations of the

hypercomplex system L] (Q, m). This bijection is given by the relation

(3.23)

where <p E P(Q) and Uris the corresponding representation of the hypercomplex

system L] (Q, m) in a Hilbert space :J{ with cyclic vector ~o.

Proof. If Ur is a bounded representation of the hypercomplex system L] (Q, m)

with cyclic vector ~o' then the function <p(r) = (Ur~O' ~o) j{ is positive definite in the

sense of (3.1). Indeed, let x ELI (Q, m). Then

by virtue of Theorem 3.10. Assume that there exists another representation U; of the

hypercomplex system L] (Q, m) in the space :;( with cyclic vector ~o such that

<p(r) = (U; ~o' ~o) J/. Let U~ be the corresponding (obviously, cyclic) representation

of the algebra L 1 (Q, m). It is clear that (Ux ~o' ~o) j{= ( U~ ~o' ~o) J/. For any x, y E

L 1, we have

Since Ux~o (U~ ~o) are dense in :J{ (in :;(, respectively), this equality implies that

there exists an isomorphism V: :J{~:;( such that V(Ux~o) = U~ ~o (x E L]). Let

us show that V maps Ux into U~, i.e., V Ux = U~ V (x E L]). Indeed, for all y ELI'

we have


Since Uy ~o are dense in :Ji, we get V Ux = U~ V. On the other hand, for any x ELI'

whence ~o = V~o. It is obvious that the representations Up and U~ of the hypercom

plex system L! (Q, m) that correspond to U x and U~, respectively, are also unitarily equivalent.

For an arbitrary continuous positive definite function <p (r) (r E Q), we construct a

cyclic bounded representation Ux of the Banach algebra L! (Q, m) in a Hilbert space

:J{ with cyclic vector ~o by applying the GNS construction to a positive functional

(x) = f <p(r)x(r)dr (x ELI (Q, m» (3.24)

(this is possible because L 1 (Q, m) contains an approximative unit). Let Up be a cyclic

bounded representation of the hypercomplex system L! (Q, m) that corresponds to U x

(XE L!(Q,m») by virtue of Theorem 3.10. Then, for all XE L1(Q,m), we have

Hence, relation (3.23) holds almost everywhere. Since the functions (Ur ~o' ~oh{ and

<per) are continuous, relation (3.23) holds for all r E Q.

• Corollary 3.7. For a normal hypercomplex system with basis unity that satisfies the

condition of separate continuity, the concepts of positive definiteness in the sense of (3.1) and (3.1)' are equivalent.

It suffices to show that if <p (r) is positive definite in the sense of (3.1), then relation

(3.1)' holds for any AI'"'' An E ([ and rl"'" rn E Q. Indeed, by virtue of (3.23) and (3.22), we have

n

L A/A/Rr;*<p)(r) i,j=!

n

= L A/AjRr;*(Ur.~o'~Oh{ i,j=! '

= ;~/';X/U'j U,;So,f;O)g{ = II~A-;u"soL ~ o .

•


Recall that properties (i)-(v) formulated in Corollary 3.2 hold for continuous functions positive definite in the sense of (3.1)' and, hence, any function <per) E P(Q) also possesses these properties.

Remark. It is easy to see that the GNS construction can also be performed for a

positive definite function <per) that is not necessarily continuous. In this case, equality (3.23) holds almost everywhere. Consequently, an arbitrary bounded positive definite

function <P (r) coincides almost everywhere with some continuous positive definite function. Since the linear span of the set of bounded positive definite functions is weakly dense in Loo (Q, m), the linear span of the set P (Q) of continuous bounded

positive definite functions is weakly dense in Loo(Q, m).

It is clear that relation (3.24) establishes a one-to-one correspondence between the set

of positive functionals on LI (Q, m) and the set of bounded positive definite (and,

hence, continuous positive definite) functions on Q.

Functions obtained by generalized translation of any continuous positive definite

function can be represented as a linear combination of four continuous positive definite functions. Indeed, assume that <per) E P (Q), p, q E Q, and Ur is the representation

of a hypercomplex system in the Hilbert space J{ with cyclic vector ~o that corre

sponds to the function <P according to Theorem 3.12. We set Up ~o = ~ and U q* ~ 0 = 11. By virtue of (3.22),

• Let <PI' <P2 E P(Q). We say that <PI majorizes 1 ~ <1>2. A

positive definite function <per) is called elementary if that corresponds to an elementary positive definite function is indecomposable. It is known that the representation of a Banach *-algebra associated with a continuous positive functional is irreducible if and only if this functional is indecom

posable. This implies that the representation of the hypercomplex system LI (Q, m)

that corresponds to an elementary function <per) E P(Q) is irreducible.

Let <p(p)= (Up~O' ~o)j[, where p ~ Up is a cyclic representation of a normal

hypercomplex system with basis unity that satisfies the condition of separate continuity.

Thefunction <PI E P(Q) is majorized by I is a positive functional on L I (Q, m) corre

sponding to , (x) = (Ux B ~o, ~ o):H (x E L,), where U x is the represen

tation of L I (Q, m) corresponding to Up, and the operator B ~ 0, II B II:::; 1, commutes

with all Ux (and, hence, with Up). This implies that <PI (r) = (UrB~, ~o):H almost

everywhere. Since both sides of this equality are continuous, we arrive at the required result.

• If 0, thereexist A" ... ,AnE ([: and sl' ... ,snE Q such that

This inequality and (3.22) imply that

for all r E Q.

• Lemma 3.6. Suppose that A is a bounded set in L 00 (Q, m) and f E Co (Q). If

<P n E A weakly converges to <P E A, then f * <P n converges to f * <P in the topology

Section 3 Elements of Hannonic Analysis for Normal Hypercomplex Systems 117

of unifonn convergence on compact sets.

Proof Clearly, f * <Pn= (<p~, Lsf). If <P n E A weakly converges to <P E A, then

( <p~, y) uniformly converges to (<p * , y) on each compact subset in L] (Q, m). Let

s n ---7 SEQ. For any function f E Co (Q), according to the condition of separate con

tinuity, we have Ls/(r) ---7 Lsf(r) for all r E Q. Since the generalized translation

operators L p (p E Q) are uniformly bounded in L] (Q, m), by the Lebesgue theorem,

we have

--~) 0.

Hence, for a given function fE Co(Q), the mapping Q 3 S ~ Lsf E LI (Q, m) IS

strongly continuous and, therefore, for any compact set K c Q, the set {Lsfl SE K}

is strongly compact in L I (Q, m).

• Lemma 3.7. Let PI (Q) be the set of continuous positive definite functions cp (r)

on Q such that IIcplloo=cp(e)=1. The weak topology cr(Loo(Q,m),L](Q,m»)

coincides on the set p] (Q) with the topology of uniform convergence on compact sets.

Proof Since II cp II 00 = 1 for any cp E PI (Q), it is clear that the topology of com

pact convergence on PI (Q) is stronger than the weak topology.

Now assume that <Po E PI (Q), K is a compact set in Q, and E> 0. Let us show

that if <P E PI (Q) lies in a certain appropriate weak neighborhood of CPo, then

I cp(r) - <po(r) I ~ E + 4-JE

for all r E K. Denote by 0 E 1Jo(Q) a neighborhood of the point e E Q such that

1<p(r)-ll<E forall rE Q. One can find a function fE Co(Q) such that f(r)?O,

supp f c 0, and II fill = 1. Consider a weak neighborhood U of the point <p 0 in

PI (Q) such that, for <p E U,

If f(r)(cp(r)-<po(r»)drl ~ E.

If <p E U, then

If f(r)(l-<p(r»)drl ~ If f(r)(l-<po(r»)drl

+ Iff(r)(cp(r)-<po(r))drl ~ Ilfl!tE+E = 2E.


On the other hand, for <p E V and r E Q, we have

If * <per) - <p I = If (LJ)(s)<p(s*)ds - <per) I

= If f(s)(Lr*<p*)(s)ds - f f(s)<p(r)ds I ~ f f(s) I Ls*<p(r) - <per) Ids. (3.25)

By applying the inequality

I K(s, r) - K(t, r) 12 ~ K(r, r)[K(t, t)+ K(s, s) - 2ReK(s, t)]

to the positive definite kernel K(s, r) = Ls* <per) and setting t = e, we get

I ( Ls* <p) (r) - <p (r) 12 ~ (Lr* <p) (r) [ <p (e) + ( Ls* <p) (s ) - 2 Re ( Ls* <p) ( e)]

~ 2<p(e)(<p(e)-2 (Re<p)(s*») = 2(1-2Re<p(s».

Here, we have used the factthat I (Lr* <p) (r) I ~ II <p ILXl = <p(e) and <pes *) = <pes). Consequently, we can extend (3.25) as follows:

By Lemma 3.6, one can find a weak neighborhood V' of the point <Po E PI (Q) such

that l(f* <po)(r)-(f* <p)(r) I ~ I:: foraB <pE V' and rE K. Hence,

for <p E V n V' and r E K.

• Theorem 3.13. Let L I (Q, m) be a normal hypercomplex system with basis unity

that satisfies the condition of separate continuity. Any continuous function on Q can be uniformly approximated on every compact set by linear combinations of elementary positive definite functions.


Proof. Assume that the theorem is proved for arbitrary functions from P(Q). Since

any function of the form a * b * (a, b E Co (Q) is a linear combination of four positive

definite functions (a ± b) * (a ± b) *, (a ± i b ) * (a ± i b )*, the statement of the theorem is true for all functions of the form

<per) = L Ak(ak * bk)(r) (alc> b k E Co(Q)· k

According to Lemma 3.2, we can uniformly approximate any function f E Co (Q) by functions of this form. This proves the theorem because any continuous function can be

uniformly approximated by functions from Co (Q) on every compact set.

It remains to show that any function <per) E P(Q) can be uniformly approximated on every compact set by linear combinations of elementary positive definite functions.

Without loss of generality, we can assume that <p (e) = II <p II 00 = 1. As is known, every continuous positive functional defined on a Banach *-algebra with approximative unit and having the norm less than 1 is a weak limit of convex combinations of indecom

posable functionals and the zero functional. Therefore, the positive functional as

sociated with i of indecompos

able positive functionals <l>i' IIdl= 1, and the zero functional. Since limll\fll = II II = 1 and II \f II ~ 1, we can assume that II \f II = 1 (this can always be realized by multiplying \f by proper scalars). If <Pi are elementary positive definite functions cor-

responding to <l>i' then <PiCe) = II <l>dl = 1. In this case, \f corresponds to the positive definite function

Consequently, the functions \jf (r) weakly converge in PI (Q) to <P (r). To complete the proof it remains to use Lemma 3.7.

• Corollary 3.8. Assume that L I (Q, m) is a commutative normal hypercomplex

system with basis unity that satisfies the condition of separate continuity. Then any continuous function can be uniformly approximated on compact sets from Q by linear combinations of Hermitian characters.

Proof. It suffices to show that every elementary normalized positive definite func

tion (x) = J <p(r)x(r)dr (XE Lj(Q,m)


corresponds to the function <po It is well known that a positive functional is indecomposable in the commutative Banach *-algebra R if and only if (x) = x(M)

(x E R), where M is some symmetric maximum ideal. This and Theorem 1.2 imply that

(x) = f x(r)x (r) dr.

Hence, <p (r) = X (r) almost everywhere. Taking into account that both functions in this equality are continuous, we arrive at the required statement.

• Let us introduce the concept of Fourier transformation for noncommutative normal

hypercomplex systems with basis unity satisfying the condition of separate continuity. Denote by ffi the set of equivalence classes of bounded irreducible representations of a hypercomplex system. The operator-valued function ffi 3 U H A(U) = U~ is called

the Fourier transform of a measure j..t E M b( Q). If a function f belongs to L 1 (Q, m),

then its Fourier transform is defined as the operator function leu) = Uf (U E ffi). By Cb(ffi) we denote the *-algebra of operator functions on ffi with ordinary multipli

cation and the norm

II flloo = sup II feU) 11q>~ VEffi

(here, II A II is the greatest eigenvalue of the operator I A I). Let Q be a compact q>~

set. The Fourier transformation M b(Q) 3 j..t H A E Cb(ffi) is a continuous injective

*-homomorphism of M b(Q) into C b(ffi). Indeed, it is obvious that II A L ~ II j..tll and the Fourier transformation is a *-homomorphism. Assume that A(U) = U~ = 0 for

all U E ffi. Then

In view of Theorems 3.12 and 3.13, this relation implies that

f f(r)dj..t(p) = 0

for any f E C(Q). Thus, j..t = o.

•

Section 4 Hypercomplex Subsystems and Homomorphisms 121

4. Hypercomplex Subsystems and Homomorphisms

The notion of normal hypercomplex system with basis unity is a generalization of the notion of group algebra of a locally compact group. As shown above, many facts of harmonic analysis for locally compact groups are true for a hypercomplex system with locally compact basis. It is natural to expect that some other facts from the group theory can be generalized to the case of hypercomplex systems. In this section, we introduce and study an analog of subgroups of a group-the so-called hypercomplex subsystems. Since a subgroup of a locally compact group, generally speaking, cannot be described in

terms of subalgebras of a group algebra (lR 1 = { 0 } x lR 1 in lR 2 is an example of a subgroup of this sort), the definition of hypercomplex subsystem is given in terms of generalized translation operators associated with the hypercomplex system but not in terms of the structure measure. We also present a brief description of the homomorphisms of hypercomplex systems. At the end of the section, we introduce the notions of the direct and semi direct products and the join of compact and discrete hypercomplex systems (hypergroups). The latter notion is of pure hypergroup nature, since, generally speaking, the join of two groups is a hypercomplex system but not a group.

4.1. Definition of Hypercomplex Subsystems. Let us now introduce an analog of the notion of subgroups for the case of hypercomplex systems Let L1 (Q, m) be a normal (not necessarily commutative) hypercomplex system with basis unity that satisfies the condition of separate continuity. A closed nonempty set H c Q is called a basis of a hypercomplex subsystem if (a) e E H, (b) H* = H, (c) the equality (Lsf)(t) = 0

holds for all t, s E H and any function f E C (Q) such that f( r) = 0 (r E H).

We equip H with the induced topology of the space Q: The sets of the form 0' = o n H, where 0 is a neighborhood in Q, are regarded as neighborhoods in H. According to condition (c), the generalized translation operators Ls (s E H) can be cor-

rectly restricted to C(H). Namely, we consider an extension 1(r) E C(Q) of an arbit

rary function fE C(H) and set (Lsf)(t)= (L s1)(t). By virtue of (c), this definition

does not depend on the choice ofthe function 1. Clearly, if H is a basis of the hypercomplex subsystem, then there exists a family of involution-consistent generalized trans

lation operators acting in Cb(H) preserving positivity and the identity, satisfying the finiteness condition (F), and the condition of separate continuity. The generalized trans

lation operators Ls (s E H) also satisfy inequality (2.7), i.e., the operators L s (s E H)

are uniformly bounded in Cb(H).

In what follows, for simplicity, a basis H of a hypercomplex subsystem is called simply a hypercomplex subsystem and we keep in mind that the family of generalized


translation operators Ls (s E H) is given on H. If the generalized translation operators

Ls (s E H) are commutative, then we say that the hypercomplex subsystem is commu

tative.

If the hypercomplex subsystem H is commutative and compact in the induced topology, then there exists a finite regular positive Borel measure f..l on H invariant under the action of the generalized translation operators L p (p E H):

J (Lpf)(q)df..l(q) = J f(q)df..l(q) (p E H; fE Cb(H)). H H

In fact, we apply the M. Krein-Rutman theorem [KrR]: If the collection r = {A} of commuting linear operators which transform a solid cone K into itself has a common

fixed vector v > 0, i.e., A v = v (A E 1), then their adjoint operators A * have a com

mon fixed vector '1'"* 0 in K*, i.e., A *", = '" ('I' E K*). Since the generalized translation operators L p (p E H) preserve positivity, the commutative operator family

{L p} p E H maps the solid cone ct (H) of nonnegative functions from C b( H) into it

self. The operators Lp have a common fixed vector 1 (r) == 1 (r E H) inside ct (H).

By the Krein-Rutman theorem, their adjoint operators have a common fixed vector f..l in

CCtCH))', i.e., L~f..l=f..l (pE H). By the Riesz-Markov theorem, (Ct(H))' is iso

morphic to the set of finite regular positive Borel measures on H. Thus, we have f..l E

Mt(Q) and

J (Lpf)(q)df..l(q) = J f(q)df..l(q) (f3 Cb(H),p E H). H H

• Here, we do not prove the existence of weakly invariant measures on noncommuta

tive locally compact hypercomplex subsystems H (this problem contains the unsolved problem of existence of the invariant measure on a hypergroup) and postulate the exis

tence of a measure f..l of this sort on H. The following argument can be regarded as a justification of this assumption: The

convolution of probability measures on H can be defined as follows: Let < 0 p * 0 q> f) = (Lpf)(q) (fE Cb(H), p, q E H). Since the generalized translation operators Lp (p E

H) preserve the identity, op * Oq is a probability measure. By virtue of the condition of

separate continuity, the operation of convolution can be extended to arbitrary probability measures by the relation

It is easy to show that this convolution in M 1 (H) satisfies axioms (G 1)-(G4) in the de-


finition of hypergroups, i.e., H becomes a weak hypergroup. If H is a hypergroup with respect to this operation of convolution (in this case, H is called subhypergroup), then the existence of weak invariant measure in compact and commutative locally compact cases follows from the results of Jewett [Jew] and Spector [Spe4]. If the hypercomplex subsystem is discrete, then as shown in Section 2, it is a hypergroup and, by virtue of [Jew], it possesses the invariant measure.

Throughout Section 4.1, we assume that the support of the invariant measure ~ co

incides with the entire H. If H is a subhypergroup, then supp ~ = H. Indeed, it is

clear that, for any measure 11 E Mt (H) with compact support, there exists convolution

11 * ~ which is a positive regular Borel measure and supp (11 * ~) = supp 11 * supp ~

Crecall that the convolution of sets A * B CA, B E ~CH)) is defined as

u supp COp * Oq), pEA:qEB

see Subsection 2.3). The property of invariance can be reformulated in terms of the con

volution of measures. Indeed, Op * ~ = ~ (p E H). Thus, it suffices to show that e E

supp~. Let p E supp~. Since

supp (Op' *~) = U supp 0p" * Oq, qEsuPPIl

we see that supp COp" * Op) C supp~. By using axiom (G5) from the definition of

hypergroups, we obtain e E supp (op" * Op)'

• Let us now describe hypercomplex subsystems in terms of the convolution of sets.

A closed non empty subset H c Q is a hypercomplex subsystem if and only if (a')

eE H, (b/ ) H*=H, and (c') H*H cH.

Indeed, assume that H have properties (a/)-(c') and let f E C(Q) be such that f(r)=O (rEH). Then supp(op*Oq)cH for any p,qEH by virtue of (c'),

whence (Lpf)(q)=(Op*Oq,f)=O and H is a hypercomplex subsystem Conversely,

assume that H is a hypercomp1ex subsystem and fE CCQ) is a function such that

f(r)=O (rE H). Then (Op* Oq,f) = (Lpf)Cq) = 0 foraH p,qE H. Since the func

tion f is arbitrary, this equality means that supp (Op * Oq) c H.

• Let us show that the invariant measure is unique to within a constant.

Theorem 4.l. Assume that H is a hypercomplex subsystem and the generalized


translation operators Ls (s E H) satisfy condition (C) in Lemma 2.2 (i.e., for any

neighborhood W of the point SEQ, there exist a neighborhood U of the basis unity such that supp(Ls'<p)(·) C W for any <p E Co(Q) with supp<p c U). If

J..ll and J..l2 are two weakly invariant measures on H, then one can find a positive

real number C such that J..l2 = CJ..ll'

Proof Denote by 1I·1I2,1l, and (',' h,Il' the norm and the scalar product in L2(H,

J..li) (i = 1,2), respectively. Note that I (LpJ. gh,Il,1 ::; IIflb,IlJI g 112,11; by (2.7). This

implies that the operators Lp (p E H) form a family of bounded generalized translation

operators in L 2 (H, J..l i) ( i = 1, 2). Clearly, the generalized translation operators L p

(p E H) satisfy the conditions of Theorem 2.2. Therefore, L~ = L p' in each of the

spaces L2(H, J..lJ (i = 1,2), i.e., J..li are strongly invariant in the sense of Subsection 2.1.

We take E > 0 and functions f, g E ct (H) such that f* 0 and g * O. By virtue of the condition of separate continuity,

lim «Lpf)(q) - (Rpf)(q» = 0 p-+e

for each q E H. In view of (2.7) and the Lebesgue theorem, this yields

Thus, there is an open neighborhood U E ~o(H) of the basis unity e such that

f I (Lpf)(q) - (Rpf)(q) I dJ..l2 (q) < ~ f f(q)dJ..ll (q) H H

f I(Lpg)(q) - (Rpg )(q)ldJ..l2(q) < ~ f g(q)dJ..ldq) H H

for all p E U. We now take hE ct (H) such that

J h(q)dJ..ll (q) > 0, H

h * = h, and h = 0 outside U. Then

J h(q)dJ..l2(q) J f(q)dJ..ll (q) = f f(P) f (Lp* h)(q)dJ..l2(q)dJ..ll (p) H H H H

Section 4 Hypercomplex Subsystems and Homomorphisms

= f f j(P)(Lq.h*)(p) dill (p)dIl 2 (q) HH

= f f (Lqj)(p) h*(p) dll l (p)dIl2 (q) HH

= J h(p)(J (RpJ)(Q)dIl2(q)) dill (p). H H

Since h = 0 outside U, we have

$ i J h(p)dll l (p) J j(q)dll l (q).

This yields

f j(q)dIl2 (q)

f j(q)dll l (q)

f h(q)dIl2(q)

f h(q)dlll (q)

The same argument is applicable to the function g. Hence,

f j(p)dIl2(q)

f j(q)dll l (q)

whence 112 = CIlI for some C ~ o.

f g(p)dIl2(P)

f g(q)dll l (q)

2

$ E,

125

• We fix a weakly invariant measure Il and, in the remaining part of this subsection,

assume that the generalized translation operators L p (p E H) satisfy the conditions of

Theorem 4.1. By virtue of Theorems 2.1 and 2.2, L I (H, Il) becomes a normal hyper

complex systems with basis unity with respect to the convolution

We define the left-invariant measure Il * Dp+ E M+(H) as follows:


(f..L * ?:Jp',f) = J (Rp*f)(q)df..L(q) (fE Co(H».

By virtue of Theorem 4.1, there exists a positive number ~ (P) such that f..L * ?:J p' = ~(p )f..L. The function Q:3 P ~ ~(p) is called modular. One can show that the modular function is a continuous generalized character of the hypercomplex system L 1 (H,

f..L). In addition, ~~ * = 1 and f..L = ~f..L *. Note that all assertions in this section hold for H = Q. Wherever possible, for the sake of simplicity, we assume that the measure f..L is unimodular, i.e., ~(p) == 1 (p E H). The corresponding hypercomplex subsystem is also called unimodular.

4.2. Fundamental Properties of Hypercomplex Subsystems. Let us establish some

properties of hypercomplex subsystems.

Assume that the separately continuous generalized translation operators L p (p E

Q) associated with the hypercomplex system L1 (Q, m) are such that the corres

ponding convolution of measures satisfies condition (G6) from the definition of hypergroups. Let H be the connected component of the basis unity e of the hypercomplex system L 1 (Q, m). Then H is a hypercomplex subsystem.

In fact, since H* is connected, we have H* = H. It suffices to show that H * H is

connected. Let H * H c V U W, where U and W are open subsets of Q such that

(H * H) n V and (H * H) n W have no common points. Assume that {e} * {e} =

{ e} c (H * H) n V. The sets P = { (t, s) I { t} * {s} E ~ 0 ( V)} and S = { (t, s) I {t} * {s} E ~o (W)} are open in Q x Q by virtue of (G6). Moreover, PUS con

tains H x H and the sets (H x H) n P and (H x H) n S have no common points.

Consequently, HxH cP, since (e,e)E P and HxH is connected.

• In what follows, we denote the set {p} * {q} (p, q E Q) simply by p * q. Note

that the operation of convolution of sets is associative: A * (B * C) = (A * B) * C (A, B,

CE ~o(Q». Denote G={rE Qlr*r*=r**r=e}. Clearly, eE G and G*=G.

The set G is a group with respect to the correctly defined operation of multiplica

tion r s = r * s (r, s E G). If, in addition, the continuity axiom (G6) from the defini

tion of hypergroups holds, then the group G is locally compact. Indeed,let rE G and pE Q. We take qE r*p. Then r**qcr**(r*p) =

e * p = p, i.e., r* * q = p. Multiplying this equality from the left by r, we obtain q =

r * p. Hence, multiplication is correctly defined in G. Assume that r, s E G. Then

rs(rs)*=r*s*s**r*=r*r*=e, i.e., G*G cG. The remaining assertions areobvious.

•


The group G is called a maximal subgroup (clearly, the case G = { e} is not excluded). We have also proved that, for each rEG, the maps Q 3 P H r * p E Q and Q 3 q H r* * q are inverse to each other.

Theorem 4.2. Assume that L1 (Q, m) is a normal hypercomplex system with

compact base and basis unity satisfying the condition of separate continuity. Let

Q 3 P H Up be a bounded representation of the hypercomplex system Denote by T

a weakly closed subgroup of unitary operators in the Hilbert space H of representa

tion.· Then S = {r E Q I Up E T} is a hypercomplex subsystem.

Proof. The closeness of S follows from the weak continuity of the representation Q 3 P H Up and from the coincidence of the weak and strong operator topologies.

Clearly, e E Sand S*= S. Let us show that S * S c S. By applying (3.22) and the definition of the convolution of measures, for any r, s E S and ~ E H, we obtain

where T1 = UrUs~. Since I (Up~, UrUs~)H I :s: (~, ~)H and the mapping supp Or * Os 3

P H (Up~, UrUs~)H is continuous, we have (~, ~)H =(Up~, Uq~)H (p, q E r * s; ~ E H). In particular, for p = q, the last equality implies that that the operator Up is

unitary for all pEr * s, whence it follows that S * S c S.

• Corollary 4.1. Assume that G' = n {r E Q I Up is unitary} (the intersection is

taken over the set 91 of equivalence classes of irreducible representations). Then G' is a maximal subgroup of the hypercomplex system L 1 (Q, m).

Proof. Let G be a maximal subgroup. For rEG, we have

(the integral is understood in Bochner's sense). Hence, G c G'. On the other hand, by virtue of Theorem 4.2, G' is a hypercomplex subsystem of L 1 (Q, m) and, for any rEG' and ~,T1 E H, we have

whence Uor*or' =Ue = I for all U E 91. By using the injectivity of the Fourier trans-


formation, we conclude that or * OrO = Oe, i.e., rEG.

• If Q is a hypergroup and H is its subhypergroup, we can introduce the quotient

space Q I H. Indeed, by definition, rH = {r} * H is a left coset in H. For any r, s E

Q, the sets rH and sH either do not intersect or coincide. Indeed, let p E rH n sH. Obviously, pH c rHo Since p n rH"* 0, by virtue of (G5), we conclude that e E

p * (r * H)*= p * (H* * r*) = (p * H) * r*, whence, by using (G5) once again, r E pH.

Therefore, rH = pH. The fact that pH = sH can be proved by using the same argument.

• As shown in [Jew], the quotient space Q I H is a locally compact Hausdorff space

and the map Q 3 rH E r ~ Q I H is continuous and open. Moreover, the same work

contains the description of the spaces Q I I H of dual cosets H rH = H * { r} * H of the hypergroup Q with respect to the compact subhypergroup H; it is also shown therein

that Q I I H is a hypergroup with respect to a naturally defined operation. We do not

present here these results but, in Section 2 in Chapter 2, we consider the set Gil H, where G is a locally compact group and H is its compact subgroup.

A hypercomplex subsystem H is called normal if pH = Hp for any p E Q, or supernormal if pHp * c H (p E Q). If Q is a hypergroup and H is its normal subhypergroup, then Q I H = Q II H is called a quotient hypergroup. In contrast to groups, the notions of normality and supernormality are not equivalent for hypergroups. If H is a supernormal subhypergroup, then H is normal. Indeed, let p * H * P * c H

(p E Q). Then p * H * P * * P c H * p. On the other hand, by virtue of (G5), we have p*Hcp*H*p**pcH*p, i.e., pHcHp. By analogy, we get HpcpH.

Generally speaking, the converse statement is not true. For example, the normal hypercomplex subsystem H = { e} is not necessarily supernormal because, otherwise, p * p * = p * e * p * = e for all p E Q, which implies that Q coincides with the maximal subgroup. In [Vre5], it was shown that if H is a supernormal subhypergroup of the hypergroup Q, then Q I H is a group with respect to the induced convolution; furthermore, the set p * p * is a group for all p E Q.

4.3. Homomorphisms. Let us define a homomorphism of a hypercomplex system

Assume that Ll (Ql' m 1) and L 2 (Q2' m 2) are two normal hypercomplex systems with basis unity satisfying the condition of separate continuity. A continuous mapping

<p: Ql ~ Q2 is called a homomorphism of Ll (Q2' m2) into Ll (Ql' ml) if

1) Lp(fo<p)(q) = Lcp(p)f(<p(q» for any p,qE Q 1 andfE C(Q2);

2) <p(p*) = (<p(P» * (pE QI)'


If, in addition, the map <p is a homeomorphism of QI onto Q2' then the hypercomplex systems Ll(QI' ml) and L 1(Q2' m2) are called isomorphic. Inwhatfollows, we prove that the measure ml coincides with m2 to within a constant.

It is more convenient to consider homomorphisms <p with certain topological properties. We say that the complements of compact sets in the locally compact space Q are

neighborhoods of the infinite point in Q. We say that a continuous map <p: Ql ~ Q2

is continuous at infinity if, for any neighborhood Uoo of the infinite point in Q2' there

is a neighborhood V 00 of the infinite point in Q1 such that <p (V 00) c U 00. Clearly, if the

continuous map <p: Q 1 ~ Q2 is continuous at infinity, then the preimage of any com

pact set in Q2 is compact in Q 1. In what follows, the map <p which realizes the homo

morphism of LI (Q2' m2) into LI (QI' m I) is assumed to be continuous at infinity. Consider a linear operator C" (Q2) 3 f f-7 Ff = f 0 <p E Coo (QI). It is obvious that

IIFII~ 1 and F(CO(Q2)) c Co(Qd. If <p maps Ql onto Q2' then KerF={O}. Let F* be the operator adjoint to F. Clearly, F* transforms M b(Ql) into M b(Q2)' with

II F*II ~ 1 and F*(on) = o<p(P). One can show that F* can be extended to the linear

operator acting from M (Qd into M (Q2). In fact, for any J! E M(QJ), we can set

(F*J!,J) = (J!, Ff) (fE Co (Q2»· In order to check that F*J! is a Radon measure, we

must show that the inequality I (F*J!,J) I ~ CK llflloo,Q2 (fE CoCK»~ holds for any

compact set K c Q2. Indeed, we have

for any f E Co (K) because <p -1 (K) is a compact set (since <p is continuous at infi

nity)' J! is a Radon measure on Ql' and II F II ~ 1.

• The equality

supp (F*J!) = <p (supp J!) (4.1)

Indeed, assume that there is p E supp J! such that <p (p ) ~ supp (F*J!). There exists a

nonnegative function f E COO (Q2) equal to zero on supp F*J! and such that f( <p(P» = 1. We have

0= (F*J!,J) = f f( <p(P» dJ!(p).

It is clear that the last integral is strictly positive. Hence, <p (supp J!) c supp F *J!. The

inverse inclusion supp (F*J!) C <p(supp J!) is also obvious.

•


Lemma 4.1. Assume that the homomorphism <p maps QJ onto Q2 and the hypercomplex system L J (Q2' m2) satisfies the conditions of Theorem 4.1, i.e., for any

neighborhood W of the point s E Q2, one can indicate a neighborhood U of the

basis unity such that supp Ls* g ( . ) c W for each g E Co (Q2) with supp g cU.

Then there exists a constant C> 0 such that F*mj = Cm2.

Proof. For any J, g E Co (Q2)' we have

(F*ml, (LqJ(p)f)· g) = J (LqJ(p)f)(<p(q»g(<p(q» dml (q)

= J f(<p(q»(LqJ(p*)g)(<p(q» dmJ (q)

Hence, the measure F*ml is strongly invariant. By Theorem 2.1, the generalized trans

lation operators Lq (q E Q2) satisfy the finiteness condition (F). Therefore, the measure F* mJ is invariant, i.e., equality (2.3) holds. It remains to apply Theorem 4.1.

• Without loss of generality, we may assume that C = 1 (otherwise, we must consider

LJ (Q2, Cm2) instead of LJ (Q2, m2»)·

Theorem 4.3. Assume that the homomorphism <p maps Qj onto Q2. The oper

ator F can be extended to an isometric *-homomorphism from the Banach algebra

L j (Q2, m2) to Lj(Qj, mj), where m2=F*mj.

Proof Since, by Lemma 4.1, the measure m2 is invariant, we have

(F(f*g»(p) = J (LqJ(p)f)(s)g(s*)dm2(s)

Q2

= (F*mj, (LqJ(p)f)· g*) = J (LqJ(p)f)(<p(q»g(<p(q*»dmj (q)

Q\

= J (Lp(fo <p)(q»(g 0 <p)(q*)dmJ (q) = (Ff* Fg)(p)


for any f, g E CO(Q2) and p E Q1' It is obvious that II FfllLt(Q l>mt) = IIfIlL t(Q2,m:z)'

Consequently, the operator F can be extended by continuity to a homomorphism of

L 1(Q2' m2) into LI(Q1' m1)' Finally, by virtue of the equality

F(fj(p) = f*(<p(P» = f(<p(p*)) = (Ff)*(p),

the operator F is involutory.

• If the homomorphism <p maps Q 1 onto Q 2' then <p (e I) = e 2, where e j is the

basis unity of the hypercomplex system L I (Qj, m i), i = 1, 2. Indeed,

As q runs over QI' <p(q) runs over the entire Q2' Hence, Lcph)=I and <p(el) = e2

by virtue of the fact that the generalized translation operators Lp (p E Q2) are reduced.

• Suppose that <P(QI) = Q2' Let H 1 C QI be a compact hypercomplex subsystem of LI (QI' m I)' Then

<p(H)) = H2 is a compact hypercomplex subsystem of LI (Q2' m2)' * Indeed, since e1 E HI, we have <peel) = e2 E H2. Moreover, it is clear that H2 =

H2. LetfE C(Q2) andf(r)=O for all rE H2. Then

since HI is a hypercomplex subsystem and f 0 <pep) = 0 for all p E HI. Consequent

ly, H2 is a hypercomplex subsystem.

• Note that this assertion is not proved for general hypercomplex subsystems because

<p(H)) is, generally speaking, not closed. Therefore, in what follows, we suppose that

the mapping <p is closed.

As usual, the kernel Ker <p of the homomorphism <p is defined as the set of points

p E QI such that <pep) = e2.

Theorem 4.4. Assume that separately continuous generalized translation operators

Lp (p E Q1) and Lq (q E Q2) associated with the hypercomplex systems LI (QI'

m1) and L I (Q2' m2)' respectively, are such that the corresponding convolution of


measures satisfies axiom (G5) from the definition of hypergroups. Then the kernel of the homomorphism <p is a normal hypercomplex subsystem of LI (QI' ml)'

Proof. We denote H = Ker <p and show that H is a hypercomplex subsystem.

Clearly, H is closed and el E H. Since <pep *) = (<p(p»*= e; = e2 for each p E H,

we have H * = H. Let p, q E H. Since

for each fE Co(Q2), we have F*(Bp* Bq) = B<p(p)* * B<p(q) = Be2 * Be2 = Be2. In view

of (4.1), this implies that <pep * q) = e2 and H * He H.

Let us show that the hypercomplex subsystem H is normal. To do this it suffices to show that

rH = <p-I(<p(r» = Hr (4.2)

for each r E Q. Assume thatthere is a point s E <p-l (<p(r» such that s e rHo By vir

tue of what has been proved in the previous subsection, s H n rH = 0. In view of the

fact that <p(s*) = (<p(r»* and axiom (G5), we have e 2 E <p(s*) * <per) = <p(s** r).

This implies that s* * r n H"* 0. By virtue of (G5) and the associativity of the convolution of sets established in the previous section, we obtain e I E (s* * r) * H * = (s >I< * r) * H = s* * (r * H). In view of (G5), this yields s n rH = 0. The first equality in (4.2) is proved. The second equality is proved similarly.

• Let separately continuous generalized translation operators L p (p E Q) associ

ated with the hypercomplex system L 1 (Q, m) be such that the corresponding convo

lution of sets satisfies axiom (G6) from the definition of hypergroups. Then. for each r

from the maximal subgroup G, the mapping Q:3 P ~ <p r(P) = r * p * r* E Q is an

automorphism of the hypercomplex system Ll (Q, m).

In fact, in the previous subsection, it was shown that, for each rEG, the mappings

Q:3 P ~ rp E Q and Q:3 P ~ r*p E Q continuous by virtue of (G6) are correctly defined and, moreover, each of these mappings is inverse to the other. Hence, the map

<Pr is a homeomorphism of Q onto Q. Further, (<Pr(P»* = ( r * p * r*)* = <Pr(P*) for

each p E Q. One can easily check that F,.* J.t = B r * J.t * B r*, whence


4.4. Direct and Semidirect Products of Hypercomplex Systems.

loin of Hypercomplex Systems

133

•

1. Direct product. Let Lp. (i = 1, 2) be generalized translation operators associ-I

ated with normal hypercomplex systems with basis unity LI (QI, m I) and L 2 (Q2'

m2)' respectively. We denote HI =L 2(QI' ml) and H 2 =L2(Q2, m2). Thedirect

product of generalized translation operators L PI and L P2 (p 1 E Q, P 2 E Q) is defined as the operator-valued function

It is easy to see that the operators Lp (pE Q) formin L2(Q, m) (m=ml ®m2) a family of generalized translation operators satisfying the conditions of Theorem 2.1. The

hypercomplex system L\ (Q, m) constructed from the generalized translation operators

Lp is called the direct product of the hypercomplex systems L \ (Q\, ml) and LI (Q2'

m2). To denote the operation of taking the direct product, we write LI (Q, m) =

L1(Q!, m})®L\(Q2, m2). For example, if Lp. (PiE [0,00» are generalized I

translation operators associated with the hypercomplex system in Example 2 of Subsec-tion 1.1, then, clearly,

(Pi,qiE[O,oo), i=I,2,jE C([O,oo)x[O,oo».

2. Semidirect product. One can also introduce the concept of semidirect product

oj a hypercomplex system and a group. First, we recall the concept of sernidirect product of groups. Let G be a locally compact group and let Aut G be the group of all


automorphisms of the group G equipped with standard topology; the base of neighborhoods of the identity in this topology is fonned by the sets

~(U,F) = {PE AutGIP(g)E Ug,p-l(g)E Ug forall gE F},

where U is a neighborhood of the identity in G, and F is a compact subset of G (see,

e.g., [HeR]). Let GA be a subgroup of Aut G. The semidirect product G >4 GA of

groups G and G A is the set of ordered pairs (g, P) (g E G, PEG A) with group

multiplication (g, P )(g', P') = (gP(g'), P P'). The element (e, id) is the identity in

G®GA, and the pair (g,p)-1=(P- 1(g_1),P- 1) is the inverse of the element (g,

P). The topology in G >4 G A is defined as the product of topologies of the spaces G

and GA. The most common situation is as follows: Let G be a normal subgroup of a

group L and let a subgroup H of L be such that GH = Land G n H = {e}. Then

L = G >4 H because

where p h j is the automorphism of the group G generated by the internal automorphism

x f---) p h j (x 1) of L.

Let L 1 (Q, m) be a nonnal hypercomplex system with basis unity that satisfies the

condition of separate continuity and let G be some topological subgroup of the group of

automorphisms of L 1 (Q, m). Let us define generalized translation operators corres

ponding to the semidirect product of L 1 (Q, m) and G . Assume that the mapping

Q x G 3 (p, cp) f---) cp (p ) E Q is continuous. For each fEe (Q x G), we set

Let us show that the operators R(p,~) (p E Q; cp E G) form afamily of generalized

translation operators. Indeed, let p, p' ,p" E Q and cp, cp', cp" E G. Then

(R(P',IP') (R f»)C ) - R(p',~') [RP f( ')] CP", ~") (p', ~') p, cp - Cp", ~") ~(p') p, cpcp

Denote h(p')= (R;J)(p, cpcp'cp"). Then

Section 4 Hypercomplex Subsystems and Homomorphisms l35

- (RCP(P') (RP I)) ( (1'\''') - (RP (RP I)) ( (1'\'(1'\" ) - cp<p'(p") CP(p') p, 't' cp cp - cp(v) <PcP'(p") p, 't' cp 't' •

On the other hand,

( R(P,<P) (R(p,<P) I)) ( ) - R(P, cp) [RP I( ')] ~1p',CP') ~1p",CP") p, cp - (cp',p') <p(p") p, cpcp

The validity of other axioms is obvious.

• One can easily construct a hypercomplex system corresponding to generalized trans

lation operators R(p, CP) (p E Q, cp E G); in this case, one should take the mapping

Q x G 3 (p, cp) H (p, cp)* = (cp-i (p*), cp-i) E Q x G

as an involution in the new hypercomplex system, and the tensor product of the right invariant measures on Q and G as the new right-invariant measure.

The semidirect product of a hypercomplex system and a group possesses standard properties of semidirect products of groups. Let us illustrate this taking the semi direct

product of a three-dimensional hypercomplex system and the group ~2 as an example. In the discrete case, the concepts of hypercomplex system and hypergroup are equivalent and, therefore, it is convenient to define generalized translation operators in terms of the

convolution of Dirac measures. Let Q 3 = { e, a, b} be a commutative hypergroup with the involution a * = b and the multiplication

1 1 1 oa * Ob = -0 + -0 + -Ob. 3 e 3 a 3

Let the action of the group ~2={I,y} on Q3 be defined by y(r)=r* (rE Q3)'

Clearly, y is an automorphism of Q 3' The semi direct product of Q 3 and ~ 2 is the

hypergroup Q= Q3 XI ~2 = {e, a, E, c, d, f}, where e = (e, 1), a = (a, 1), E = (b,I), c=(e,y), d=(a,y), and f=(j,y), with the involution a*=E, c*=c,

d * = d, f * = f and the multiplication law given by the following Cayley table:

136

0-a

General Theory of Hypercomplex Systems

111 - 0- + - 0- + - Ob-3 e 3 a 3

0-c

2 1 -0- +- Ob-3 a 3

of

Chapter 1

1 2 -0- + -0" 3 a 3 0

111 - 0- +-0- +-Ob-3 e 3 a 3

As expected, the hypergroup Q is noncommutative. Moreover, the hypergroup Q 3

and group ~2 are isomorphic to the subhypergroup H = { e, a, b} and group ~ 2 =

{ e, c}, respectively. Note that ~2 is the maximal subgroup of the hypergroup Q. Since c*H*c={c, d, ]}*c=H, d*H*d={c, d, ]}*d=H, and

] * H * ] = { c, d, ]} * ] = H, the subhypergroup H is supernormal and the

quotient group Qf H is isomorphic to ~2.

3. Join. Finally, we define the join of two hypergroups (this construction was proposed by Jewett [Jew D. Let J be a compact hypergroup and let K be a discrete hypergroup. We identify the identities of the hypergroups J and K so that J n K = {e},

where e is the identity for both hypergroups. The join of the hypergroups J and K is defined as the hypergroup J v K = J U K with the topology in which J and K are closed subspaces (note that such a topology is unique) and with the operation • defined by the following equalities:

(iv) if p E K, p'* e,

Op * Op' = L cror. rEK

and a is a normalized (a(J) = 1) invariant measure on J, then


()p.()p' = ceO' + L Cr()r·

rEK

137

The validity of the axioms of hypergroups is obvious. Note also that J is a compact

subhypergroup of J v K, but K is a subhypergroup if and only if either J or K is

equal to {e}.

Let us give an example of the join of countably many discrete hypergroups. Let b~ (n ;::: 1) be a sequence of numbers satisfying the inequalities ° < b n~ 1. We set Co = 1

and define C n (n;::: 1) by the recurrence relations C n = (co + C 1 + ... + C n-l ) / b n. For

each n;::: 1, we consider the hypergroup K n = {O, n} with the identity e = ° and multi

plication () n * () n = b n()O + (1 - b n) ()o. Consider the infinite join K = K 1 v K 2 v ... = {O, 1,2, ... }. Multiplication in K is defined as follows:

and

The join K is an Hermitian hypergroup with the identity ° and the invariant measure

Clearly, each set Hn = {O, 1, ... , n} is a subhypergroup of K. One can easily establish

that the characters of the hypergroup K have the following form: X 0 (n ) = 1 (n E K),

Xk(n)=l for O~n<k-l, and Xk(k)= -bk and Xk(n)= ° for n>k. The set of

characters X = X h is countable, but the topology of the space of maximal ideals

transforms X into a compact set, e.g., Xk -7 XO' k -7 00 in the topology of X.

Note that, generally speaking, the union of two groups is not a group, but a hyper

group (to verify this, it suffices to consider the case where J = SO(2) and K = ~2).


5. Further Generalizations of Hypercomplex Systems

In the previous sections, we studied hypercomplex systems with nonnegative structure measure. For hypercomplex systems of this sort, we proved numerous results from harmonic analysis including an analog of Pontryagin's duality for commutative hypercomplex systems. If we omit the condition of commutativity of a hypercomplex system, then we encounter certain difficulties connected with definition and investigation of the dual object. For noncommutative locally compact groups, problems of this sort are solved by using specially introduced objects such as Kac algebras (originally, ring groups) or, equivalently, by using Hilbert bialgebras (see Subsection 1.7) with certain relations between the operations in them (see [Kac]). If one operation in a Hilbert bialgebra is commutative, then this bialgebra is generated by a locally compact group. Here, the main role is played by the fact that comultiplication is a homomorphism, i.e., the Kac algebra is a Hopf algebra or, in the language of Hilbert bialgebras, the binding operator W is unitary and satisfies the equality

( W 0 J)( J ® W)( W* 0 J) = (/0 W)( - 0 J) (/0 W)( - 0 J),

where J is the identity operator in H and - (a 0 b) = b 0 a (a, b E H). In Subsections 5.1 and 5.2, we study a class of bialgebras related to unimodular hy

percomplex systems in a similar way. These bialgebras are called quantized hypercompIe x systems. In this case, the most important thing is that, generally speaking, the operation of comultiplication (generalized translation) is no longer a homomorphism for hypercomplex systems. The bialgebras associated with unimodular hypercomplex systems were studied in [Vai8]. This case is more complicated and requires special technique for its investigation; therefore, we do not consider it here.

As can be shown by examples, the generalized translation operators associated with differential operators and orthogonal polynomials do not always preserve positivity and finiteness. Hence, it is reasonable to extend the concept of hypercomplex systems to the case of hypercomplex systems with alternating structure measure. A generalization of this sort have already been presented in Subsection 2.3. However, for hypercomplex systems considered in Subsection 2.3, one fails to construct the duality theory because the dual object of a hypercomplex system of this kind no longer satisfies the axioms of Subsection 2.3. Therefore, in Subsection 5.3, we consider hypercomplex systems with real structure measure satisfying the axioms of quantized hypercomplex systems. Fundamental facts from harmonic analysis and duality theory remain true for such hypercomplex systems. At the same time, one can construct natural examples for which the axioms of quantized hypercomplex systems are not satisfied (see Subsection 3.5 in Chapter 2). Therefore, in 5.4, we describe another class of compact or discrete hyper-

Section 5 Further Generalizations of Hypercomplex Systems 139

complex systems with real structure measure (quite good from the viewpoint of harmonic analysis and duality theory) which was introduced by Vainerman [Vai2], [Vai3].

5.1. Properties of Hilbert Bialgebras. Assume that.JS) is a Hilbert bialgebra in H

with respect to the operations a U b and a n b (a, b E J.9) and the corresponding in

volutions aU, an (a E J.9). Let J:;u and J:;n be the left W*-algebras of the bialgebra

J.9, Le., they are the weak closures of the *-algebras of the operators J.9 3 b H

Lu(a)b=a U band J.9 3b H Ln(a)b=a n b (aE J.9), respectively. Denote by

J:; u * and J:; n *, the corresponding preduals, Le., the Banach spaces J:; u * and J:; n *

such that J:;U and J:;n are dual to ~ U * and ~ n *, respectively. Each element 0) of

~ u * (~n *) admits the representation

n

< " 0) = L ( . uk' v k) H' k=l

where uk> Vk E H. Hence, it can be continued to a weakly continuous form on the W*

algebra of all bounded operators in H.

Let J.9 U J.9 (J.9 n J.9) be the linear span of all vectors of the form a U b (a n b),

where a, b E J.9. By Eu and En we denote the Banach spaces which are the comple-

tions of J.9 n J.9 and J.9 U J.9 with respect to the norms II· II u = II LU ( . ) II and

1I·lIn = 'II Ln(')II, respectively.

The inequalities

(5.1)

and

(5.2)

hold for any a, b E J.9.

In fact, we have

= (a®c,W(bv (8l(aUb)nc))


for any a, b, c E J.9. This implies that

which is equivalent to (5.1). Inequality (5.2) is proved similarly.

• In view of (5.1) and (5.2), the products U U v and un v can be defined by continu

ity for all u, v E H. They lie in En and E u' respectively. Clearly, En (Eu) is iso-

metrically isomorphic to the closure of the linear span {Ln (a U b) I a, b E J.9}

({LU(anb)1 a, bE J.9}) in the uniform norm. We denote this closure by

Ln (J.9 U J.9) (Lu C J.9 n J.9 )), and the corresponding isomorphism as 1tn : En-?

Ln(J.9UJ.9) (1tu:Eu-?LuCJ.9nJ.9)). We also denote LuCunv) = 1t~}(unv) and LnCuUv) = 1t~!(uUv) (U,VE H).

Any functional ro E ~ n * is associated with an element of the Banach space E U by

the formula

n n

~n*3ro(·) = LC'Uk,Vk) ~ pu(ro) = LUknV~E Eu· k=! k=!

The map Pn (clearly, linear) is defined in a similar way. Let us show that the maps

Pu and Pn are defined correctly. In other words, if

n n

L (AUk' Vk)H = ro(A) = L (AUk,vDH, k=1 k=l

then

First, we prove the identity

(LU(u n v) u', v') H = (u, Ln(v' U u'U)Vn)H (u, v, u', v' E H). (5.3)

Let vn and u~ from J.9 approximate v and u', respectively, in the norm of H. By

virtue of (5.2), the sequence of operators L U (u n vn ) uniformly converges to


LU(unv). Since unvn=Rn(Vn)UE HnEu, where Rn(a)b = bna (a,bE.I9) is

an operator of the right n-multiplication in H, this yields relation (5.3). Indeed,

Let us return to proving the correctness of definition of the map p u. By virtue of (5.3),

we have

n

L (Lu{UknV~)V, u)H k=l

n n

= L (Uk,Ln(uUVU)Vk)H = L (L~(uUVU)Uk,Vk)H k=l k=l

n

= L (Ln{uUVun)UbVk)H = ro(Ln(un U vun )) k=l

n' n'

= L (Ln{un U vun)Uk,vk)H = L (uk,Ln(uUvU)v")H k=l k=l

for any u, v E H, whence


The correctness of the definition of the map P n can be proved similarly.

• The range Pu (X n*) of the map Pu coincides with the linear span H n H of all

possible n -products of elements from H and belongs to E u· The set Pn ( Xu *) has

similar properties. We summarize the assertions established above in the following diagram:

It U /

Xn

U

.L9U.L9

5.2. Quantized Hypercomplex Systems. In this subsection, we define quantized hy

percomplex systems as bialgebras .L9 with certain relations between the operations U and n in .L9. Quantized hypercomplex systems can be regarded as a generalization of ordinary hypercomplex systems in the sense that they satisfy the theorem on realization

(Theorem 5.1): If one operation in .L9 is commutative (e.g., n), then X n* can be

realized as an object close to a hypercomplex system with locally compact base. Then

we present axioms on .L9 that guarantee the multiplicativity of the measure m, the existence of the basis unity, and the preservation of positiveness for the indicated hyper

complex system, provided that one of operations in .L9 is commutative. Assume that the operations U and n satisfy the following axioms:

(A2) Ker Pu = KerPn = {O}.

Since


one can introduce involution U in the Banach space E U by setting (a n b)U = = aU n bU (a, b E 49), and then extending it by antilinearity onto the entire 49 n 49

and by the closure in the norm 1I·lIu onto the entire Eu. The mapping 7tU: EU-7

LU (49 n 49) is, clearly, a *-isomorphism; furthermore, H n H is invariant under the

involution U and (u n v)U = uU n v U. Similar assertions are true for HUH and En.

If the bialgebra 49 satisfies (Ai), then in order that (A2) be satisfied, it is neces

sary and sufficient that the linear span of the operators Ln(a U b) (a, b E 49) be

weakly dense in ~ n and that the linear span of the operators Lu(a n b) (a, b E 49)

be weakly dense in ~ u.

Indeed, it suffices to prove this claim for p U.

Sufficiency. Let

n

PU(ill) = L uk n V~ = o. k=1

Then, by virtue of (5.3), for any a, b E 49, we have

n

ill(Ln(aUb)) = L (Ln(aUb)UbVk)H k=!

Hence, ill (A) = 0 for any A E ~ U' i.e., ill = O.

Necessity. We rewrite (5.3) as follows:

(5.4)

144 Ge!leral Theory of Hypercomplex Systems Chapter 1

where (', .) denotes the duality between an W* -algebra and its predual space. Clearly,

(5.4) remains true forarbitraryfonns ffiuE .I;u* and ffinE .I;n*. Assume the contrary,

i.e., let the set of operators Pu(ffin) be nondense in .I; U in the weak: operator topology.

Then one could find ffi~ E .I; u * orthogonal to this set in the sense of the duality (.,.).

Then it would follow from (5.4) that < Pn( ffi~), ffin) = 0 for any ffi n E .I; n*' i.e.,

P n (ffi~) = O. However, by virtue of Ker P n = 0, we have ffi~ = O.

• Since the mappings P U and Pn are bijective, one can identify 1; U* and 1; n* with

HUH and H n H, respectively, i.e., elements x E 1; U * and y E 1; n * can be re

garded as linear combinations of elements of the fonn u U v and u n v (u, V E H),

respectively. We also identify En with Ln (J.9 U J.9) and E U with LU (J.9 n J.9 ). Then the diagram presented above takes the form

1;n

/U

Taking into account that 1; n * ( .I; u *) is imbedded into .I; U (1; n), we introduce the

operations of multiplication n in 1; U * and U in .I; n *.

We say that a Hilbert bialgebra J.9 is a quantized hypercomplex system if it satisfies (Al), (A2), and

(A3) 1;U* and 1;n* are subalgebras in 1;n and 1;u, respectively.

Also note that, since H n H is invariant under the involution U, it follows from

(A3) that 1;n* is a *-algebra; similarly, 1;U* is a *-algebra with respect to the opera

tion n and involution n. A normal hypercomplex system with basis unity L 1 (Q, m) that satisfies the condi

tions of Theorem 3.5 generates a quantized hypercomplex system Indeed, in this case, we have H = L2 (Q, m) and, according to Theorem 1.8, J.9 = LIn Loo is a Hilbert bialgebra with respect to convolution (the operation U), multiplication (the operation n),

and relevant involutions. The W* -algebra 1; n is isomorphic to Loo (Q, m), and 1; u is

Section 5 Further Generalizations of Hypercomplex Systems 14S

the weak (in L2) closure of convolution operators with functions from Ll n Loo. It is

obvious that ~ n* = L 1 (Q, m) and the range of P n coincides with the Fourier algebra

A(Q). By virtue of Lemma 3.2, we have En = Coo (Q). Axiom (AI) is evidently

satisfied. By virtue of Lemma 3.2, A(Q) is weakly dense in Loo(Q, m) and, hence,

Ker P u = 0. In order to check that Ker P n = 0, it suffices to show that the convolution

with a function from L 2 can be weakly (or strongly, which is the same) approximated

by convolutions in Co (Q) with functions from L, n L 00. This fact was established in

Lemma l.2. It remains to verify (A3). Since Ker Pu = 0, we have ~ u* = A (Q). It

follows from Theorem 3.6 that A (Q) is a subalgebra of ~ n = Loo (Q, m) and, by virtue

of Theorem 1.1, L, (Q, m) is a subalgebra of .r; u.

• Note that, since ~ u * = A (Q) is isomorphic to L 1 ( Q, m) and A (Q) is isomorphic

to L,(Q,m), we have ~u=Loo(Q, m) and Eu=Coo(Q).

The Banach spaces Eu and En are completions of the *-algebras ~ u* and ~ n * in the operator norm. Therefore, En and E u are C* -algebras. Furthermore, the fol

lowing inequalities hold:

(S.S)

Ilxlln ::; II Wllllxll~ (XE ~u*)· u*

(S.6)

Indeed, let a, b E H and let x = u n v E .r; n *. Then, by virtue of (S.3) and (S.l),

we get

This implies (S.S). Inequality (S.6) can be proved similarly.

• Lemma 5.1. The C* -algebras En and E u contain left ideals J.9 nand J.9 u,

respectively, which belong to the intersection H n En n E u and are dense in each of

these spaces.


Proof. Let us prove Lemma 5.1 for En. By J9 n we denote the linear span of the

set {u n v I U E En n H (here, n denotes intersection), v E J9 U J9 }. Consequently,

J9 n is a left ideal in En' and J9 n c En nH n EU'

Let us show that J9 n is dense in En. It suffices to show that (J9 U J9 ) n (J9 U J9 )

is dense in En. For this purpose, in tum, it suffices to approximate an arbitrary element

x = U U V E J9 U J9 by elements of (J9 U J9 ) n (J9 U J9) in the norm II· II n' Let us

represent x as x = xl + iX2' where

and n

= x-x E J9UJ9. 2i

Thus, it suffices to consider the case where x is self-adjoint (i.e., x = xn). Since x E En' one can represent this element as the difference of nonnegative elements x+ and

x_, i.e., x=x+-x-, X+,X_E En' x+~o, x_~o, x+nx_=o. Let y+E En(Y+~O)

be such that x+ = y+ n y+ and let Yn E J9 U J9 be a sequence which approximates y+

in the norm 11·11 n' Then Yn n Yn approximates x+ in the norm 1I·lln because

as n --7 O.

By analogy, we find a sequence zn E J9 U J9 which approximates Y_ in the norm

II· lin' Then, clearly, the sequence Xn = Yn n Yn - Zn n zn approximates x.

Let us show that J9 n is dense in EU' It suffices to approximate any element u nv E

J9 U J9 in the norm 11·lIu by elements from J9 n. Since (J9, U) is a Hilbert algebra,

one can find sequences Un E J9 U J9 and v n E J9 U J9 that approximate u and v,

respectively, in the norm II·IIH . Then, according to inequality (5.2), u~ vn approxi

mates un v in the norm 1I·lI u' Finally, let us show that J9 n is dense in H. Note first that the operators L n (a U b)

(a, b E J9) are strongly dense in ~ n. Indeed, since Ln (J9 U J9) is dense in L n' we

conclude that the C* -algebra En is weakly (and, hence, strongly) dense in ~ n. It

Section 5 further Generalizations of Hypercomplex Systems 147

remains to note that the operators Ln (a U b) are uniformly dense in En. Any element

of H can be approximated in the norm II· II H by elements of the form a n b (a, b E

.L9 ). As proved above, the element L n (a) can be approximated by the operators

Ln(uUv) (U,VE .(9). Let us approximate bE.L9 by elements of the form u'Uv'

(U',V'E .(9). We obtain

This implies that (.L9 U .L9 ) n (.L9 U .L9 ) is dense in H. It remains to note that

(.L9 U .L9 ) n (.L9 U .(9) is a part of .L9 n .

• Theorem 5.1. Let .L9 be a quantized hypercomplex system with commutative op

eration n. Then there exist a locally compact space Q, a Borel measure m

regular and positive on open sets, and an involutory homeomorphism Q 3 P H P * E Q such that

(i) the C *-algebra En is isomorphic to the C *-algebra Coo (Q) of continuous

functions on Q vanishing at infinity;

(ii) this isomorphism can be continued to an isometry between Hand L 2 (Q, m)

and between ;C n * and L I (Q, m );

(iii) m (A) = m (A *) for any Borel set A c Q.

Proof. By the Gelfand theorem, the commutative C*-algebra En is isometrically

isomorphic to the algebra En of continuous functions on the Hausdorff locally compact

space Q of maximal ideals of the algebra En. By In we denote the image of .L9 n under this isomorphism. Let us introduce involution * in Q by setting t*(x) = t(xnu ) (t E Q, x E En).

Below, we use the Weil-Krein-Godeman theorem (see Subsection 3.1).

We define a functional f on the ideal .L9 n as follows:

f(x) = lim (x,fn) (x E .L9 n), n~oo

where fn= f{;' E .L9 n is an approximative unit in the Hilbert algebra (.L9, n). Note

148

that

Then

General Theory of Hypercomplex Systems

X= LUknvkEJ9n (ukEHnEn, vkEJ9nJ9). k

lim (x,fn)H = L lim (Uknvk,fn)H n-too k n-too

Chapter 1

Therefore, the functional I is well defined. Let us show that I satisfies the conditions of the Weil-Krein-Godeman theorem. Clearly, f is real. Indeed,

f(x n) = lim (Xn,fJH = lim (In,x) = I(x). n-too n-too

Since

n-too n-too

the functional I is positive. Hence, one can find a regular Borel measure dm(p) = dp on Q such that

lim (x n y,fn)H = f x(p )y(p )dp n-:;.oo Q

for any XE P, yE En.

Let us show that J9 n c L2 (Q, m). It is easy to see that, for any x E JS) n, the ele

ment x n xn is positive with respect to f Hence,

Consequently, we get X(p)E L2(Q,m) because y(p)E Coo(Q) is arbitrary. Since

(x, y)H = I(yn nx) = (x, y)z (x, Y E JS) n), f


we have H f = H. By analogy with the proof of Lemma 5.1, one can prove that

~ n n ~ n is dense in H. Therefore, H is isometrically isomorphic to some closed

subspace F(H) of L2(Q, m).

Let us verify that m is positive on open sets from Q. Let 0 c Q be an open set such that m(O) = O. Assume that Po E O. Obviously, one can find a function x(p) E

Coo(Q) such that x(po) = 1, 0::; x(p)::; 1, and x(p) = 0 outside of O. Denote by x

the preimage of x(p) under the Gelfand isomorphism. Clearly, x 7:- O. Since ~ n is

an ideal, we have x n x = x n x E J9 n for any x E ~ n. Therefore,

IIxnxllt = J I(xn x)(p)1 2dp = J Ix(p)1 21 x(p)1 2dp = 0 Q Q

because, by assumption, m ( 0) = O. Thus, we have x n x = 0 for any x E ~ n. Since

~n is dense in H, we get Ln(x) = 0, whence x = O. We arrive at a contradiction.

Let us show that F(H) = L2 (Q, m). It suffices to establish the denseness (in the

sense of L2(Q, m)) of the ideal ~n in the space Co(Q) of finite continuous func

tions. Assume that z (p) E Co (Q), A = supp z, and U is a neighborhood of A with

compact closure. Let us choose Y (P) E Coo (Q) such that

yep) = {~ if

if

pEA,

PEQ \ U.

Since J9 n is dense in Coo (Q), one can find a sequence Y n E ~ n that approximates Y

in the norm 11·1100. Then xn(P)=Yn(P)Z(P)E ~n approximates Z inthenormof

L2(Q, m).

For any x(p) E L) (Q, m), we have

lim J x(p )fn(P) dp = J x(p) dp. n~oo

Q Q

Indeed, the function x(p) admits the representation x(P) = yep )z(p), where Y (p),

z (p) E L2 (Q, m), e.g., x(P) = I x(P) 1112 (I x(P) 1112 exp [arg x(p )]). Therefore,

J x(P)fn(p)dp Q

= (y,znnfn)H --~) (Y,Zn)H = Jy(p)z(p)dp = Jx(p)dp. n~oo

Q Q


Denote by I the image of J9 under the considered isometry. Clearly, I is dense in

L2(Q,m). Let us show that m(A)=m(A) for AE $o(Q). Let KA(t)E L2(Q,m)

be the indicator of A. There is a sequence x J t) E I that converges to K A in the quad

ratic mean. Clearly, x n( t *) ~ K A .( t) in L2 (Q, m). This implies that

= lim Ilxn II~ n--t oo

for any A E $o(Q).

The weak closure (in L2 (Q, m») of the C*-algebra of operators of multiplication by functions from Coo (Q) coincides with the W *-algebra of operators of multiplication by functions from Loo(Q, m). As is known, Ll (Q, m) is the predual of Loo(Q, m). There

fore, the considered isometry can be continued to a mapping of I; n * onto L I (Q, m ).

Let us show that this mapping is also isometric. Since I· I is dense in L2 (Q, m), by

using the decomposition L I (Q, m) 3 x(p) = yep )z(p) (y(P), z (p) E L2 (Q, m », we establish that I· I is dense also in L I (Q, m ).

Let x = u n v E J9 n J9 (u, V E J9). Then

= sup A(p)ELooCQ,m)

J A(p )u(p )v(p) dp Q

II A(p) Iloo = J lu(p)v(p)ldp = IIx(P)IILdQ,mY

Q

The inequality II u n v 11;r; ~ II willi u II H II v II H implies that J9 n J9 is dense in I;n *. n·

This and the fact that I· I is dense in L I (Q, m) yield that the mapping of I;n* onto

L 1 (Q, m) is an isometry.

• By using the second assertion of the theorem, we can equip the space L I (Q, m )

with convolution (generally speaking, noncommutative) and involution by transferring

them from I; n *. The measure m is a Plancherel measure and, therefore, the Plancherel

formula and the inversion formula (see Subsection 5.3) hold for this convolution. If one of the operations is commutative, then, by Theorem 5.1, the quantized hyper-


complex system is realized as an object similar to a hypercomplex system with locally compact basis. However, generally speaking, this object may not have the properties of

the multiplicativity of the measure m, the existence of the basis unity, and the preservation of positivity under convolution. To guarantee that it possesses these properties we impose the following additional condition on the quantized hypercomplex system:

(A4) the mappings m H m(l), where I is the identity operator and either mE 1; n* or m E 1; u *, can be extended to characters of the C *-algebras En and E u'

respectively.

Lemma 5.2. Suppose that the mappings m H m(l), where mE 1;u* or mE

1; n*, are multiplicative on 1; U* 0 r 1; n *. If there exist approximative units

e a E 1; u * and f ~ E X; n * such that II e a II 1; u. ~ 1, II f ~ II 1; n. ~ 1, and P n (e a) -7 I

and P U (f~) -7 I * -weakly in 1; nand 1; u' respectively, then these mappings can

be extended to characters of the C *-algebras En and EU' respectively.

Proof. It suffices to check the inequalities

By virtue of (5.4), we have

where mu E 1;u*. The second inequality can be proved similarly.

• Condition (A4) is satisfied for the normal hypercomplex system with basis unity

L 1 (Q, m) satisfying the conditions of Theorem 3.5. Indeed, one can take the approxi

mative unit e 11 of the algebra L 1 (Q, m) constructed in Lemma 1.3 as f U' and the se

quence

1 xn(t) = --c(Qn, A, t),

meA)

where A E ~o (Q) is an arbitrary open set and Q n C Q n+ 1 is a sequence of compact


sets that covers the entire space Q, as e ex'

Let a quantized hypercomplex system satisfy (A4) and let the operation n be com

mutative. Then there exists a basis unity, i.e., a point e = e *E Q such that (j, g)2 =

(j * g *)( e), where f, gEL 2 (Q, m), (".) 2 is the scalar product in L 2 (Q, m), and

the measure m is multiplicative, i.e.,

(j*g,m) = (f, m)(g, m) (f,gE L2(Q,m)).

Indeed, since the functional G(x) (XE EU)' which is an extension of (/, cou)

(coU E X U*)' is multiplicative on En = Co(Q), there exists a point e E Q such that

G(x) = x(e) (x E Co(Q». It is easy to see that, on X U* = HUH, the functional G

has the form (/, u U v) = (u, v U) H' Therefore, for any f, g E H, we have (j, g) 2 = G(fUgU)= (j* g*)(e). Since

we establish that x(e *) = x(e) for all x E .is) U .is), which implies that e *= e because

.is) U.is) is dense in En.

To prove the multiplicativity of m it suffices to note that any element x(p) E

LJ(Q,m) can be represented as x(P)=u(p)v(p) (u(p),V(p)E L2(Q,m» and to use the multiplicativity of the functional

(I, cox> = (u, vn)H = J u(p)v(p)dp = J x(p)dp.

As a result, we get

Q Q

J x * y(p)dp = (I, coxU COy) = (I, coX>(I, COy).

Q

• If we interchange the pairs of operations U and n in the quantized hypercomplex

system .is), we obtain a quantized hypercomplex system .is) (dual to .is). Thus, if we

pass from the pair (*,.) to the pair C, *) in the hypercomplex system L] (Q, m) in Theorem 3.5, then the corresponding quantized hypercomplex system L] (Q, m ) n Loo(Q, m) is realized with respect to the operation * according to Theorem 5.1. As a

result, we obtain the dual hypercomplex system L J ( Q, m). Clearly, (i> r = .is). The class of quantized hypercomplex systems satisfying (A4) is closed with respect to duality. All results presented above remain true if we require the commutativity of the

operation U instead of n.


In a quantized hypercomplex system with commutative operation n satisfying (A4) and realized, according to Theorem 5.1 in the space L2(Q, m), we introduce generalized translation operat~rs: (LpJ, gh = (f* g*)(p), where J, g E L2(Q, m) and p E Q.

This definition is correct because f * g * E Coo (Q). Let us show that these operators

have the following properties:

(i) Le = I;

(ii) (Lpf)(q)E L2(Q,m) m-almosteverywherein q;

(iii) (L'b(Lqf))(r) = (~(Lpf))(r) mXm xm-almosteverywhere in p,r, and q;

(iv) (Lpf)(q) = (Lq.j*)(p*) m x m-almost everywhere;

(vi) II Lp II ::; II wlI;

(vii) the operator function Q 3 P H Lp is weakly continuous;

(viii) (LqJ)(p) = (Lqf)(p) m xm-almosteverywhere in p and q;

(ix) the measure m is invariant, i.e.,

J (Lpf)(q)dq = J f(q)dq

Q Q

for m -almost all p.

Indeed, (i) follows from the definition of basis unity, (vi) follows from inequality (5.1), and (v) is implied by the axioms of Hilbert algebras. Let gEL 1 n L2, h E L2,

fnE I, bea sequence which approximates fE L 2. It is easy to show that fn*g* con-

verges to f* g* in mean square. It follows from inequality (5.1) that fn* * h uniformly

converges to 1* * g. In view of the Lebesgue theorem, this yields

J (Lpf)(q)g(q)dqh(p)dp = (f* g*, h)2

Q

= lim (fn * g*, h)2 = lim (g* ,fn* * h)2 n---700 n~oo

154 General Theory oj Hypercomplex Systems

= f g(q*) f (Lqf)\p)h(p*)dpdq Q Q

= f f (Lq./)(p*)g(q)h(p)dpdq; QQ

since g E L1 n L2 and hE L2 are arbitrary, we arrive at (iv).

Chapter 1

Let us check the associativity relation (iii). For all g, hELl n L2 and J E L2, we have

f J (Lt(Lqf))(r)g(r) h(q)drdq

Q Q

= J g(r) J (Lqf)(r) Lp.h(q)dqdr

Q Q

= J (f* g*)(q)Lp.h(q)dq

Q

= «(f* g*) * h*)(p) = (f* (h * g)*)(p)

= f (Lpf)(r) J (Lrh)(p)g*(q)dqdr

Q Q

= J J (Lpf)(r)(Lr*g*)(q)h(q)drdq

Q Q

= J J (0;(Lpf))(r)g(r)drh(q)dq,

Q Q

whence we get (iii) because g and h are arbitrary and LIn L2 is dense in L2. We note that (ii) immediately follows from (iv). Property (vii) is a consequence of the fact

that J* g E Coo(Q) (f, g E L2(Q, m» and property (viii) is obvious.

Finally, let us prove (ix). For f, g E L1 n L2, we have

J J J(p)dpg(q)dq = QQ

J (f* g)(p)dp

Q

= J J (Lpf)(q)g(q*)dqdp = J J (Lq*J*)(p*)g(q*)dqdp Q Q Q Q

= J J (Lq/)(p*)g(q)dqdp = J J (LqJ*)(p)dqg(q)dp. Q Q Q Q


Hence, for almost all q,

f (Lqf*)(p)dq = f f(P)dp = f f*(p)dp,

Q Q Q

whence

J (Lqf)(p)dp = J f(p)dp.

Q Q

• The element A E ~ U is called n-positive (A Q 0) if (A, ro n ron) ::?: 0 for any ro E

U ~U*. The inequality A::?: 0 is understood similarly.

We introduce the axioms

n n (AS) A,B::?:O => AB::?:O (A,BE ~U);

u U (AS') A,B::?:O => AB::?:O (A,BE ~n).

Theorem 5.2. Let.J.9 be a quantized hypercomplex system which satisfies (A4)

and (AS) and let the operation n be commutative. Then L1 (Q, m) is a Banach

* -algebra and the convolution of nonnegative functions is nonnegative.

Proof. We first show that the equality

(Pn(x),y) = J x(P)y(P*)dp

Q

holds for any x E ~ n* and y E ~ u*. In fact, for any a E".J.9 and y = u U v U E ~ U*, we have

(5.7)

= (a, (u U VU)U)H = (a, yU)H = J a(p)y(p*)dp. (5.8)

Q

Since I is dense in L 1, one can approximate an arbitrary function x E L1 (Q, m) by

functions a E .J.9. Then Lu(a) --7 Lu(x) in ~ U. Hence, by passing to the limit on

both sides of (5.8), we obtain (5.7).


It follows from (5.7) that if x(p) E Ll (Q, m) is almost everywhere nonnegative, n

then LU(x) ~ O. Indeed, for any a E .1.9 U .1.9, we have

(LU(x),ynyU) = J x(p)ly(p*)1 2dp ~ 0 Q

by virtue of (5.7). Since J.9 U .1.9 is dense in Coo (Q), the converse assertion is also true.

Let x, YELl (Q, rri) are real-valued functions. The inequality

(5.9)

holds for almost all p.

In fact, we have x = x+ - x_ and y = y+ - y_, where x:b y± E Ll CQ, m) and are

nonnegative almost everywhere. Therefore, LU(x±) ~ 0 and LUCY±) ~ O. Since

x± * y± can be represented as a linear combination of the functions 1 x 1 * 1 y I, 1 xl * y,

x* Iyl, and x*y from Lt(Q,m), we have X±*Y±E LlCQ,m). By virtue of (AS),

Therefore, Cx+ * y+)(p) ~ 0 almost everywhere. Hence, for almost all p, we have

By integrating (5.9) and using the fact that the measure m is multiplicative, we obtain

If the functions x, YELl (Q, m) are complex-valued, then we represent them in the

form x = Xl + iX2 and y = Yl + iY2' where Xk and Yk are real-valued functions, and easily conclude that

•


Thus, for quantized hypercomplex systems which satisfy the conditions of Theorem 5.2, the Banach *-algebra L1 (Q, m) possesses all properties of a hypercomplex system with locally compact basis except, maybe, the fact that the convolution of compactly supported functions may be not compactly supported. In order that a class of quantized hypercomplex systems of this sort be closed with respect to the operation of transition to the dual quantized hypercomplex system, one must require that both (AS) and (AS') be satisfied. A class of finite-dimensional quantized hypercomplex systems satisfying (A4), (AS), and (AS') is contained in the class of bialgebras considered by Vainerman [Vai4], [VaiS]. He also presented an example of a finite-dimensional quantized hypercomplex system noncommutative with respect to both operations.

5.3. Harmonic Analysis in Quantized Hypercomplex Systems with One Commutative Operation. In this subsection, we briefly describe harmonic analysis in quantized hypercomplex systems with commutative operation n. For these systems, we prove analogs of the Plancherel theorem and inversion formula and establish duality in the sense of Pontryagin.

Let J.9 be a quantized hypercomplex system satisfying (A4) and such that the operation n is commutative. By using the results established in Subsection 5.2, we construct a locally compact space Q, involution Q:3 pH P *E Q, multiplicative measure m, and a basis unity e = e *E Q such that the space L 1 = L 1 (Q, m) is a * -algebra with respect to the (noncommutative) generalized convolution f * g.

Let us show that J.9 = Lin Loo is a Hilbert algebra in L2 with respect to the con

volution f* g and involution 1* (t) = f(t*) (the first pair of operations) and with re

spect to the pointwise multiplication offunctions and complex conjugation (the second pair of operations).

It is easy to see that .19 is a Hilbert algebra with respect to ordinary multiplication

and complex conjugation. Let us show that (19, *) is a Hilbert algebra. Denote the linear span of L2 * L2 by A(Q). Obviously, A(Q) is an algebra dense in Coo(Q) be-

cause A(Q) is isomorphic to X;U*' Let a,bE .19. By virtue of (A3), a*bE L1, but - -

since .19 c L2, we have a * b E A(Q), and, hence a * bE Loo- Thus, .19 is a * -al-

gebra with respect to convolution and involution a * (p) = a(p*). Let us verify that .19 satisfies conditions (i)-(iv) in the definition of Hilbert algebra (see Subsection 1.7). In-

deed, condition (i) follows from the unimodularity of m (m(A j = meA»). Condition

(ii) follows from the equality L~(x)=Lu(x*) (XE X;n*), namely,

(a*b,ch = (Lu(a)b,ch = (b,LU(a*)ch = (b,a**ch (a,bE 19).

Condition (iii) follows from (5.5), namely,


Let us show that condition (iv) is also satisfied. Let I be the image of the original Hil

bert bialgebra J.9 under the isomorphism H H L2 (Q, m). Then Ie Loo and I· Ie - - -

L 1. This implies that I· I c J.9. Hence, to show that J.9 * J.9 is dense in L 2 it suf-

fices to establish the denseness of (I. I) * (I. I) in L2(Q, m) or, which is the same,

the denseness of (J.9 n J.9 ) U (.19 n .19) in H. The last assertion is already proved in

Lemma 5.1. Thus, (19, *) is a Hilbert algebra and it remains to prove the continuity of

the binding operator" W II. Let ai, a2, b l , b2 E 19. By virtue of relation (5.5) and the Schwartz inequality, we have

where W is the binding operator of the original quantized hypercomplex system.

• Let 19 = LI n Lao be the Hilbert bialgebra introduced above. Consider the Hilbert

- -algebra (.19, *). The set of elements f E L2 such that the operator .19 3 a H L u (a)f =

f * a is continuous in L 2 forms a Hilbert algebra .19' that contains .19. The algebra

.19' is called perfect. Let X; U be the left W*-algebra of the Hilbert algebra (19, * ) (or, which is the same, (.19', *» and let X;~ be the set of its positive elements. For

AE X;~, we set cp(A) = (a,a)Lz if A I / 2=Lu(a), where aE .19', and cp(A)=+oo,

otherwise. Then cp is a semifinite faithful normal trace on X;~, i.e., the function cp :

X;~ --'? [0, 00] has the following properties:

(i) cp(A +B) = cp(A)+cp(B) (A,BE X;~);

(ii) cp(AA) = Acp(A), (A;?: 0, A E X;~; we suppose that 0· 00 = 0);

(iii) cp(UAU- I ) = cp(A) (AE X;~; U is a unitary operator in X;u);

(iv) cp(A) = sup{cp(T)IT~A and cp(T)<+oo} (AE X;~);


(v) <p(A) = 0 <=> A=O;

(vi) <p(A) = sup {<p(B) I B E IDC} for any increasing family IDC c ~~ with the up

per bound A E ~~.

The trace <p on I;~ is called canonical. We extend <p to the entire ~ U by line

arity. Clearly, <p(LU(b)*LU(a» = (a,b)2 (a,bE J9).

By applying the central decomposition (see, [Dixl]), we expand L2 into the direct integral

L2 = f HAdp(A) z

of Hilbert spaces HI.. so that I; U can be represented as a direct integral of lV-algebras:

~u = f ~U(A)dp(A), z

where Z is the quasispectrum of the W*-algebra ~ U (see, e.g., [Dixl]), i.e., a spe

cially topologized set of representations of ~ U such that the W*-algebra spanned by

the operators of each representation is a factor.

The Fourier transform of the function a E J9 is defined as an operator function

a(A) = f a(s)Ls(A)dm(s) (A E Z), Q

(5.10)

where Ls(A) are the components of the central decomposition of generalized translation

operators L s E ~ U constructed for the quantized hypercomplex system J9 in Subsec

tion 5.2. As follows from the results obtained in Subsection 5.2, Theorem 1 in [VaL]) is applicable to the generalized translation operators L s' As a result of its application, we obtain the following assertion:

Theorem 5.3. The following Plancherel formula and the inversion formula hold for the Fourier transform (5.5):

(a, b)L2 = f <PA(LA(b)* LA(a»dp(A), Z

a(s) = f <PA(Ls(A)a(A»dp(A), z

(5.11)

(5.12)


where <pA, are the components of the canonical trace <p on X u and q(a) are the

components of the operator X u (a) in the central decomposition of XU,

Note that the measure p in (5.11) and (5.12) is defined uniquely up to equivalence.

If the generalized translation operators L s are commutative, then the measure p is

unique. The Fourier transform (5.10) and the inversion formula (5.12) establish a unitary

isomorphism between L2 (Q, m) and the space H <p constructed from the trace <p on

X u by using the G NS -construction.

Remark 1. In the case of ordinary commutative hypercomplex system, Theorem 5.3 turns into Theorems 3.2 and 3.3. Thus, Theorems 3.2 and 3.3 can also be proved by using Theorem 1 in [VaL]. For simplicity, in Section 3, we gave more elementary proofs

of these theorems.

Remark 2. The Plancherel theorem and inversion formula are also true for nonunimodular generalized translation operators (see [VaL]) and unbounded Hilbert algebras generated by unbounded generalized translation operators (see [Vai6]-[Vai8]).

It is c<;mvenient to express the Pontryagin duality in terms of quantized hypercomplex

systems. Let the operations U and n be commutative. Realizing a quantized hyper

complex system with respect to the operation U, we obtain a locally compact space Q dual to Q. Denote the operation of convolution in L J ( Q, m) by * (x * y (p) is the

image of x ny E X u*). The operator A E X n is called a U-character (or the charac

terofthehypercomplexsystem LJ(Q,m)) if (A,xUy) = (A,x)(A,y) (X,YE X n*)

and A"* O. Similarly, we define the notion of n-character. By X and X h we denote

the sets of U-characters and Hermitian U-characters, respectively (a U-character A is

called Hermitian if A=Aun). Similarly,wedefine X and Xh . Clearly, Q CXh.

If we realize a quantized hypercomplex system with respect to the operation n, then

the W*-algebra X n turns into Loo(Q, m). Hence, the U-characters have the form

X(p) (pE Q); similarly,the n-charactershavetheform X(p) (pE Q). Since

Q c X, we conclude that p (X) = X (p) (X E Q) is a n -character. If a quantized hypercomplex system satisfies (A4), (A5), and (A5'), then, clearly, there is a one-to-one

correspondence between the set of U-characters (n-characters) and the set of maximal

ideals of the Banach algebra L 1 (Q, m) (L 1 ( Q, m)). One can now easily formulate an analog of Theorem 3.5 for the Fourier transformation defined in the standard way (we do not present it here).

If X n * has a weak approximative unit, i.e., a net e a E X n *, II e a 1I,n ::; 1, such n*

that e a U x --7 X weakly in X n *, then an analog of the Bochner theorem is true for the

quantized hypercomplex system with the commutative operation n. The proof of this


statement, in fact, coincides with the proof of Theorem 3.1 and is thus omitted.

5.4. Real Hypercomplex Systems with Compact and Discrete Bases. In this subsection, we briefly discuss hypercomplex systems with compact and discrete bases introduced by Vainerman (see [Vai2], [Vai3], and [Vai5]). Generally speaking, these hypercomplex systems do not belong to the class of quantized hypercomplex systems. Nevertheless, they also admit rich harmonic analysis. The hypercomplex systems introduced by Vainerman are defined by analogy with hypergroups as follows: a hypercomplex

system is a set of Radon measures on Q (Q is the basis of this hypercomplex system) with the structure of a *-algebra with identity I (here, it is not assumed that the identity

belongs to the basis, i.e., it is not required that 1= Oe, where e E Q). Rigorous definitions are independently formulated for the cases of discrete and compact bases.

Let Q be a compact space satisfying the second axiom of countability and equipped with involutory homeomorphism Q 3 P I---) P * E Q, let M = M b( Q) be the space of

bounded Radon measures equipped, in addition to the standard norm, with the topology of ordinary convergence on a subspace T which is dense in e (Q) and invariant under

the mappings fer) I---) fer) and fer) I---) f*(r) (this topology is called T-topology; for

T = e (Q), it coincides with weak topology). Assume that the set of Dirac measures

J9 (Q) = { 0 d t E Q} is total in the T-topology (for T = e (Q), this condition is clearly

satisfied).

The space M is called a real hypercomplex system with compact basis if the following conditions are satisfied:

(i) the space M equipped with T-topology is a topological associative *-algebra with involution * and identity 1;

(ii) the function Q x Q 3 (t, s) I---) (Ot * Os,/) is separately continuous (this condi

tion implies that the form (W * 11,J) with fixed f E T and fl E M is T -con

tinuous on M and, hence, is given by a certain function fl * f E T);

(iii) there exists a unique (to within a positive factor) positive Plancherel measure,

i.e., a measure mE M+, supp m = Q, such that m* = m and 11 * fm = (11 * f)m (fE T,11 EM);

(iv) II 11 * fib ~ ell 11l1l1flb, where fE T, e ~ 0, 11 E M, and 11.11 2 is a norm in

L2(Q, m).

If 1= Oe for some e E Q, Ox * Oy E MI (Q), and I (x) E T (l (x) == 1), we obtain an object similar to a compact hypergroup; such a hypercomplex system is called a gen

eralized hypergroup and the Plancherel measure turns into an invariant measure (which is unique). If Q is connected, the uniqueness of the Plancherel measure follows from


axioms (i), (ii), and (iv). If Q is disconnected, then the Plancherel measure can be non

unique; for example, in the case M = ([n, Q = { ], ... , n}, 0 i * OJ = O~Oi (oj is the

Kronecker symbol), the Plancherel measure is nonunique.

The operations f* g = fm * g (J, gET) and f* (x) = f(t*) define the structure of

a *-algebra on T. Moreover, axiom (iv) implies that IIf* giL::; Kllfll211 g 112 (K ~ 0).

This estimate allows us to extend the convolution to arbitrary functions from L 2 (Q, m ).

In this case, L2(Q, m) is a complete Hilbert algebra (i.e., J9 = L2 (Q, m». Therefore, it can be decomposed into a direct sum of at most countably many minimal closed ideals

H k [Dix1]. One can prove that each H k is finite-dimensional. This enables us to establish the following analog of the Peter -Weyl theorem for the hypercomplex system M: All irreducible *-representations of M are finite-dimensional, the set of their equivalence classes is at most countable, the matrix elements of all irreducible representa

tions of M form a complete orthogonal system in L 2 (Q, m), and the linear span of this system is uniformly dense in C(Q).

For a real hypercomplex system, by analogy with Theorem 5.3, one can prove the Plancherel theorem and inversion formula for the corresponding Fourier transformation; moreover, one can prove an analog of the Bochner theorem and study the Fourier algebra

A (Q) following [Eym]. Note that, in this case, A (Q) is an algebra with respect to convolution (however, it is not an algebra with respect to pointwise multiplication). This

and the fact that, generally speaking, L 1 (Q, m) is not a *-algebra explain that a real hypercomplex system with compact basis is not necessarily a quantized hypercomplex system.

Let us define a real hypercomplex system with discrete basis. Assume that Q is a

countable set with discrete topology and M is a C * -algebra with identity 1 which consists of Borel measures on Q and is a completion of II (Q) in some topology generated

by the C* -norm II· II which is weaker than the topology of [I (Q) but stronger than the

topology of pointwise convergence of measures on Q. The form (J, iI* * v), where f has a compact support, is continuous in v and, hence, is given by a bounded function

J.l * f with compact support. A Borel measure m is called a Planche rei measure on Q

if m * = m and J.l * fm = (J.l * f)m (J.l E M,J E Co(Q». We say that M is a real

hypercomplex system with discrete basis if the following conditions are satisfied:

(i) Ilvll=llvll and Iv(Q)I::;cllvll (vE/I(Q), C~O);

(ii) there exists a unique positive Plancherel measure m = (m k) kE Q whose support

is the entire set Q;

(iii) Ilf112::; CllfJ.l1I (C~O, fE Co(Q), and 11.112 is the norm in 12(Q,md).

Ordinary hypercomplex systems with discrete bases satisfy the axioms of real hyper

complex systems with discrete bases (in this case, M is the enveloping C* -algebra for


the Banach *-algebra II (m), i.e., the completion of II (Q) in the norm

II x II' = sup 111t(x) II, ItEffi

where x E II (m) and ffi is the set of representations of II (m); it is clear that, in the

case of commutative hypercomplex systems, we have II x II' = II x(x) 1100), Note that one can use the following equivalent definition of real hypercomplex sys

tems with discrete bases in terms of C* -algebras: A real hypercomplex system is a C *

algebra M with identity 1 such that there exists a countable total subset Q = {8d of

M invariant under the involution * and satisfying the following conditions:

(i) there exists a unique (to within a constant factor) finite faithful trace p- on M

suchthat p-(8~*8i) = 8,p-i/ (P-k>O, 8i,8k E Q));

(ii) the numbers p- (8 k* * 8 i * 8j ) are real and there exists a continuous Hermitian

form 1: on M such that 118k II ~ C1:(8k) for any 8k E Q.

If M is commutative, then, applying the Gelfand theorem to the C* -algebra M, we establish that a commutative real hypercomplex system with discrete basis is isomorphic

to a C* -algebra C( Q) with identity, where Q is a Hausdorff compact set that satisfies A

the second axiom of countability. There exists a finite Borel measure p- (supp p- = Q)

and an at most countable set Q = {(PiCt)}:1 of functions from C(Q) mutuallyortho

gonal with respect to the scalar product in L 2 ( Q, m ). These functions form a total set

in C ( Q) and satisfy the following conditions:

(i) if <Pi(t) E Q, then <Pi(t) E Q;

(ii) the numbers f <Pi(t)<P/t)<Pk(t) dp-(t) are real;

(iii) there exists a finite real Borel measure 1: on Q such that

for all i.

One can show that the form s(p-) = (11, I), where p- E M and (', .) is the scalar

product in H=12(Q, l/mk)' defines a finite faithful trace on the C*-algebra M.

Denote by L (P-)f = p- * j the left regular representation of the C* -algebra M in H. By


applying the construction suggested in [VaL] to the W* -algebra generated by the operators of this representation and following Subsection 5.3, one can prove the Plancherel theorem and inversion formula for the corresponding Fourier transformation. For such hypercomplex systems, one can also prove an analog of the Bochner theorem and the following analog of the Pontryagin duality principle: Commutative real hypercomplex systems with compact and discrete bases are in duality.

2. ExamPLES OF HYPERCOmPLEX SYSTEmS

As indicated in the introduction to Section 2 of Chapter 1, the appearance of generalized translation operators is, as a rule, connected with the existence of a Fourier-type transformation satisfying the Plancherel theorem and the inversion formula. These generalized translation operators often possess additional properties which enable one to construct a hypercomplex system. In view of the existence of developed harmonic analysis for hypercomplex systems, it is possible to consider, from the general point of view, numerous results of harmonic analysis obtained in various special cases. Note that the application of duality theory to these cases sometimes gives new results (see, e.g., Vainerman's version of the inverse problem for the Sturm-Liouville equation [Vai3] in Subsection 4.5 of this chapter).

In Chapter 2, we consider four situations of this kind. In Section 1, we investigate a hypercomplex system isomorphic to the center of the group algebra of compact group and the dual hypercomplex system. In Section 2, we study hypercomplex systems associated with Gelfand pairs, i.e., with pairs of locally compact groups G and their compact subgroups H such that the sub algebra of the group algebra of G which consists of H-biinvariant functions is commutative. In particular, these hypercomplex systems contain hypercomplex systems studied in the Section 1. Section 3 is devoted to the analysis of hypercomplex syst((m associated with orthogonal polynomials defined on bounded subsets of the real axis. In Section 4, we consider hypercomplex systems constructed by using the Sturm-Liouville equation. Here, we do not consider hypercomplex systems

connected with association schemes, hypergroups K II H, where K is a locally compact hypergroup and H is a compact subhypergroup of K ([Jew]), and some other examples of hypercomplex systems.

1. Centers of Group Algebras of Compact Groups

We can consider the center of the group algebra of a finite group G as a commutative finite-dimensional hypercomplex system whose basis coincides with the set of conjugacy

classes of the group, i.e., the collection of sets of the type {aga- 1 I a E G}. In this section, we show that, for a compact topological group, the center of its group algebra is a commutative hypercomplex system whose basis coincides with the compact set of con-

165

166 Examples of Hypercomplex Systems Chapter 2

jugacy classes of the group. By applying the results obtained in Chapter, 1 we obtain fundamental principles of the theory of representations of compact groups. In the second part of this section, we present some results from [BeG] devoted, in particular, to the investigation of hypercomplex systems whose basis is the compact set of conjugacy classes of compact semisimple Lie groups and dual hypercomplex systems.

The hypercomplex systems studied in this and subsequent sections are subalgebras of the group algebra.

1.1. General Construction of Hypercomplex Systems Corresponding to Locally Compact Groups. First, we describe a general method aimed at the construction of new examples of generalized translation operators based on the reduction of the operation of

ordinary translation on a locally compact group G. Let G be a locally compact Hausdorff group satisfying the second axiom of count

ability. Consider its group algebra, i.e., a set L1 (G, dg) of functions defined on the

group and summable with respect to the left Haar measure d g with norm

II x II = J I x (g ) I d g G

and convolution

(x*y)(g) = f x(a)y(a-1g)da (X,YE L1(G,dg); gE G). G

Let K be a closed subalgebra of the algebra L J ( G, d g). Assume that there exists a decomposition of G into disjoint compact classes of elements p, q, r, ... such that every function from K can be obtained as the limit in the norm of the space Ll (G, d g) of a sequence of continuous finite functions defined on the group which take constant values

on every class p, q, r, ... and, vice versa, every limit of this sort belongs to K. We equip the set Q with the factor topology, namely, a set A c Q is open in this topology

if and only if p-J (A) eGis open (here, P is the natural projection which maps every

g E G into the class p which contains g). Assume that P is a closed mapping. Then Q is a locally compact Hausdorff space with the second axiom of countability. This implies that every continuous function on G taking constant values on classes from Q can be regarded as a continuous function on Q, and vice versa.

In the space Co (Q), we consider a positive linear functional

/(!) = J f(g)dg (fE Co(Q»· G

By virtue of the Riesz-Markov theorem, this equality defines a unique nonnegative regu

lar Borel measure m(E) (E E ':S(Q» such that

Section 1 Centers of Group Algebras of Compact Groups 167

J f(g)dg = J f(p)dp (fE CO(Q») (1.1) G Q

(integration with respect to the measure m is denoted by dp). It follows from (1.1) that there exists an isometric mapping x (g) ~ x (p) of the entire K onto the entire space

LI (Q, dp) such that

J x(g)dg = J x(p)dp (X(g)E K). G Q

Note that meA) = A(~) (A E ~o(Q), ~ E ~o(G), ~ is a preimage ofthe set A, and A is the Haar measure). It is not difficult to show that the convolution of two functions x, y E K can be written in the form

(x*y)(r) = J x(p)dpJ y(q)dqc(EpEq,r) (r,p,qE Q), Q Q

where c(A, B, r) is the structure measure given by the formula

c(A, B, r) = A(g-I ~ n ~-I) = X2r * X)8(g) (1.2)

(A, B E ~o(Q), g ErE Q, and ~ and ~ are, respectively, the preimages of the sets

A and B), and m is the multiplicative measure. Thus, the subalgebra K c L, (G, d g) can be regarded as a hypercomplex system

with locally compact basis Q.

1.2. Centers of Group Algebras of Compact Groups. Let us show that the center of the group algebra of a compact group is a compact hypercomplex system.

We apply the general constructions of Subsection 1.1 to the case where K is the

center Z L 1 (G) of the group algebra of a compact group G, i.e., K = Z LI (G) = {x E L J (G, dg) I x * y = y * x for all YELl (G, dg)}. It is well known that ZL1 (G) consists offunctions x(g) E L J (G, dg) such that x(gh) = x(hg) for almost every

(g, h) E G x G (these functions are called central). Indeed, for all YELl (G, dg) and

almost all g E G, by virtue of unimodularity of the group G and commutativity of

ZL1 (G), we have

0= x*y(g) - y*x(g) = J x(h)y(h-Ig)dh - J y(h)x(h-Ig)dh

= J x(gh)y(h-I)dh - J x(hg)y(h-I)dh

= J [x(gh)-x(hg)]y(h-l)dh.


Since the function y is arbitrary, we conclude that x E Z LI (G) if and only if x (g h) = x(hg) for almost all (g, h) E G x G.

• Let Q be the set of conjugacy classes of the group G. Denote by p, q, r,... ele

mentsof Q (recall that p = {aga-1IaE G}). Note that the equality x(gh) = x(hg)

(g, h E G) is equivalent to the equality x(h- I gh) = x(g). Thus, continuous functions from ZL1 (G) take constant values on the conjugacy classes of the group G and, vice versa, every continuous function which takes constant values on these classes belongs to

ZLj (G). Since ZLj (G) is a closed subalgebra, the limit of these functions in the norm

of the space LI (G, dg) also belongs to ZL1 (G). Conversely, if x(g) E ZL1 (G), then

it is possible to construct a sequence of functions which are continuous on G such that

II x - fnlll ~ 0 as n ~ 00. This means that continuous functions

f~ (g) = f fn(a- l ga)da E ZLj (G) G

approximate x (g). If we equip the set Q with the factor topology and apply the general

construction of Subsection 1.1, then we conclude that the center Z Ll (G) of the group algebra of a compact group is a commutative hypercomplex system with compact basis

Q of conjugacy classes. This hypercomplex system becomes a normal hypercomplex system if we introduce the involution * as follows:

Q 3p = {a-lgalaE G} H {a-lg-1alaE G} = P*E G.

The identity e of the group is a basis unity of the obtained hypercomplex system.

Consequently, ZLI (G) is semisimple. It easy to see that the hypercomplex system

ZL1 (G) satisfies the condition of separate continuity. It contains the identity if and only

if G is finite.

1.3. Elements of the Theory of Representations of Compact Groups. Prior to the

description of the space of characters of the hypercomplex system ZL1 (G), we recall some principal notions and facts from the theory of representations of groups. A strongly

continuous operator-valued function G:3 g H 1t (g), where 1t (g) is a bounded (uni

tary) operator in a separable Hilbert space H, is called a representation (unitary repre

sentation) of a group G if 1t (gh) = 1t(g)1t(h) (g, h E G). The space H is called the space of the representation and its dimension is denoted by dim 1t. If this space is

finite-dimensional, then the representation also is called finite-dimensional. By choos

ing an orthonormal basis fl' ... ,fn in H and expanding 1t (g)h in this basis, we obtain

dimlt

1t(g)h = L 1tij(g)fi· i=1


Thus, the matrix (1tij (g) )~~:; is associated with the unitary representation 1t (g). The

elements of this matrix are continuous functions 1tij(g) = (1t(g)fj,J;)H satisfying the equations

dimlt

1tij(gh) = I. 1tik(g)1tkj(h). k=1

( )dim It Since 1t (g) is unitary, the matrix 1ti;Cg) i,j=1 is also unitary, i.e.,

Further, it easy to see that

dimlt

L 1tij(g)1tkj(g) = j=1

dimlt

L 1tij(g)1tjk(g-t) = &i k' j=1

The trace of the matrix of a finite-dimensional representation is called the character

of this representation and is denoted by

Since

dimlt

Xlt(g) = I. 1tii(g)· i=1

we conclude that the character of a unitary representation is a continuous function on G which is constant on conjugacy classes.

We say that representations 1t(1)(g) and 1t(2)(g) ofagroup G in the spaces HI

and H 2, respectively, are equivalent (unitary equivalent) if there exists a linear (uni-

tary)operator A: HI -7H2 with abounded inverse A-I such that 1t(2) = A1t(I)A- 1•

If representations 1t (I) (g) and 1t (2) (g) are equivalent (1t (1) - 1t (2), then, for properly

chosen bases, their matrices coincide. Indeed, if we choose a basis {fk} in the space

HI of the representation 1t ( I ), then it suffices to choose the basis {A f k} in the space

H 2 of the representation 1t (2 ).


It follows from the properties of the trace of a matrix that the character of a representation does not depend on the choice of a basis in the space of this representation. This implies that the characters of equivalent representations coincide.

A subspace HI of the space of a representation 1t (g) is called invariant if, for any <p E HI' we have 1t (g)<p E HI for any g E G. Representations that have no nontrivial invariant subspaces (i.e., other than the null space or the entire space of a representation) are called irreducible. Otherwise, representations are called reducible.

The representation 1t (g) of a group G in the Hilbert space H = ®f=l H k (N::; 00 )

given by the equality

N

1t(g)<p = L 1t(kl(g)<Pk k=l

(this representation is obviously reducible) is called the direct sum of unitary representa

tions 1t(k)(g) of the group G in the spaces Hk and is denoted by 1t(g) = ®f=l1t(k)(g).

A representation 1t (g) is called completely reducible if it can be decomposed into a direct sum of irreducible representations.

All finite-dimensional unitary representations are completely reducible. Indeed, if HI is an invariant subspace of a unitary representation 1t(g), then, by virtue of the

equality (<p,1t(g)'V)H=(1t(g-l)<p''V)H =0 (<pEHl' 'VE Hf, gE G), the sub

space H 2 = Hf is also invariant. Since the restrictions 1t (1 land 1t (2 l of the repre

sentation 1t (g) onto the subspaces HI and H 2' respectively, are also unitary, we get

1t(g) = 1t( 1 leg) ® 1t(2 l(g). By repeating this procedure, after finitely many steps, we arrive at irreducible unitary representations.

• This statement remains true for infinite-dimensional unitary representations of com

pact groups, i.e., every unitary representation can be decomposed into a direct sum of finite-dimensional irreducible representations (see, e.g., [Nai2]). In particular, this implies that irreducible representations of compact groups are finite-dimensional. Obviously, in the space of any finite-dimensional unitary representation, one can choose a basis in which the matrix ofthis representation takes a block-diagonal form, namely,

o

o


Here, the matrices of irreducible representations 1t( a k) (g) of the group G are repeated

on the main diagonal Mk times, where M k is the multiplicity of 1t(ak )(g) in the de

composition of the representation 1t (g) into irreducible representations. This decomposition can be written as follows:

( 1.3)

where all ak are different, i.e., the representations 1t(ak) are pairwise nonequivalent.

Denote by G the set of equivalence classes of irreducible unitary representations of

a group G. Let Xa be the character of a representation 1t(a) E {; and denote d a = dim 1t (a). Below, for simplicity, we write (J E {; instead of 1t (a) E {;. It follows from (1.3) that

X1t(g) = M1Xa (g) + ... + MnXa (g). I n

(1.4)

If a group G is compact, then decomposition (1.3) is unique in the following sense:

If, in addition to (1.3), the representation 1t (g) admits a decomposition

where 1t"('tk) are pairwise nonequivalent, then there exists a permutation (0 of the

set {I, ... , n} such that n = m, 't k = a ro(k)' and N k = M ro(k). Before proving this statement, we establish several properties of unitary representations of compact groups.

A unitary representation 1t is irreducible if and only if each bounded operator A

commuting with all 1t (g) (g E G) is a multiple of the identity operator. Indeed, let A1t(g) = 1t(g)A for any g E G and let the representation 1t(g) be irreducible. Then

1t(g)A * = (A1t(g-I»)* = (1t(g-I)A)* = A *1t(g) and, since any bounded operator A

can be represented in the form A = A I + iA2' where A I = (A I + A') /2 and A 2 = (A - A ') /2, it suffices to prove this statement for self-adjoint operators. Let A be a self-adjoint operator. Consider its spectral decomposition

.A = f AdE(A).

Then E(A)1t(g) = 1t(g)ECA), whence, by virtue of the irreducibility of 1t(g), we get

either E(A) = 0 or E(A) = I. Since the function (E(A)<p, <P)H is nondecreasing, one

can find a real number AO such that E(A) = 0 for A < AO and E(A) = I for A ~ AO. Consequently, A = Ao!. Let us prove the converse statement by contradiction. Let HI

be an invariant subspace of the space ofthe representation 1t (g). Denote by P the op-


erator of projection onto H l' Since Hf- is also invariant, we have n (g)P = Pn(g),

i.e., the operator P commutes with neg). This implies that either P = 0 or P = I, and,

hence, the representation n (g) is irreducible.

• The well-known Schur lemma (see, e.g., [Nai2]) appears to be quite useful in the

theory of representations. We present this lemma without proof:

Let n (1) (g) and n (2) (g) be finite-dimensional irreducible representations of a

group G in the spaces HI and H 2, respectively, and let an operator A : HI ~

H 2 be permutable with these representations, i.e., An(l leg) = n(2)(g)A (g E G).

Then either A is the zero operator or it has the inverse operator (and, consequently,

the representations n(l) and n(2)(g) are equivalent}. In this case, A is uniquely defined (up to multiplication by a scalar).

A representation neg) = n(l)(g) Q9n(2)(g) in the space HI Q9 H2 is called the

tensor product of representations n(l leg) and n (2)(g) in the spaces HI and H 2,

respectively. If {in and {ff} are bases in the spaces HI and H 2, respectively,

then the matrix of the representation n(l leg) Q9n(2)(g) in the basis fi(l) Q9 fF) has

the form n ij, km = n)l) (g) n)~ (g). It easy to see that the character of the tensor product

of two finite-dimensional representations is equal to the product of their characters

Let us prove the so-called orthogonality relations for matrix elements of irreducible

representations of a compact group G, i.e.,

J nija) (g)n)!t) (g) dg = 0,

G

if n(a) and n(ll) (cr, f..l E G) are not equivalent and

1 = -0' kO.[. d I, 1. a

J nija)(g)n~7)(g)dg G

To prove (1.5), we consider the matrix

Aij = J 1t(a)(g)E jj i- Il )(g-1 )dg

G

(1.5)

(1.6)


where Eij is a matrix unit, i.e., all elements of this matrix are equal to zero except the

element e ij which is equal to 1. Obviously, n(a)(g)Aij =AijItIl)(g) (g E G). By

virtue of the Schur lemma, this means that the matrix A ij is equal to the zero matrix. The elements of this matrix are integrals of the form

J n~~lcg)nt1(g-1)dg = J n~~)(g)nW(g)dg, G

whence we arrive at (15). To prove (1.6), we consider the matrix

Since A ij commutes with the matrix of the representation n (a)(g), by using the Schur

lemma, we conclude that Ai) is a scalar matrix, i.e., Aij =1..1. Consequently,

J n~~)(g)n~j)(g)dg = AOkp' G

where A does not depend on k and p. In order to find A, we note that Tr Ai) = d at...

On the other hand, this trace is equal to the trace of matrix E ij' i.e., to 0 ij"

• Let us prove the uniqueness of decomposition 0.3) stated above. It follows from the

orthogonality relations (1.5) and (1.6) that

J Xa(g)XIl(g)dg = {I, G 0,

if n( a) - n(ll) ,

otherwise. (1.7)

Here, the representation n(a) is written in the matrix form by the formula G 3 g H

(nij(g))~~:~. Obviously, if n(a) is irreducible, then n(a) is also irreducible. This im

plies that if the representation n (g) can be decomposed into a sum of irreducible representations by formula (1.3), then it follows from (1.4) that

if (j coincides with (j j'

otherwise. (1.8)

Since the character of the representation X It depends only on the equivalence class

of the representation n, it follows from relations (1.8) that (j 1, ... , (j nand M l' ... , M n


are determined by the equivalence class which contains the representation 1t. This proves uniqueness of decomposition (1.3).

• 1.4. Peter-Weyl Theorem. In this subsection, we show that, up to a constant, the

characters of the hypercomplex system ZLI (G) coincide with the characters of irreducible representations of the group G and establish fundamental statements of the theory of Peter and Weyl.

Theorem 1.1. The set of characters X (p) of the hypercomplex system ZL1 (G)

is in one-to-one correspondence with the set of characters Xa(g) of irreducible uni

tary representations of the group G. Moreover, these characters satisfy the relation

X(p) = -j-Xa(g) (gE p, X(p)E X, crE G). a

(1.9)

Proof. Let cr E G. Since the function X a (g) is constant on conjugacy classes of

the group G, we can introduce the function

1 X(p) = dXa(g) (g E P E Q)

a

defined on the set of conjugacy classes. Let us show that X (p) is a character of the hy

percomplex system ZL1 (G). It is obvious that I X (p) I ::; 1 (p E Q). Prior to checking equality (1.6) from Chapter 1, we show that, for all x E Z L 1 ( G),

tik = f x(g )1t~f)(g)dg = 0ik -j- f x(g )xa(g) dg GaG

( 1.10)

(i,k =l, ... ,da ),

where 1t~f)(g) are matrix elements of the irreducible unitary representation 1t(a)(g).

Indeed, consider the matrix T = (tik )1"=1. Since x(gh) = x(hg) (x E ZL1 (G)) al

most everywhere on G x G, by virtue of unimodularity of the group G, we obtain

L tik1t~(P(g) k

= f x(a) L 1t~f)(a) 1t~)(g)da = f x(a)1t~a)(ag)da G k G

= f x(ag- I )1t~a)(a)da G

= f x(g-I a)1t~ja)(a)da G

Section 1 Centers of Group Algebras of Compact Groups

= J x(a)nijcr)(ga)da = J x(a) L n~f)(g) n~j)(a)da G G k

= L n)f)(g)t k),

k

175

whence Tn(cr)(g) = n(<J)(g)T. Since the representation n(<J)(g) is irreducible, we

have T = 'AI. This yields (1.10). By virtue of (1.10), for all A, B E 'Bo(Q), we obtain

J c(A, B, r)x(r)dr

Q

= : J Kl2r(a)Xcr(a)daT J Ksn(g)Xcr(g)dg = X(A)X(B), cr G cr G

where ~r and Q3 are preimages of the sets A and B, respectively, under the natural mapping G 3 g ~ P E Q (g E p).

Conversely, let X (p) be a character of the hypercomplex system ZL 1 (G). We re-

gard it as a function on G (note that X (g h) = X (hg) and X (g-I) = X(g» and consider an integral operator

Since

L 2 (G, dg) 3 <peg) ~ (K<p)(g) = J X(gh- I )<p(h)dh.

G

(K<p, 'I1)L 2 = f f X(gh- I )<p(h)\\f(g)dhdg

G G

= f <p(h) f x(hg-I)\\f(g)dgdh = (<p, K\\f)L2'

G G

the operator K is self-adjoint. Since I X (g) I :::; 1 and the group G is compact, the operator K is compact. Let 1.1 *- 0 be one of its eigenvalues and let <p 1 (g), ... , <p ,lg)

be an orthonormal basis of the corresponding eigenspace H 11' Note that the functions


<pj(a) are continuous because <Pj = (l!~)<Pj * X. Since <peg) E Hll and <p(ga) E Hll

(a E G), we have

<pj(ga) = L 1tjk(a)<Pk(g) (a, g E G; j = 1, ... , n). (1.11) k

It is easy to see that the matrices 1t (g) = (1tik (g) )j,k=l form a unitary representation of

the group. Indeed, in view of the invariance of the Haar measure,

1tjk(a-1) = J <pj(ga- 1 )<Pk(g)dg = J <Pk(ga)<p /g)dg = 1tk/a) , G G

1tjk(gh) = J <pj(agh)<Pk(a)da = J <p/ah)<pk(ag-1)da

G G

n n

= L 1tjp(h) 1tkq(g-l) J <pp(a)<pq(a)da = L 1tjl(h) 1t lk(g)· p,q=l G l=1

It follows from the continuity of the convolution <P * \j1 (g) (<p, \j1 E L2 (G, d g») with

respect to g and the equality 1t j k( a) = <P1: * <P j(a) (<p1: (g) = <Pk (g-I») that the func

tions G:3 a f---7 1t j i a) are continuous. The irreducibility of 1t is proved somewhat

later. Let us show that X (p) = d;1 X1t(g), where X 1t(g) is the character of the repre

sentation 1t. Let 6 * X be any other character of the hypercomplex system ZL1 (G). For any <P (g) E H 11' in view of the equality X * 6 = 0 and the fact that the characters

of the hypercomplex system ZL1 (G) are Hermitian, we obtain

J <peg )6(g)dg = ~ J J X(gh- 1 )<p(h)dh6(g)dg

G ~GG

= ~ J <p(h)J X(h- 1g)6(g-1)dgdh ~ G G

= ~J <p(h)(X*8)(h- 1)dh = O. ( 1.12) )lG

By virtue of X(gh) = X(hg) (h, g E G), it is not difficult to show that every <Pj (ag) regarded as a function of g belongs to H w This, (1.11), and (1.12), enable us

to conclude that


f Xn(g)8(g)dg = L f f Cf'/hg)Cf'j(h)dh8(g)dg G j G G

= L f f Cf'j(hg)8(g)dgCf'j(h)dh = O. j G G

177

(1.13)

Since the characters of the hypercomplex system ZL, (G) form a complete system in

L2(G, dg), it follows from (1.13) that Xn(g) = CX(g). At the same time, Xn(e) = dim1t and x(e) = 1. Consequently, Xn(g) = dim1tx(g).

Let us show that the representation 1t is irreducible. Indeed, we assume the opposite

and decompose 1t into a direct sum of irreducible representations as follows:

This gives

1 1 v X(g) = -d·-Xn(g) = -d'- L MkXcr (g).

Im1t Im1t k=' ..

By virtue of the first part of the theorem, the functions

are characters of the hypercomplex system Z LI (G) but this leads to a contradiction

since the characters of the hypercomplex system ZL1 (G) are orthogonal. The established correspondence is one-to-one in view of the orthogonality of the

characters of the group G and the same property of the characters of the hypercomplex

system ZL1 (G).

• As a consequence of this theorem, we establish fundamental results of the theory of

Peter and Weyl (see, e.g., [Nail]).

Since characters of an arbitrary hypercomplex system with compact basis form an Asystem of functions, by virtue of Theorem 3.8 in Chapter 1 and relations (1.8) and (1.9), the Plancherel measure on the set of classes of equivalence of irreducible representations is given by the formula

(1.14)


(here and in what follows, instead of m (1t (<J), c(1t (<J), 1t (J.l), 1t Ct), etc., we write m (cr),

c(cr, J.l, -r), ... , respectively).

The system of irreducible unitary representations 1t (<J)(g) (cr E G; g E G) is

complete, i.e., for any g 1 *" g2. there exists a representation 1t (<J) such that

1t(<J)(gl) *" 1t(<J\g2) (the Gelfand-Raikov theorem for compact groups). This statement follows from Theorem 3.1 in Chapter 1.

Denoteby TrigG the linear span of {1t&<J)(g)lcrE G, i,j= 1, ... ,d<J}. By de

composing the tensor product 1t(<J)(g) 01t(J.l)(g) (cr, ~ E G) according to (1.3), we

conclude, in view of the fact that (1t(<J\g) 01t(J.l)(g») ij,kl = 1t&<J\g) 1t<t)(g), the lineal

Trig G is an algebra with respect to the ordinary operations of summation and pointwise

multiplication of functions. It is called the representing algebra of the group G.

In view of Theorem 3.13 from Chapter 1, Trig G is dense in C(G) in the uniform

norm. We can now prove the main Peter-Weyl theorem which completely describes the

structure of L2 (G, dg).

Theorem 1.2. Let 1t (<J)(g) (cr E G) be a complete system of pairwise inequiva

lent irreducible unitary representations of a compact group G. Then the functions

{d;; 1t~?) (g) (i, j = 1, ... , d <J; cr E G) form an orthonormal basis of the space

L 2(G, dg). Every function fE L 2(G, dg) can be expanded into a convergent (in

mean square) Fourier series with respect to the functions 1t&<J\g) , namely,

da

f(g) = L L aij<J) 1t~?)(g). <JEG i,j=l

Its coefficients are given by the formulas

aij<J) = d<Jf f(g)1t&<J)(g)dg

G

and the Parseval equality is true

( 1.15)

( 1.16)

(1.17)

Proof. The orthonormality of the system of functions {d;; 1t~y)(g) follows from


the orthogonality relations (1.5) and (1.6). The completeness of this system of functions

follows from the denseness of Trig {; in C ( G) in the uniform norm, the fact that C ( G) is dense in L 2 ( G, d g), and the fact that the Haar measure is finite. The other statements follow from the elementary theory of Hilbert spaces.

• Let us now present several properties of characters which are used in what follows.

Assume that x E Ll (G, dg) and denote

x~ (g) = f x(aga- 1 )da.

G

Clearly, x ~ E ZL 1 (G). It is not difficult to show that

f X(g)x(g)dg = f X(g)x~(g)dg = f X(p)xlf(p)dp, (1.18)

G G Q

X(h)X(g) = f x(haga-1)da (g, h E G) (1.19)

G

for any character X of the hypercomplex system ZL1 (G). Indeed, it suffices to prove

(1.19). For any x, YELl (G, dg), we have

f x(g)X(g)dg f y(h)X(h)dh

G G

= f x~ (p)X(p)dp f ylf (q)X(q)dq Q Q

= f (x~ *y~)(p)X(p)dp Q

= f X(g) f x~ (h)y~(h-l g)dhdg

G G

= f f f X(g)J x(aha-l)y(bh-Igb-I)dhdadbdg

G G G G

= f f X(g)x(h) f J y(ba- 1 h- I gab-1)dadbdgdh

G G G G

= f f X(g)x(h) f y(a- 1 h- l ga)dadhdg

G G G

= J J J x(haga-I )dax(h)y(g)dhdg. G G G


Relation (1.19) holds for almost all x and y owing to the arbitrariness of g, h E G. By virtue of the continuity of X(g), this implies (1.19) for all g, h E G.

• It is not difficult to understand what modifications should be made in this reasoning

to obtain the converse assertion: If a function f E L1 (G, d g) sati:;fies the functional

equation (l.19) for all g, h E G, then either f= 0 or f(g) = x(g) for some character X (g) of the hypercomplex system ZL\ (G).

Let us now describe the dual hypercomplex system. By virtue of Theorem 1.1 and the fact that every character of the hypercomplex system with compact basis is essential,

it is possible to consider the set G of equivalence classes of unitary representations of

the group G as the basis Q = X h of the dual hypercomplex system. Since Q is a

compact set, G is discrete. We can find the structure measure c ( a, Il, 't) (a, Il, 't E A

G) of the dual hypercomplex system by using relation (3.6) from Chapter 1. Indeed,

By virtue of (1.4),

Xa(g)x~(g) = X1t(a)®n;(loL)(g) = L M(a, J.L, 't)X1:(g)· (1.20) 1:

Taking (1.8) into account, we conclude that

ifthe representation n(1:)(g) belongs to expansion (1.3) of n(a)(g) EB n(J..L)(g) into ir

reducible representations; otherwise, c( a, Il, 't) is equal to zero. Thus, c( a, /l, 't) 2 0 and its finiteness with respect to 't is a consequence of (1.3). It follows from (1.20) and Corollary 3.1 in Chapter 1 that the product of any two positive definite continuous func

tions on the hypercomplex system Z L 1 ( G) is positive definite. As already mentioned (see (1.14)), the multiplicative measure of the dual hypercomplex system is given by the

equality mea) = i; (a E G). By virtue of Theorem 3.5 in Chapter 1, duality takes

place for the hypercomplex system ZL\ (G), i.e., the support of the Plancherel mea

sure of the dual hypercomplex system is homeomorphic to the compact set of conjugacy

classes of the group G.

1.5. Tannaka-M. Krein Duality Theorem. As is known, the set G of irreducible unitary representations of an Abelian locally compact group G coincides with the group of characters of the group G and, therefore, it is the object dual to the group G. If G


is a compact, generally speaking, noncommutative group, then G is not a group but a hypergroup. Nevertheless, according to the Tannaka-M. Krein theorem, it is possible to reconstruct a group if the tensor product of any pair of irreducible representations and the adjoint of each irreducible representation are known (in a certain sense). In the present subsection, we prove the Tannaka-M. Krein theorem by using the duality theorem for hypercomplex systems. To do this, we apply the results of Section 3 in Chapter 1, to the

hypercomplex system ZL1 (G) and establish the Tannaka-M. Krein duality principle for compact groups ([Tan], [Krel], and [Kre2]).

If G 3 g H (ni/g) )~~:~ is a unitary representation of G in the matrix form, then

the mapping G 3 g H (nij(g»)~~:~ is also a unitary representation of G. Consequent

ly, together with any function, the representing algebra Trig G contains its complex conjugate.

Since every finite-dimensional unitary representation is equivalent to a direct sum of finitely many irreducible representations, the matrix elements of irreducible representa-

tions n~a\g) (aE G; i,}==l, ... ,da ) form a basis in TrigG. In particular, by ex-

panding the tensor product n(a) ® nOll of any two irreducible representations according

to (1.3), we obtain the matrix equality

( 1.21)

where Sail is a unitary numerical matrix which does not depend on g. The character of

representation n(a) ® n(ll) is computed according to (1.20). By integrating (1.20), we

get

k I Xa(g)XIl(g)dg == L M(a, 11, 't i ) I X1)g)dg. (1.22) H i=1 e

By virtue of (1.7), the left-hand side of (1.22) is equal to 1 or 0 if the representation

n (Il) is equivalent or not equivalent to the representation n(a), respectively. The inte

gral Ie X't/g)dg is equal to 1 or 0 if the representation n('t;)(g) is trivial (one-di

mensional and identically equal to one) or nontrivial, respectively.

Thus, the multiplicity of the trivial representation in decomposition (1.21) is equal

to zero if n(a) is not equivalent to n(ll) and is equal to one whenever n(a) - n(Il).

A commutative algebra ~ with identity e and involution * is called a Krein alge

bra (or a square block algebra according to M. Krein) if there exists an at most count

able basis which can be split into disjoint sets (blocks) U a with d; elements (d a is

called the rank of a block) which have the form of square matrices Ua = (uija))ti=1

such that following conditions are satisfied:


1. The collection of sets U 0 contains a set with a single element e.

2. Each block Vo is associated with a numerical matrix So of order do such that

the set S~l V;Sa' where V; = (uija)*)l,i=l' is also a block from the indicated collec

tion of sets.

3. For any two blocks Va and VIl of orders da and d ll , respectively, there exists a unitary numerical matrix Sail of order da xd ll such that

( 1.23)

4. The block {e} either does not belong to the right-hand side of (1.23) or appears

there exactly once; the latter takes place if and only if V Il = S~ I V; S a' where S a is a unitary numerical matrix.

5. Each block Va = (uija))l,i=l satisfies the condition

(1.24)

where 0 i, k is the Kronecker symbol and U J is the relevant transposed matrix. Note that relation (1.23) defines, in fact, the operation of multiplication of basis elements of the algebra 2t (and, hence, of its arbitrary elements).

The reasoning presented above demonstrates that the representing algebra Trig G of a compact group G is a Krein algebra.

Every element <p of the Krein algebra 2t can be represented in the form

(1.25)

where each Fa is a square matrix of order da and da is the order of the block '-h (the sum in (1.25) is finite). This fact enables us to introduce a norm in 91 by the formula

( 1.26) a

(recall that I Fa I = ~ FaF;). It is not difficult to show that 91 with norm (1.26) is a

normed space (not necessary complete). Let us prove that 91 with this norm is an invo-

Section 1 Cente rs of Group Algebras of Compact Groups 183

lutive normed algebra, i.e., II <P'l' II ::; II <P 1111 'I'll and II <p*1I = II <P II for all <P, 'I' E%'(.

It suffices to establish these relations for <P = Tr (A Uo ) and 'I' = Tr (B U~). By using (1.23), we obtain

p

= L Tr(CkUO)'

k=!

where C k (k = 1, ... ,p) are matrices composed of elements of the matrix C = So~ (A ® B)S;;~ and located in C at the same places as the matrices UOK in the sum

p 87 U o. Taking into account the fact that the set of matrices U OK may contain equal

k=! K

matrices, we obtain

II <P'l' II ::; f Tr I Ck I = Tr Il~! Ck I ::; Tr I C I k=!

= Tr I A ® B 1= Tr I A I Tr I B I = II <P 1111 'I' II

(the proof is based on the use of the well-known properties of traces).

In order to check the equality II <p*1I = II <P II, we note that <p* = Lo Tr (XaU;). According to condition 2 in the definition of the Krein algebra, every block U 0 is asso

ciatedwithablock U~ suchthat U; = S~U~S~I. As a consequence <p* can berepre

sen ted in the form

<p* = L Tr (S~IXaS~UJ.l)' o

Consequently,

11<p*1I = L Trl S~!AoS~1 = 11<p11· o

• By completing the Krein algebra ~I with respect to norm (1.26), we arrive at a com

mutative Banach *-algebra ffi(~I) which is called a complete Krein algebra. The alge

bra ffi (%'() can be regarded as a set of infinite formal sums (1.25) for which series (1.26)


is convergent. The Krein algebra 2! is a dense part of m( 2!) and consists of sums (1.25) with finitely many terms.

Let us show that the involutive Banach algebra ffi(2!) is symmetric. To do this,

it suffices to prove the equality g ( V;) = g(V cr)' where Vcr is an arbitrary block and

- = ( ( (cr»))da g(Vcr ) gUy i,j=I'

for every maximal ideal g of the algebra ffi (~o. Indeed, the ideal g satisfies the equal

ities g(e)=1 and Ig(<p)I~II<p1I (<pE ffi(2!»). Applying the last inequality to the el

ement <p = Tr (A Vcr) and using the known fact that Tr (AB) ~ Tr IA I· II B II (see, e.g.,

[ReS]), we get II g(Vcr ) II ~ 1. Further, by virtue of condition 5 in the definition of a

Krein algebra, we have V crV;T = (Oi,je )1,i=I' whence g (Vcr )-1 = g (V;) T. Conse

quently, by virtue of condition 2,

This implies that the matrix g ( Vcr) is unitary and, hence, g ( V;) = g ( ( Vcr /) -I =

g(Vcr ) .

• A complete Krein algebra generated by the representing algebra Trig (; of a com

pact group G coincides with the Fourier algebra A (g) of the group G (see, e.g., [Eym] and [HeR]); moreover, it also coincides with the linear span of the set of continuous positive definite functions on G.

Let us introduce the notion of a representing group of a Krein algebra 2!. A functional g (a) (a E ~I) is called elementary if it is a symmetric homomorphism of the algebra 2! in the field of complex numbers, i.e., if g(ab) = g(a)g(b) and g(a*) =

g( a) (a, b E 2!). Every square block Vcr = (uij cr) )1,i= 1 with elements from 2! can be

associated with a numerical matrix

where g is an elementary functional. The matrices g ( Vcr) are unitary by virtue of (1.24) and satisfy the same conditions (1.23) as the blocks themselves. Conversely, it is

obvious that every mapping of the blocks Vcr into a set of unitary matrices of the same

orders dcr that satisfy conditions (1.23) defines an elementary functional. Therefore, we

can introduce a group operation in the set G?l of all elementary functionals of the Krein

algebra 2!. Namely, we set (gh)(Vcr) = g(Vcr)h(Vcr ). It is obvious that relations

(1.23) are invariant under the mapping Vcr f---7 gh(Vcr) and, therefore, it generates an


elementary functional. One can now easily check that G~ is a group. The group G~(

is called the representing group of a Krein algebra 2{.

There is a one-to-one correspondence between the set of elementary functionals G ~

and the set of maximal ideals of the complete Krein algebra Dl(2{). Indeed, since the algebra Dl( 2{) is symmetric, it is clear that any maximal ideal is an elementary func

tional. Conversely, by virtue of the inequality 1 Tr AB 1 $; II A II . Tr 1 B I, we have

for any <p E 2{ and g E G~. Since 2{ is dense in Dl(2{) we conclude that the func

tional g is bounded and, furthermore, II gil$; 1. Thus, it is a maximal ideal of the complete Krein algebra Dl (2{ ).

• Note that the group G 2( is compact in the topology ofthe space of maximal ideals.

As mentioned above, the representing algebra Trig G of a compact group G is a

Krein algebra. Let G denote the representing group of the Krein algebra Trig G. Since the mapping Trig G 3 f H f (g) (here, g is a fixed element of the group G) is

an elementary functional, we conclude that G c G. It is clear that G is a closed sub-A A

set of G by the definition of topology in the group G. Let G 3 g H 1t(g) be a finite-dimensional unitary representation of the group G.

The correspondence {; 3 g H 1t( g) = g(1t) = (g(1tij))1.j~f = (1tij(g»)~T=!r associ

ates every g E G with the unitary matrix (1tij(g»)~~~f. Obviously, this mapping is a A

unitary representation of the group G. If the representation 1t (g) of the group G is ir-

reducible, then the representation 1t ( Ii) of the group G generated by the representa

tion 1t (g) is also irreducible. Note that representations of the group G generated

by all irreducible unitary representations 1t (O)(g) (0' E G) of the group G form a

complete system of representations of G. Indeed, let g J "* g2; this means that there exists a unitary representation 1t (g) of the group G such that g I (1t) "* g 2 (1t). By decomposing 1t (g) into irreducible representations according to (1.3), we establish that

gJ(1t(O)"* g2(1t(G) at least for one O'E G.

• Below, we present the well-known Tannaka-M. Krein duality theorem (see [Tan] and

[Krel], [Kre2]).


Theorem 1.3.

(i) If G is a compact group and Trig G is its representing algebra, then the re-A

presenting group G of the algebra Trig G is isomorphic to the group G.

(ii) If 2t is a Krein algebra and G 2! is its representing group, then there exists

an isomorphism between the algebra 2t and the representing algebra

Trig ~ of the group G W. This isomorphism associates the block-basis U fJ

of the algebra 2t with the complete system of irreducible unitary representa

tions of the group G 2! .

A A

Proof. Assertion (i) follows from the fact that the set G of elementary functionals

of the algebra Trig G is homeomorphic to the space of maximal ideals of the Fourier algebra A (g), which is homeomorphic to the group G (see, e.g., [Eym] and [HeR]).

We prove assertion (ii) in several steps.

1. Let 2t be a Krein algebra and let L denote the set of indices cr of the blocks U fJ

equipped with discrete topology. Each block U fJ (cr E L) generates an irreducible unitary representation of the group

Gw according to the formula n (fJ)(g) = g(UfJ)' To prove this it suffices to show that

the representation n(fJ) is irreducible. Assume the contrary. Then one can find a

bounded operator A such that A"* AI and An(fJ)(g) = n(fJ)(g)A. For any gE G 2X,

i,j = 1, ... ,d fJ , we have

Since G 2( coincides with the space of maximal ideals of the Banach algebra ~ % the last equality yields

L. (aikukj) -akjU~)) = 0 k

for all i, j = 1, ... , d (J" Since A "* AI, it follows from these equalities that the vectors

U&fJ) E U fJ (i, j = 1, ... , dfJ) are linearly dependent. On the other hand, as elements of a

basis of the algebra 2t, they must be linearly independent.

2. The system of irreducible unitary representations n (fJ)(g) = g (U fJ) (cr E L,

g E Gw) of the group G w is complete, i.e., for every g I "* g2' one can find cr E L

such that n(fJ)(gl) "* n(fJ)(g2)' This statement obviously follows from the facts that


G 2( coincides with the space of maximal ideals of the Banach algebra ~ (~) and the

elements u&cr) E Vcr (crE:E, i,j = 1, ... ,dcr) form a basis in ~(~). By using the results of steps 1 and 2, one can establish that there is an isomorphism

between the Krein algebra ~ and a certain subalgebra ~' of the representing algebra

Trig ~ of the group G2( and, furthermore, this isomorphism associates the block

basis Vcr of the algebra ~ with the complete system of irreducible unitary representa

tions of the group G 2(. To complete the proof it suffices to verify that ~' coincides

with Trig G2(. We prove this fact by using the duality theory for hypercomplex systems.

3. We set c(cr, Jl, 't) = M(cr, Jl, 't)dcrdll(d,)-I, where M(cr, Jl, 't) is the multi

plicity of the block V" in expansion (1.23) of the product Vcr Q9 VIl in the "block-basis"

(M(cr, Jl, 't) = 0 if the right-hand side of (1.23) does not contain V,,). Also denote

m ( cr) = d;. One can easily verify that c ( cr, Jl, 't) is a structure measure and m ( cr) is

a multiplicative measure. Denote the obtained hypercomplex system by ZI] (:E). We

introduce the involution :E 3 cr H cr*E :E in the hypercomp1ex system Z1 1 (:E), where

cr * is the index of the block S~1 V~S cr in condition 2 in the definition of Krein al

gebras. Obviously, the hypercomp1ex system Z/ 1 (:E) is normal and possesses the basis

unity (the index of the block Ve = {e}). It is easy to see that the hypercomplex system

Z1 1 (:E) is isometrically isomorphic to a closed sub algebra of the Banach * -algebra

~ (~) which consists of vectors of the form

where

Fcr = x(cr)dcr1cr' L I x(cr) I d; < 00

cr

According to the results obtained in [FeT], the hypercomplex system ZI] (:E) is sym

metric because ~ (~ ) . is symmetric and Z /1 (:E) contains the identity.

4. Denote by Q the space of characters of the hypercomplex system Z 11 (:E). Let

us show that Q is homeomorphic to the compact set Q 2( of conjugacy classes of the

group G 2(. Indeed, assume that g ( cr) (cr E :E) is a character of the hypercomplex sys

tem Z/](:E). As indicated above, ZI 1(:E) is isometrically isomorphic to a closed

subalgebra of the Banach algebra m (2l). Therefore, according to the Shilov theorem on

the extensions of maximal ideals [Shi], the character g (cr) can be extended to a maxi

mal ideal g E Gjl{ of the algebra ~(~) and, obviously,

1 1 g ( cr) = - Tr g ( Vcr) = -d X cr (g ),

dcr cr (1.27)


where Xu is the chara~ter of the irreducible representation 1t(U) of the group Gilt cor

responding to the block U U' We associate the character g ( cr) with the conjugacy class

p of the group G \}{ that contains the element g. It follows from (1.27) that p does not

depend on the choice of g.

Conversely, let p E Q~{. We choose an arbitrary element g E P and define a char

acter g ( cr) of the hypercomplex system Z 11 (L) by relation (1.27); the fact that g ( cr)

thus defined is a character follows from (1.27), the definition of c(cr,~, t), and condi

tion 3 in the definition of Krein algebras. Obviously, g (cr) does not depend on the

choice of g E p. Thus, there is a correspondence between the space of characters Q of

the hypercomplex system Z II (L) and the compact set Q ~( of conjugacy classes of the

group G\}{. This correspondence is bijective by virtue of (1.27) and the completeness of

the system of irreducible representations 1t (a) of the group G Il( generated by the

blocks U u (cr E L). Since Q and Q ~r are equipped with the same topology as the

spaces of maximal ideals of the algebra m (2:( ), we conclude that this correspondence is a homeomorphism.

5. Let us show that duality takes place for the hypercomplex system ZII (L). First,

note that the measure in on the space Q Il{ of conjugacy classes of the group G ~( gen

erated by the Haar measure of G 2t according to (1.1) coincides with the multiplicative

measure of the hypercomplex system dual to Z II (L). Since supp m = Q ~1' by virtue

of Theorem 3.7 in Chapter 1, it suffices to check that, for any g I' g 2 E Q, the product

g I (cr) g 2 ( cr) is positive definite. Let PI and p 2 be the conjugacy classes of the group

G \}{ that correspond to the characters gland g 2' Let g I E P I and g 2 E P 2 be arbitrary elements of these classes. Taking into account (1.27), (1.19), and the invariance of

the Haar measure of the group G ~(, we get

gl(cr)g2(cr) = X(gl)X(g2) = f x(glag2a- 1)da

~(

= f x(bglag2a-lb-l)da = f f X(bg[b- 1ag2a- 1)dadb,

~( Gw G)(

where X(g) = Xu(g)/da is the character of the hypercomplex system ZIl(G~{) cor

responding to the character Xa of the irreducible representation 1t (u) generated by the

block U a (cr E L). Let us define a measure d ~ P \' P2 (p) on Q by the equality

f f(p)d~pl'p2(p) = f f f(agla- 1bg2b- 1)dadb (fE C(Q)). Q ~(~(

Then, by virtue of Theorem 3.1 in Chapter 1, the product g [ ( cr) g 2 ( cr) of characters


£I 1 and £12 of the hypercomplex system ZII (I,) is positive definite.

It is obvious that the hypercomplex system Z II ( G 9() is dual to the hypercomplex

system Z/ 1 (I,).

6. Let us complete the proof of Theorem 1.3. Assume that 7t is an arbitrary irreducible unitary representation of the group G 9f and

1 X(p) = -d.-X1t(g) (gE p; pE Qw)

Im7t

is the character of the hypercomplex system Z1 1 (G w) that corresponds to the character

X 1t of the representation 7t. Since duality takes place for the hypercomplex system

Z II (I,), there exists a E I, such that X (p) = £I ( a) (here, 9 is the character of the

hypercomplex system Z/ 1 (I,) that corresponds to the class p E Qw). By virtue of

(1.27), we have X1t(g) = Xcr(g), where Xcr is the character of the representation 7t(cr)

of the group G 91 generated by the block U cr' This means that the representations 7t

and 7t(cr) are equivalent. Since the matrix elements of all irreducible representations of ~

the group G w form a basis of the representing algebra Trig Gw of the group G w, we

conclude that ~ I coincides with Trig G2r. •

1.6. Elements of the Theory of Semisimple Groups and Lie Algebras. Below, we describe the hypercomplex system Z/I (G) in the case where G is a compact semisimple Lie group. These results were obtained by Berezin and Gelfand [BeG] and Gelfand [Gell]. First, recall basic definitions and results concerning semisimple Lie groups and Lie algebras.

A set G is called a Lie group if it is a topological group and a smooth manifold

such that the mapping <p: G x G -7 G defined by the equality <p (x, y) = xy-I is

smooth. A linear space £I with a bilinear operation 9 x 9 3 (X, Y) 1--7 [X, Y] Egis called a Lie algebra if the following conditions are satisfied:

(i) [X, Y] = -[Y,X] for all X, YE g;

(ii) the. Jacobi identity

[[X, Y], Z] + [[ Y, Z], X] + [[Z, X], Y] = 0

holds for all X, Y, Z E g.

With every Lie group, one can associate a finite Lie algebra. Namely, denote by

A (g) (fJ E G) an inner automorphism of the group G; every inner automorphism is a


diffeomorphism of the group G. By 9 = TeG we denote the tangent space at the

neutral element e of the group G and consider the tangent mapping A d g = dA (g) ( e) :

9 --7 g. The mapping G 3 g ~ Adg is a homomorphism of the group G into the

group Aut 9 of linear invertible transformations of the space 9 and is called the ad

joint representation of the group G. Since the adjoint representation is a smooth mapping of G into Aut g, one can define the tangent mapping ad = dAde of the space 9

into the space End 9 of all linear transformations of 9 (which is obviously a tangent

space to Aut g). For every X, Y E g, we set [X, Y] = ad X (Y). One can easily verify that the tangent space 9 with such multiplication is a Lie algebra and the mapping 9 3 X ~ adX E End 9 is a representation of the Lie algebra 9 (it is called the regular

representation of the Lie algebra). Note that we can define a regular representation of

an arbitrary Lie algebra 9 by setting adX(Y) = [X, Y] (X, Y E g). For each X E g,

the mapping adX is a differentiation of the Lie algebra g, i.e., adX([Y, Z]) = [ad X (Y), Z] + [ Y, ad X (Z)] for every X, Y, Z E g. Note that, with every homomorphism of Lie groups, one can associate a homomorphism of their Lie algebras and with every Lie subgroup H of the Lie group G, one can associate a Lie sub algebra 1) c g. In this case, the sum of the Lie subalgebras corresponds to the product of Lie subgroups, the intersection of Lie subalgebras corresponds to the intersection of Lie subgroups, and ideals correspond to normal subgroups.

Every finite-dimensional Lie algebra uniquely (to within a local isomorphism) determines a local Lie group (roughly speaking, it determines a rule of multiplication in a certain vicinity of the neutral element of a Lie group). Moreover, every finite-dimensional

Lie algebra 9 can be associated with a unique connected simply connected Lie group G for which 9 is a Lie algebra. All connected Lie groups with this property have the

form G / D, where D .is a discrete normal subgroup which belongs to the center of the

group G. Let 9 be a Lie algebra over the field of the characteristic 0 and let B (X, Y) =

Tr (adX ad Y) (X, Y E g) be the Killing form on g. A Lie algebra 9 is called semi

simple if the Killing form B of the algebra 9 is nondegenerate. A Lie group is called

semisimlple if its Lie algebra is semisimple. Let 9 be a complex Lie algebra. A maximal Abelian sub algebra I) is called a Car

tan subalgebra if its adjoint representation I) 3 H ~ ad H is completely reducible in

g. Every semisimple complex Lie algebra contains a Cartan subalgebra. By I)' we denote the space dual to the complex space I) of a given Cartan subalgebra in g. For

every a E I)', we set

gIX = {XE gl [H,X]=a(H)X foraH HE I)}.

If gIX"# (0), then the functional a is called a root (of the algebra g with respect to

I) and gIX is called the root subspace. Obviously, I) = gO. By ~ we denote the set of nonzero roots. Any complex semisimple Lie algebra g can be represented as a direct sum


such that

(i) dimga = 1 (a E ~);

(ii). if a, ~ E ~, a + ~ * 0, then ga and g~ are orthogonal with respect to the Killing form;

(iii) the restriction of B to ~ x ~ is nondegenerate and, for every a E ~, there exists

a unique element Ha E ~ such that BCH, Ha) = a(H) (H E ~);

Let

The Killing form is real and positive definite on ~ 0 x ~ 0 and the Cartan subalgebra can

be written as a direct sum, i.e., ~ = ~o + i~o.

Since the restriction of the Killing form B to ~ x ~ is nondegenerate, for every

'A E ~/, there exists a unique element H'J.., E ~ such that 'A(H) = BCH, H'J..,)' Therefore,

on ~/, we can define a nondegenerate scalar product by the formula

For any 'A, Jl E ~/, we have

('A, Jl) = L ('A, a)(Jl, a) aE.6.

and (a, a) > 0 for a E d (since a E ~ ~ and the restriction of the Killing form to

~ 0 x ~ 0 is positive definite).

Let a, ~ E ~. Then the set of all roots of the form ~ + no. (n E :l) is called the 0.

series which contains the root ~. If a, ~ E d, then the a-series containing the root ~

has the form ~ + no. (p ~ n ~ q) and

-2 (a, ~) = p + q (0.,0.)


(obviously, p ~p + q ~ q). The only roots proportional to the root a are -a, 0, a. For any a E ~, there exists an element Ea E ga such that

[H,Eal=A(H)Ea (HE 1),

if a + B is a nonzero root,

if a + B :;t: 0 is not a root,

where N a, ~ = -N -a,-~ E IR. The set {Eal a E ~} is called the Weyl basis (modulo 1) ).

Let HI' ... ,Hr be a basis of the space 1) 0 and let F I , ... ,Fr be a basis of the space

1)~ dual to 1)0 and biorthogonal to HI,,,,, Hr (i.e., (F i , H» = 0i,) for all i,j =

I, .. , ,r). In the space 1)~, we introduce lexicographic ordering with respect to the basis

F1, .. , ,Fro namely, we say that F > E (E, F E 1)~) if F(H j) = E(H j ) for i = 1, ... ,

k and F (H k+ I) > E (H k+ 1)' It is clear that the space 1) ~ is completely ordered. A

root a E ~ is called positive if it is a positive element of 1)~. Obviously, a > 0 if and only if 0> -a and, therefore, ~ = ~+ U (-~+), where ~+ is the set of positive roots. A root a E ~+ is called simple if it cannot be represented as a sum of positive

roots. Let n be a space of all simple roots. Then n = {a l' ... , a r} forms a basis in

the space 1) ~ and every root B E ~ admits a representation in the form B = n) al + ... + nrar, where n j E ~ and either all n are nonnegative or all n j are nonpositive. The number

is called the order of the root B. Let cij =-2(aj, a)/(aj' a j> (ai' aj E n; i,j = I, ... , r). Then CijE ~ and

C jj ~ 0 for i :;t: j. The numbers C ij are called Cartan numbers and (cij )[,j=1 is called

the Cartan matrix. The set E a1 , ... ,Ea,., E_a1, ... ,E-ar, where akE nand Eak

are elements of the Weyl basis corresponding to simple roots, generates g.

For every A E 1)~, we define a mapping SA E GL(1)~) by the formula

Section 1 Cente rs of Group Algebras of Compact Groups 193

S A is the operator of reflection with respect to the hyperplane PA = {~E f) ~ I < t.., ~) = a}. Obviously, Sa~ E 11 for ~ E 11. The subgroup of O(f)~) generated by the set

{ sal a E 11} is called the Weyl group of the system 11 and is denoted by Ad 11. Since every element (Q E Ad 11 is an orthogonal transformation, its determinant det 0) =

± 1. Obviously, det Sa = -1. We set (f)~) - = f) ~ \ U P a' The connected compo-aELl

nents of the set (f)~) - are called Weyl chambers. Let IT = {a l' ... , a r} be the set

of simple roots. Then Wo = {~E q~l(al,~»O, ... ,<ap~»O} isaWeylcham

ber. The Weyl chamber Wo is called dominant for the system IT. The Weyl group

acts transitively on the set of Weyl chambers and the elements Sal"" , S ar are genera

tors of the group Ad 11. Every root ~ E 11 admits a representation

where ajp' aj E IT (p = 1, ... , k~ Let Autl1 = {S E O(f)~) I SI1 = 11} be the group

of automorphisms of the system 11 and let Aut IT = {S E 0 (1)~) I SIT = IT} be the group of automorphisms of the system IT. Then Aut 11 is a finite group and Aut 11 = Ad 11 XI Aut 11. The groups Aut g/ Ad g, Aut 11/ Ad 11, and Aut IT are isomorphic.

Let 1t (g) be a finite-dimensional representation of the Lie group G in a real or complex space V. Then the operator-valued function G 3 g H 1t (g) is differentiable

and its derivative d1t at the point e determines a representation of the Lie algebra g of the group G in the space V. If G is a connected simply connected Lie group, then every representation of g can obtained from a certain representation of G in this way (i.e., the problem of description of finite-dimensional representations of a connected simply connected Lie group reduces to a similar problem for its Lie algebra). Every finitedimensional representation of a connected semisimple Lie group is completely reducible.

Let 1t be a linear representation of a semisimple complex Lie algebra g in a space

V. Forevery t..E q', we denote VA = {VE VI1t(H)v=t..(H)v foraH HE q}. If

VA. "* (0), then t.. is called a weight of the representation 1t, the subspace V A. is called

a weight subspace, and the number n A = dim VA is called the multiplicity of the

weight A. Let

be subalgebras of g. A vector v E V is called a dominant vector of the representation 1t if v E VA. for some 'weight t.., 1t (n +)v = 0, and the least subspace of V which is


invariant under the action of all operators 1t (X) (X E 9) and contains the vector v coincides with V. The relevant weight A is called the dominant weight of the representation 1t. If A is the dominant weight of the representation 1t, then the weight subspace VA is one-dimensional and any weight !-l of the representation 1t admits a representation

r

!-l = A - I, mi(Xi

i=!

where (Xi E nand m i ~ 0 (whence, in particular, it follows that A is the maximal

weight in the collection of all weights of the representation 1t). Moreover, V = EB Vjl = 1t (U_) v, where U_ is a universal enveloping algebra of n -' Every finite-dimensional representation 1t of the algebra 9 possesses a dominant vector which is determined uniquely (to within multiplication by a scalar). The set of all weights of the representation 1t is finite and invariant under the action of the Weyl group Ad.1.. An irreducible representation is uniquely determined by its dominant weight. A linear functional A E f)' is the dominant weight of a certain finite-dimensional representation of the Lie alge

bra 9 if and only if all numbers 2 (A, a i) / (a j' a j) (a j En) are integer and nonnegative.

Let 9 be a complex Lie algebra. Obviously, we can consider 9 as a real Lie algebra \h~ whose dimensionality is twice as large. Moreover, there exists a complex

structure on 9IR, i.e., a linear mapping 1: ~ ~ 9IR such that 12 = -id and [X,

1Y] = 1 [X, Y] (X, Y E 9 IR)' We define a real form of a complex Lie algebra 9 as a sub algebra 90 c 9IR such that 9IR = 90 + 190' A mapping cr: X +iY H X - iY (X, Y E 90) is called the conjugation of a complex Lie algebra 9 with respect to its real form 90' On the other hand, for every real Lie algebra 90, it is possible to construct a

complex Lie algebra 9 = (90)<r' Indeed,weset 9 = {x+iYIX,YE go} withthe operation of commutation defined by the formula

[X+iY,Z+iT] = [X,Z] - [Y,T] + i([Y,Z] + [X, Tn (X,Y,Z,TE 90)'

The algebra (90) <r is called the complexification of 90' A real Lie algebra is called compact if it possesses an invariant (positive definite)

scalar product. A subalgebra f of the real Lie algebra 9 is called compactly embedded if there exists a f-invariant scalar product on 9. Obviously, a compactly embedded subalgebra is compact. In order that a real sernisimple Lie algebra be compact, it is necessary and sufficient that its Killing form be negative definite, i.e., B (X, X) < 0 for X 7:- O. Every sernisimple complex Lie algebra possesses a compact real form. Let 90 be a real semisimple Lie algebra and let 9 be its complexification. The decomposition 90 = f 0 + Po of a Lie algebra 90 into the direct sum of a subalgebra fo and a vector subspace Po is called the Cartan decomposition if there exists a compact real form u of the algebra

9 such that f 0 = 9 n u and Po = 9 n i u. The mapping 't: X + Y H X - Y (X E f 0 '


YE lJO) is an involutive automorphism of the algebra ~; [fo,fo] efo, [fo,lJo] e

lJo, and [lJo,lJo] efo· For any XE fo and YE lJo, wehaveB(X,X)<O (X=FO),

B(Y, y) > 0 (Y '* 0), B(X, Y) = 0, and fo + ilJo is a compact real form of the algebra g. The subalgebra f 0 is a maximal compactly embedded sub algebra of go.

Let go = fo + lJ 0 be a Cartan decomposition of the real semisimple Lie algebra go and let 9 be the complexification of go. By a we denote the maximal Abelian sub

space of lJ o. The dimension of a is called the real rank of the algebra go. Let 1) a

be a maximal Abelian subalgebra of the algebra go that contains a and let 1) be the subalgebra of 9 generated by 1) a. Then 1) is a Cartan sub algebra of g, and a is a

sub algebra of

We choose a basis Xl' ... ,X r in lb so that X b ... , X dima forms a basis in a, and

order 1) ~ lexicographically with respect to this basis. As a result, we obtain an order on

the set of roots 8. Let P_ = {aE 8+la=0 on a} and let P+ = 8+\P_. We construct the subalgebra

of the algebra 9 and denote no = go n n. Then n and no are nilpotent Lie algebras, and go can be decomposed into the direct sum of subspaces: go = fo + a + no-This

decomposition is called the lwasava decomposition.

A global version of Cartan and Iwasava decompositions is also true. Let G be a

connected semisimple Lie group with finite center and Lie algebra go. Let K be an analytic subgroup (i.e., connected Lie subgroup) of G with Lie algebra fo and let P be the image of the vector space lJ 0 under the exponential mapping. Then the set P K is everywhere dense in G. If G is a connected semisimple Lie group with Lie algebra go and if K, A, and N are the analytic subgroups of G that correspond to the subalgebras fo, a, and n () respectively, then the mapping q>: (k, a, n) H kan (k E K, a E A, n E N) is an analytic diffeomorphism of the manifold K x A x N onto G, and

the groups A and N are simple connected. The group G also admits the Cartan de

composition G = KA + K, where A + = exp a+ and a+ is the closure of the dominant Weyl chamber in the algebra a. The Cartan decomposition becomes unique if the left

factor K is replaced by K / M, where M is the centralizer of A in K. Let G be a noncompact semisimple Lie group with finite center and let K be its

maximal compact subgroup (the group K corresponds to the Lie algebra fo in the Iwa

sava decomposition Bo = fo + a + fo). Let M be the centralizer of the subalgebra a in K, let rn 0 be the Lie algebra of the group M, and let Io be the orthogonal comple

ment to rno in fo with respect to the Killing form B of the algebra Bo. We identify


10 with the tangent space to the manifold K / M at the point e. Let d k be the Haar measure on K normalized by the condition

f dk = 1 K

and let d g be the invariant measure on G such that

f f(g)dg = f f f(gk)dkdgK

G G/K K

for every f E Co ( G) (here, dg K is the Riemannian measure on the manifold G / K). The following integral equality holds for any f E Co ( G) :

f f(g)dg = Vo~~~~) fill sinh <x(H) I f f f(k I expHk2) dk 1dk 2dH, (1.28) G I a exeP+ K K

where vol (K/ M) is the volume of the manifold K/ M in the K-invariant metric induced by the restriction of the form B to 10,

Let U be a connected simply connected compact semisimple Lie group, let u be the corresponding Lie algebra, and let t be a maximal Abelian subalgebra of u. Then

u = U Ad(g)t. geU

The connected Abelian subgroup T that corresponds to the algebra t is called the

maximal torus. The set t e = {H E t I exp H = e} is called the unit lattice for U. De

note by g the complexification of the Lie algebra u of the group U. Then the subal

gebra ~ = t + it is the Cartan sub algebra of g, and

For any f E C( U), the following equality is true:

f f(u)du = IA~~I f f f(utu-I)O(t)dtdu, U U T

(1.29)

where d u and d t are the Haar measures on U and T normalized by the conditions


f du = f dt = 1, U T

3(t) = II sino:C:) aELl

(t = expH, HE i).

Denote by A the set of all weights of finite-dimensional representations of the group

U. For every 1-1 E A, we define the function e)l on T by the equality e)l (exp H) =

e)l(H) (HE i) (this definition is correct because A = {AE f)'IA{te) c21ti;l}). Let

1t A be an irreducible representation of the group U with dominant weight A. Then the

character 'X A and dimension dim 1t A of this representation are given by the equalities

where

'XA(expH) =

I (detro)ero(A+P)(H)

ro E AdLl

I (detro)eroP(H)

ro E AdLl

dim 1tA = n (A+P, 0:),

aELl (p,o:) +

1 P = - I 0:.

2 aELl+

(H E i), (1.30)

( 1.31)

Equality (1.30) is called the Weyl formula for the characters of representations of compact groups.

Taking into account (1.29), (1.30), and the denseness of linear combinations of characters in C (Q) (here, Q is the set of conjugacy classes of the group U), we conclude that the algebra ZL 1 (U) is isomorphic to some algebra of functions on i which are symmetric with respect to the Weyl group Ad I:!.. and periodic with respect to the unit lattice ie.

Let n)l = n (1-1, A) be the multiplicity of the weight 1-1 in the representation 1t A with

dominant weight A.. Then, obviously,

. 'XA(expH) = tr1t A(H) = I n)le)l(H).

)l

The numbers n J.1 can be determined, e.g., by the reccurent Freudenthal formula

(1.32)


2 00

n(ll, A) = (A A ) ( ) L L n(1l + ka, AHIl + ka, a) + p, + p - Il + p, Il + P A k-l

aELl+ -

or the following simpler reccurent formula:

n(ll, A) = - L (detro)n(ll+p-roP,A). WEAd8

w"'e

Let us illustrate the notions introduced above, taking the Lie algebra sl(n, <C )

(n ~ 2) as an example (s I (n, a::) is the algebra of complex matrices with trace zero).

Let E ij be the matrix units. The complex dimension of s I (n, a::) is equal to n 2 - 1,

and the elements Eij (i:t= j) and Hij = Eii - Ejj form a basis in s I(n, a::). The Killing

form on the algebra sl(n, a::) is given by the formula B(X, Y) = 2ntr XY. The algebra

l) = {H I H = diag (Xl'''' , x n), Xl + ... +Xn= o} is the Cartan subalgebra of sl(n, a::).

Let ci E f)' (i = 1, ... , n) be defined by the equality ci(diag(xI'"'' xn» = Xi' Then

L1 = {aij=ci-cjli:t=j}, L1+ = {aijli<j}, IT = {ai=Ci-Ei+11 i= 1, ... ,n-l},

and Ha = Hii+ 1/ 2n. Since (ai' a) = B(H a' H a) = 2ntr Ha H a' it is obvious that I I} J J

1 i = j,

n

(ai,a)= -2n' li-jl=l, (i, j = 1, ... , n -1).

0, li- jl ~ 2

The Weyl group Ad L1 of the algebra sl(n, a::) is isomorphic to the symmetric group S n' Indeed, it is easy to find the form of the generators Sa. of the group Ad L1.

I

We have

{aj'

Sa;aj = ai + aj'

-ai'

Ii - jl > 2,

Ii - j 1= 1,

i=j

(i, j = 1, ... , n -1).

By using the equality cl+"'+cn = 0, we express ci (i=I, ... ,n) in terms of ai

( i = 1, ... , n - 1). As a result, we get

Ci = -.!.. [a I + 2a2 + ... + (i -l)a i-1 - Un - i)ai + (n - i-I )ai+ 1+ ... + an-I)] n

(i = 1, ... , n - 1),

Section 1 . Centers of Group Algebras of Compact Groups 199

It is easy to see that the transformations Sa.. transpose the vectors £- i and £ i + r I

• Since Hii+1 = Hex.l(ai,aj> is a basis in 1), the dominant weight A oftheirreduc-

I

ible representation 1t is uniquely determined by the set of nonnegative integers (k l' ... ,

k n-I)' where k i = A (H j i + I)' For what follows, it is convenient to define the weight A in terms of the set of nonnegative integers m;;::: m; ;::: ... ;::: m~ ;::: ° such that k i = mi -

mi+1 (i = 1, ... ,n - 1). It is obvious that the set (m;, ... , m~) is uniquely determined

by A up to a summand constant for all m~ (k = 1, ... , n ).

We can take su(n) = {X E S len, a::) I x* = -X} as a compact real form of the alge

bra s 1 (n, a::). Let us now present the Cartan and Iwasava decompositions for the Lie al

gebra s 1 (n, lR) of reai matrices with zero trace. Clearly, s 1 (n, lR) is also a real form

of the algebra s len, a::). We set fo = so(n) = {X E sl(n, lR) I XT = -X} and Po =

{YE sl(n, lR)1 yT = Y}. It is easy to see that so(n) = sl(n, lR) n su(n) and Po =

sl(n, lR) n isu(n). Hence, sl(n, lR) = so(n)+ Po is the Cartan decomposition for

sl(n,lR). The decomposition sl(n,lR) = so(n)+a+n, where a= {XE sl(n,lR)1

xij=O, i:;t:j} and n ={XE sl(n,IR)lxij=O, i;:::j} is the Iwasava decomposition

for the algebra s 1 (n, 1R). Let us determine the explicit form of relation (1.29) for the compact semi simple Lie

group S U(n) -the group of unitary unimodular matrices. The Lie algebra su(n) is the Lie algebra of the group S U (n) and T is the set of diagonal matrices of the form

diag (e it1 , ... , eitn ), where tl + ... + tn = 0. Obviously, it = {(tl"" , tJ I tl + ... +

t n = O} and He = {(t I' ... , t J E it I t k = 21t n h n k E ~, k = 1, ... , n }. Consider a

manifold D = {(tl'''' , tJ I tl + ... + tn = 0, -1t ~ tk~ 1t, k = l,n}. It is clear that

T = exp iD. We introduce coordinates in D as follows: Let

-1t ~ t 1 + ... + tn-I ~ 1t, -1t ~ t i ~ 1t, i = 1, ... , n - I}.

The mapping <P n is a diffeomorphism of D onto a closed convex bounded region M c

lR n- 1 .. h symmetnc WIt respect to t I' ... , tn_I'

For every z (t I' ... , t n) E C (D), we set


Obviously, Zn(tI' ... ,tn-I) E C(Mn)' If the function Z(tI' ... , t,J is symmetric (skewsymmetric), then Zn(tI' ... , tn-I) is also symmetric (skew-symmetric) and Zn satisfies

the additional condition of symmetry (S):

(condition of antisymmetry (A):

Let d t 1 ... d tn-I be the Lebesgue measure on 1R n-I. We introduce a measure d t on

D by the formula

f Z(tI' ... , tn)dt = f Zn(tI' ... , tn_1)dt 1 ... dtn-l (z E C(D»).

D Mn

Clearly, d t is the Haar measure on T. By virtue of (1.29), for any function x (t) E

ZL I (S U(n»), we can write

f x(g)dg

SU(n)

= c f ( ) II 2( itp itq )2 d - x t I , ... , tn e - e t, n!

(1.33) D p<q

where

Let us now determine the explicit form of relations (1.30) and (1.31). Since a·· = IJ

1 P = 2[(n-l)a1+2(n-2)a2.+ ... +k(n-k)ak+ ... +(n-l)an_d,

whence


By using the equality t 1 + ... + t n = 0, we obtain

(rop )(H)

This yields

i = -[(n-l)(t00(1)-tOO (2))+···+k(n-k)(t00(k)-t OO (k+l))+··· 2

L (detro)eOOP(H) = L (_1)0"(00) e- i (tro(2) + ... +(n-l)tro (n)),

00 E Ad!'. 00 E Sn

201

where cr (ro) is the parity of the permutation ro. It is easy to see that the right-hand side of this formula is the Vandermonde determinant. Therefore,

OOE Ad!'. p>q p<q

Similarly, we can find the explicit form of

L (det ro) eW(P + A)(H).

OOE Ad!'.

One should only note that

Hence, denoting m k = mk + n - k, we conclude that the irreducible representations

SU(n) are enumerated by the collections ml > m2 > ... > mn which are determined


uniquely up to a constant summand for all m k (k = 1, ... , n). The explicit form of relation C 1.31) can easily be found and we leave the relevant calculations to the reader. Finally, for every character

1 X(g) = d' X",(g) (g E SU(n»),

Im1t",

of the hypercomplex system Z L j (S U (n »), we conclude that the following equality is true:

( ( l! ... (n-l)! X(g) = <Pm t) = ... >mn

are integers which uniquely determine the weight A up to a constant summand for all

m k (k = 1, ... , n ).

1.7. Center of the Group Algebra of Compact Semisimple Lie Groups. In this subsection, we study the hypercomplex system ZLj (G), where G is a compact connected simply connected Lie group. First, we consider the case where G = S U (n) and then the general case.

By virtue of (l.33), the hypercomplex system ZLj (SU(n») consists of symmetric functions x(t) on D summable with respect to the measure

The functions <Pm(t) from equality (1.34) are characters of the hypercomplex system

ZLj (SU(n»). We set

!let) = nceitp _eitq ) (tE D).

p<q

Clearly, !l(t) is a skew-symmetric function. Consider an isometric operator


where ALI (D, I ~(t) I dt) is the space of skew-symmetric functions defined on D and summable with respect to the measure

c -lil(t)ldt. n!

The operation * of multiplication in the space ALI (D, I il (t) I d t) is defined by setting

(Ax*Ay)(t) = A(x*y)(t), x,yE ZLl(SU(n). It is obvious that the Banach algebras

ZLI (SU(n) and ALI (D, lil(t)ldt) are isomorphic. The homomorphisms of the al

gebra ALI (D, I ~(t)ldt) in the field of complex numbers are given by the relation

where

() () A() l! ... (n-l)! \j1 m t = <Pm t u t = Il (mp - mq)

(t ED). (1.35)

p<q

Consider a differential operator L in Coo (D) defined by the formula

where

(Lz)(t) = 1!2! ... ~n-l)! Il (L~~Zn(tl, ... ,tn-l) (tE D), p<q

L(n) = ! (~ - ~) ( 1 ~ p < q ~ n - 1 ), pq i at at

p q

L(n) = pn (p = 1, ... , n - 1 ).

It is clear that the operators (L pq z) (t) : = (L~~Z n) (t l' '" , t n- 1) commute with each

other. Moreover, the operators Lpq are symmetric in L 2(D, dt) (to prove this fact, it

suffices to apply the Stokes theorem). We now introduce other coordinates in D and

write L in these coordinates. Indeed,

-1t~tl+"'+ ik+ ... +tn~1t, -1t~ti~1t, i=l, ... ,k-l,k+l, ... ,n}


(the sign A over a coordinate tk means that this coordinate is omitted in the corres

ponding expressions). Denote Zk(t l , .•• , tk , ... ,tn) = z(t) (t E D). We have

(for p"* k). Similarly,

Consequently,

and

L(k) - ~(~-~) (1 ".5.p<q".5.n, p"* k, q"* k), pq - i atp atq

4~ = if- (q? k + 1), otq

L(k) = pk (p".5. k- 1),

Lz(t) = 1 n (L~JZk )(t) (t ED). I!. .. (n -I)!

p<q

Lemma 1.1. Every skew-symmetric solution of the equation Lz (t) = 0 ( tED)

is identically equal to zero.

Proof. We rewrite the equation under consideration in the form

n (L~JZn)(tI' ... , tn-I) = O. p<q


We set

n Z(l) = IT L(n) IT L(n) Z

n pq Iq n' 2~p<q~n q=3

It is obvious that the function Z~l) is a solution of the equation

az(l) az(l) _n _ _ _ n_ = o. atl at2

The general solution of this equation is given by the fonnula

(1.36)

On the other hand, the function Z n is skew-symmetric with respect to the variables

t I' ... , t n-I' Let us show that Z~l) is skew-symmetric with respect to the variables t I

and t 2.

Indeed, it follows from the equality

that

and

This yields

which contradicts (1.36). Therefore,

n n n Lpq IT L1qz = n L(n) n L(n)Z = Zn(l) = O. pq Iq n

2~p<q~n q=3 2~p<q~n q=3


By repeating this reasoning (for Lpn' it is necessary to use the coordinates (tl' ... ,

tk' ... , t n)' k 7:- n), after finitely many steps, we conclude that z == O.

• Clearly, the operator L transforms any symmetric function on D into a skew-sym

metric function, and vice versa. Denote ~ = {z (t) E COO (D) I z is symmetric} and let

P denote the hyperplane t I + ... + t n = O.

Lemma 1.2. Assume that a measurable function w E Loo (D, d t) is skew-symmet

ric. If

f (Lz)(t)w(t)dt = 0 D

for any function Z E ~C, then w (t) = 0 almost everywhere.

Proof. Consider a fixed smooth function (s) (s E [0, 00 )) such that (s) = 0

(s;::: 1), 1;::: (s);::: 0 (s < 1). For any function z E Loo(D, dt), we set

where

Zh(tl,···,tn) = h:- l f Z('tI, ... ,'tn)(i)d't (tE P, h>O), D

n

r = LCtk-'tk)2 and K = (f ('t)d't tl. k=l D

(1.37)

It is obvious that the functions Zh are smooth and equal to zero outside the set of points

of the hyperplane P such that the distances from these points to a certain point of the set

D do not exceed h. Relation (1.37) readily implies that the function zh is symmetric if z is symmetric and w h is skew-symmetric by virtue of the skew-symmetry of w. The functions w h are standard regularizations of the distribution w. Therefore, w is the

weak limit of wh as h -? ° in the sense of duality (CflJ, Ca(P)).

If z I aD = 0, then, by using the Stokes formula, we obtain

(LZh)(tI' ... , t n) = h:- l f (LZ)('tI' ... , 't n) <1>( i )d't = (Lz)hUI' ... , tn)·

D

Let z E ~, z I aD = 0. Then zh E ~ and we conclude that


o = f (LZh)(t)w(t)dt = f (LZ)h(t)w(t)dt. (1.38) D D

It follows from (1.37) that the transition from z to zh is an integral operator with symmetric kernel. Therefore, we can continue equality (1.38) as follows:

o = f (LZ)h(t)w(t)dt = f (Lz)(t)wh(t)dt = (_I)n(n-I)/2 f z(t)(Lwh)(t)dt.

D D D

The function L w h is symmetric because w h is skew-symmetric and L transforms

skew-symmetric functions into symmetric functions. Therefore, (L w h) (t) = 0 for all

tED. At the same time, by virtue of Lemma 1.1, wh( t) = 0 (t ED).

• Lemma 1.3. The set of allfunctions Lz (z E 8) is dense in ALI (D, I .:l(t) I dt).

Proof Assume the opposite. Then one can find a nonzero functional F such that

F(Lz) = 0 (ZE @S). This means that there exists a nonzero function wl(t)E Loo(D,

I .:l (t) I d t) such that

F(z) = !!.... f z(t)w I (t).:l(t)dt (z E ALI (D, I .:l(t)1 dt)). ·n!

By assumption,

f (Lz)(t)w 1 (t).:l(t)dt = 0 (z E @S). D

We set w (t) = WI (t) I .:l (t) I (t ED). Since the function Lz (t) is skew-symmetric, we

can assume that the function w(t) is also skew-symmetric. Indeed, if this is not true,

we replace w (t) by the alternating expression

~ _ 1 ~ a(ro) ( ) wet) - - L..J (-1) w tro(l),···,tro(n)' n' . roESn

where a(ro) is the parity of the permutation ro E Sn. By virtue of the skew-symmetry

of the function z (t), we have

F(z) = !!....f z(t)w(t)dt = !!....f z(t)w(t)dt. n! n!

Consequently,


F(Lz) = ~ f (Lz)(t)w(t)dt = 0 (z E @?). n!

It follows from Lemma 1.2 that w(t) = 0 (t ED), whence F = O.

• Lemma 1.4. Let 'l'm(t) = CPm(t)L1(t) be a function defined by equality (1.35).

Then

(L'I'm)(t) = L ei(mOl(l)tl+ .. ·+mOXn)tn) (t ED).

filESn

Proof. Let d denote the determinant on the right-hand side of (1.35). Clearly,

d = L (_If(fil)ei(mOl(l)tl+···+mOXn)tn~

filESn

'Dn = L (_If(fil)ei[(mOl(I)-mOl(n))tl+ .. ·+(mOl(n-l)-mOl(nj)tn-l]

filESn

(recall that 'Dn(tl'"'' tn_I) = d(tl'''' ,tn) (t ED)). This implies that

(1.39)

L(k)'D = ~ (_l)O(fil) (m _ m )ei[(mOl(l)-mOl(n»)tl+ ... +(mOl(n_1 )-mOl(n»)tn_1 ] pn n LJ fil(p) fil(n) ,

filESn

for 1::; p < nand

for 1::; p < q ::; n - 1. Consequently,

Ld = n L~J'Dn = L (_l)O(fil) D(mfil(l), m fil (2), .. , ,mfil(n)) p<q filE Sn

where


D(ml"'" mn) = n (mp-m q). l5.p<q5.n

Since D (m I' ... , m n) is a skew-symmetric function, we have

This implies that

(Ld)(t) = D(ml'"'' mn) I ei[(mW(I)-mw(n»tl+···+(mw(n_I)-mw(n»tn_l]

ro e Sn

= n (mp-mq) I ei[mW(I)tl+ .. ·+m~n)tn] (tE D). p<q roeSn

209

• We introduce a new operation over symmetric functions. For any Z I' Z 2 E G, we

set

ZI 0 Z2(t) = c J ZI(tl-'tI'"'' tn-'tn)Z2('t I ... ·, 'tn)d't (tE D). ( 1.40)

D

Note that addition in this formula is performed modulo 21t. Obviously, G is an algebra

with respect to the multiplication defined by (l.40). Let GLI (D, dt) denote the completion of G in the norm

IIzlI10 = ~f I(Lz)(t)~(t)ldt. n!

D

Theorem 1.4. The space GLI (D, dt) is a Banach algebra with multiplication

(l.40) isomorphic to ALI (D, I ~(t)l, dt). The homomorphisms of GLI (D, dt) in the field of complex numbers are given by the formula

GLI (D, dt) :3 Z H C f ZI(tI'"'' tn)ei[mltl+ ... +mntnHt E <C, D

(1.41 )

where ml > ... > mn are integers uniquely defined (to within a common summand).

Proof. With every function Z E G, we associate a function

zet) = (_l)n(n-I)/2(Lz)(t)


and show that the correspondence z H Z is an isomorphism of the algebra @S into

ALl CD, I ~(t)I, dt). First, we establish the equality zl 0 Z2(t) = z] * Z2' For every

z E @S, in view of the symmetry of the operator L, equality (1.39), and symmetry of the function z (t) E @S, we obtain

( It(n-I)/2 .E. J z(t)'I'm(t)dt = - c J (Lz)(t)'I'm(t)dt n! n!

D D

The mapping

= .E. J z(t)(L'I'm)(t)dt n!

D

= c J z(t)ei(mltl+ .. ·+mntn)dt.

D

AL] CD, I ~(t)1. dt) 3 z H .E. J z(t)'I'm(t)dt n!

D

(1.42)

is a homomorphism of ALl CD, I ~(t)1. dt) in <C and multiplication (lAO) is nothing but ordinary convolution. Therefore, by using the fact that the right-hand side of (1.42) is an ordinary Fourier transform, we obtain

Therefore,

= c f (z] 0 Z2)(t)ei(mlfl+···+mnfn)dt

D

= c J Zl (t)ei(mlfl+ .. ·+mnfn)dtc J Z2(S)ei(mlsl+ .. ·+mnsn)ds

D D


f[(ZIOZ2)(t)-CZI*Z2)(t)]\jfm(t)dt=O (ml>··.>m n• miE~). (1.43) D

Note that (Zl ° Z2)" - Zl * Z2 E ALl (D, I L1(t) I, dt). Therefore, (Zl ° Z2)" = Zl * Z2

because (1.43) means t\1at (Zl ° Z2)" - Zl * Z2 belong to all maximal ideals of the Ba

nach algebra ALI (D, I L1(t) I, dt).

Thus, the mapping @3 3 Z H Z is a homomorphism. Let us show that this mapping

is injective. Let z = O. Then, by virtue of (1.42), the Fourier transformation of the

function Z is equal to zero and, consequently, z = O.

We now show that the product ° is continuous in the norm II· 115. For any Z I. Z2 E

@3, we have

:::; (:J2 f I Zl (t)L1(t) I dt f I Z2(s)L1(s) I ds = liz I 11511 z2115· D D

This implies that @3L] (D, dt) is a Banach algebra. The remaining statements of the theorem are obvious.

• Consider a differential operator

L = 1 n ~ (J.- _ J.-) I! ... (n -I)! p<q i atp atq

acting in Co ( IR n). It is obvious that

for any k and, hence, the restriction of L to c"" (P) coincides with the operator L. Consider the equation

~G(t,"c) = L L O(t-'tro(l)+2kl1t,···,tn-'tro(n)+2kn1t) (kl,···,kn)E~n roESn

= L 8(t-oo('t») (t,'tE IR n), roESn

(1.44)


where 3(XI' ... , x n) is the n-dimensional &-function on the torus (eitl, ... , e itn )

By repeating the reasoning used in the proof of Lemma 1.1, we conclude that every

skew-symmetric solution of the equation (iz)( t) = 0 is identically equal to zero in any

convex region 0 c ]R n. Consequently, equation (1.44) possesses a unique solution in

the class of distributions skew-symmetric with respect to 't I' ... , 't n- Since the right-

hand side of (1.44) is symmetric with respect to t l , ••• ,tn and Lr does not act upon

these arguments, the function G (t, 't) is symmetric with respect to t l' ... , tn. Finally,

the function G (t, 't) is periodic in each argument t i and 't j (j, j = 1, ... , n) because the right-hand side of (1.44) is periodic. By using relation (1.35), we extend the function

'JI m (t) from the hyperplane P to the entire space IR n. By introducing obvious modifi

cations in the proof of Lemma 1.4, we conclude that

(L\vm)(t) = L ei(mOl(I/I+···+mOXn)tn) (tE ]Rn). (l.45) OlESn

Let us show that the expansion of the function G in the Fourier series has the form

G(t,'t) = L e-i(mltl+···+mnfn)'JIm('t).

mE<iZn

(1.46)

Indeed, by applying the operator Lr to both sides of (1.46), we see that the function

G (t, 't) given by (1.46) satisfies equation (1.44). Moreover, the fact that G (t, 't) is

skew-symmetric with respect to 't follows from the skew-symmetry of 'JI m ('t) and this

implies that the solution of equation (1.44) is determined uniquely.

• Lemma 1.5. Thefunction GU, 't) satisfies the equation

(itG)(t,'t) = (_It(n-l)/2 L L (-1)0"(0l)8(tl-'t 0l (1)+2k]1t, ... ,tn-'t 0l (n)+2k n1t)

kE'iI!." OlESn

= (_It(n-:-I)/2 L (-l)0"(0l)3(t-ro('t)) (t,'tE IRn) (1.47) OlESn

and can be represented in the form

(l.48)

Proof Let us apply the operator it to relation (1.46). This gives


4e-i(mttt+ .. ·+mntn\vmCt) = (_l)n(n-I)/2 L (_l)O"(fO)e-i[mt(tt-tW(t)+···+mnVn-tW(nj)].

fOESn

Carrying out summation over all m E IR n, we arrive at (1.47).

To obtain (1.48), we rewrite (1.46) in a somewhat different form: Let t * = t j + ... + tn ,1:*=1: j + ... +1: n, t[ =ti-t; and 't; ='ti-'t*(i=l, ... ,n). Then t;+ ... +

t~ = 1:} + ... + 't~ = 0 and

G(t, 't) = L e-iht{ + ... +mnt~)\jIm(1:/)e-i(mt+···+mnXt*-'t*).

mE~n

Note that this formula easily follows from the explicit expression for \jI m ('t). In the last

equality, we first carry out summation over m j + ... + m n and then over the collections

(m j, ... , m n) such that one can never indicate two collections of this sort which differ from each other by a common summand. As a result, we obtain (l.48).

• It follows from (l.48) that G (t, 1:) I D xD = G (t, 't) and it follows from the proof of

Lemma 1.5 that

(1.49) m

where the summation is carried out over all collections (m j, ... , m n) such that one can never indicate two collections of this sort that differ from each other by a common summand.

It follows from (1.46) that

Therefore,

(1.50)

where K is a constant. Since the function \jim (t) is symmetric with respect to m and

the function G(s, t) is symmetric with respect to s, one can rewrite relations (1.49) and (1.50) in the following equivalent form:


G(t, <) = L (L e'(m."", + ... m.""')}i1 m«) (t, H D), (1.51)

m, > ... >mn roESn

'If m (t) = (21t~n-1 f ( L ei(m(J)(l)s, + ... m(J)(n)sn) ) G (s, t) ds (t ED), (1.52)

D+ roESn

where D+ = D n {s E IR nl sl > ... > sn} (note that, since tED, we can assume

that all mj~O in (1.51)). Since the set ml> ... >mn~O determines an irreducible

representation of S U (n), we can assume that the outer sum in (1.51) is taken over all

irreducible representations of S U (n ).

Let us determine the structure measure of the hypercomplex system Z L I (S U (n )).

By virtue of (1.47) and (l.48), the operator inverse to L exists and is determined by the Green function

By applying the operator L to both sides of (l.40), we get

Lz(t) = L(zl °Z2)(t) = c f (Lz,)(/-'t)Z2('t)d't (tE D).

D

The functions Lz and Lz, belong to AL, (D, I Ll(t)1, dt). Therefore, we can rewrite the last equality as follows:

(y, *Y2)(t) = (-It(n-\)/2 c f y,(t-'t)(C'Y2)('t)d't

D

= :! f f G (t - 't I, 't 2) Y I ('t I ) Y 2 ( 't 2) d't , d't 2

DD

(t E D; Yi = (_1)11(11-')/2 Lzi ; i = 1, 2) .

( 1.53)

Here, the last equality holds by virtue of the periodicity of the integrands. Let us alter

nate the function G (t - 't " 't 2) with respect to each variable and denote the function obtainedby a('t,,'t2,t). We substitute a('t,,'t2,t) in (1.53) for G(t-'t\,'t 2 ). Obviously, the integral on the right-hand side of (1.53) does not change, i.e.,


(YI *Y2)(t) = :!f f YI(1: j )Y2(1:2)a(1:I,'t2,t)d'tId't2' DD

Recall that the operator of multiplication by

,1(t) = n (eitp_eitq)

p<q

215

isomorphically maps ZL 1 (SU(n)} onto ALl (D, 1,1(t)ldt). Therefore, the structure

measure of ZLj (SU(n» is determined by the function

Thus, we have proved the following statement:

Theorem 1.5. The function

determines the convolution in ZLj (SU(n)} according to theformula

(x] *x2)(t) == f f Xj('tj)x2('t2)c(1: j,1:2,t)d'tjd1:2 (tE D). (1.54)

DD

Here, c= 1/ fD dt, the function a(1:I' 1:2' t) is obtained by alternating G(t-1:],1:2)

with respect to all variables, and the function G (t - 1: \, 1: 2) is determined by the char

acters '" >mn filESn

where the summation is carried out over all collections m j > ... > m n such that one

can never indicate two collections of this sort that differ from each other by a common

summand (or, which is the same, the summation is carried out over all irreducible

representations of SU(n».


The results obtained in this subsection remain true in the case of an arbitrary compact semisimple connected simply connected Lie group G. We present here only main statements. The hypercomplex system ZLj (G) is isomorphic to the Banach algebra of

functions on i that are symmetric with respect to the Weyl group Ad L1, periodic with

respect to the lattice ie' and integrable with respect to the measure o(t)dt (here,

~() I II . a(iH) I u t = Slll--aE~ 2

(t = exp H, H E i)

and d t is the Haar measure on the maximal torus T normalized by the condition

J dt = 1). The algebra ALl is the Banach algebra of functions on t that are skew

symmetric with respect to Ad L1, periodic with respect to ie' and integrable with re

spectto the measure I L1 (t) I d t, where

A( ) II' a(iH) Ll t = Slll--.

aE~ 2

Finally, @3 L I is the completion of the set of functions having the same properties as

functions from ZL I with respect to the norm

liz"@:) = _1-J ILz(t)"~(t)ldt, IAd~IT

where the operator L is the product of operators of differentiation along positive roots

a. The following statement is true:

Theorem 1.6. The function

determines the convolution in ZL j (S U (n )) according to the formula

Here,

(XI *x2)(t) = J J xI('tj)x2('t2)c('tI,'t2,t)d'tjd't2' T T

A( ) II' a(iH) Ll t = Slll--

aE~_ 2 (t = expH, HE i),

(1.56)


the function a('tj, 't2, t) is obtained by alternating the function G(t-'t!, 't2) with

respect to all variables, and G (t, 't) is determined by the relation

G(t, 't) = L L ei<rom,H>Xm(t)~(t), (1.57) 1tEG roEAd~

where

Xm(t) = -d. 1 X1t(t) (t E T, 1t E G) Im1t

are the characters of the hypercomplex system ZL! (G), m = 'A, + p, 'A, is the domi

nant weight of the representation, and

1 P = - L <l.

2 aE~ +

The outer sum in (1.57) is taken over all equivalence classes of irreducible representations of G. The function G (t, 't) is skew-symmetric with respect to t and is a

solution of the equation

L.rGU, 't) = L 8(t - rot) (rot = exp roH, t = exp H, HE 0. roEAd~

1.8. Algebra G of Equivalence Classes of Irreducible Representations of a Com

pact Semisimple Lie Group G. In this subsection, we study the hypercomplex system

Zl! ( G), where G is the set of equivalence classes of irreducible representations of a compact semisimple connected simply connected Lie group G. This hypercomplex system is dual to the hypercomplex system considered in Subsection 1.7. As in

Subsection 1.7, we carefully analyze the case of the group S U (n); the results obtained can easily be extended to the general case.

We introduce the following equivalence relation on the integer-valued lattice :l n: vectors m = (m j, ... , m n) and m' = (m;, ... , m~) are regarded as equivalent if they

differ from each other by a vector with equal coordinates, i.e., m i = mi + k for all

i = 1, ... , n. Let ~n = :l n /-. In what follows, we consider vectors m E :l n such that

m j > ... > m n. Let Q c ~n be the set of equivalence classes of such vectors. Relation

(1.34) enables us to identify SU(n) with the set Q. Let fl E Q. The dimension dJl

of an irreducible representation 1t Jl is determined by the equality

dJl = I!. .. (n - I)! '

(1.58)


the multiplicative measure of the hypercomplex system ZL 1 (SU(n) is equal to

m (f.1) = dJ, and the characters X (f.1) = X t (f.1) = <j> (t, f.1) (t ED, f.1 E Q) are defined

by (1.34). Denote by M(f.1I' f.12; f.1) (f.1I' f.12, f.1 E Q) the multiplicity of the represen

tation 1t 11 in the decomposition of the tensor product 1t III ® 1t 112 into irreducible repre-

sentations. Then the structure measure of ZL I ( S U(n» is equal to

Our aim is to find the explicit form of the function M(f.1I' f.12; f.1).

Let us extend a function x E· ZL 1 (SU(n» to a symmetric function on ~n accord

ing to the formula x(mf.1) = x(f.1) (m E S n' f.1 E Q). Further, we extend d ll according

to (1.58). For any function x E ZL 1 (SU(n), we set Ax(f.1) = x(f.1)d 11 (f.1 E ~n). Obviously, the function A x (f.1) is skew-symmetric. One can easily verify that the op-

erator A realizes an isometric isomorphism between the Banach algebras Zll (SUcn»

and A II ( ~n), where A II (~n) is the space of skew-symmetric functions on ~n sum

mabIe with respect to the norm I dill / n! (f.1 E ~n) with the multiplication

and M(f.1I' f.12; f.1) is a function skew-symmetric with respect to each variable and such

that M(f.1I, f.12; f.1) = M(f.1I' f.12; f.1) (f.1I, f.12, f.1 E Q). It is obvious that M(f.1I' f.12; f.1)

is uniquely determined by this condition and, furthermore, M(m 1f.11' m2f.12; m3f.13) = (-If(OOI OO2CO:J)M(f.1I,f.12;f.13) (f.1iE Q, i=1,2,3). The homomorphisms of the Banach

algebra A II (~n) in the field of complex numbers are determined by the relation

where

8(t,f.1) = n (eit P - eitq )

p<q

(t ED),

(t E D, m E f.1 E ~n) (1.59)


(the right-hand side of (1.59) is well defined because t 1 + ... + t n = 0 for all tED).

Let us introduce difference operators A ± in the space of functions on ~n as follows:

(l 5, P < q 5, n ),

where

An (mE /lE ~ ).

We set

(A±x)(/l) = II (A~qx)(/l) (/lE ~n). p<q

It is easy to see that the operators A ± map the set Co (~n) of finite functions into it+

self. Moreover, A-map functions symmetric in m l' ... , m n into skew-symmetric and

vice versa. By direct calculation, one can easily verify that

L (A ±x)(/l)Y(/l) = L x(/l)(A+Y)(/l) (1.60) ~E;ln ~E;ln

for arbitrary functions x and Y on ~n such that the series in (1.60) are convergent. Properly modifying the reasoning in the proofs of Lemmas 1.1-1.3, we conclude that

every skew-symmetric solution of the equation (A ±Z)(/l) = 0 is identically equal to

zero, and the set of functions A ±z (z E @S = {z E co(~n) I Z is symmetric}) is dense

in A II (~n) (in this case, the proofs are simpler, e.g., it is not necessary to regularize

the functions well) as in Lemma 1.2). One can easily check that

(A +S)(t, /l) = L ei(mltro(l)+···+mnt~n», (J)ESn

(A -S)(t, /l) = (_1)n(n-1)/2 L ei(mltro(l)+···+mnt~n» (t ED, /l E ~n). (J)ESn

Let us introduce multiplication 0 in @S. We set

(1.61)


Zl °Z2 = L Zl(ml-k), ... ,mn-kn)Z2(k 1,···,kn) (kE K, mE ~E iln) Ke~n

for any Z), Z 2 E @S. Clearly, @S is an algebra with respect to the multiplication o. De

note by @S/) (iln) the completion of @S with respect to the norm

Following the proof of Theorem 1.4, we conclude that @S/) (iln) is a Banach algebra

with respect to the multiplication ° and is isomorphic to A 11 ( ~ n). The corresponding

isomorphism has the form

An The homomorphisms of @S/ 1 (~ ) into the field of complex numbers are given by the relation

@S/l(iln) 3 Z H L z(~)ei(mjfj+ ... +mnfn) (t ED).

Ile~n

Consider the equation

(A~G)(~, K) = L Bo(m) - k W (1),· .. , mn - kw(n) weS.

( 1.62)

where BIl(K) = 1 for K=~, and BIl(K) = 0 for K*~. Equation (1.62) has a unique

solution in the class of functions skew-symmetric in k l' ... , k n Since the right-hand

side of (1.62) is symmetric with respect to m l, ... , m n and A~ does not act on these

variables, the function G (~, K) is symmetric with respect to ~. Let us show that the thefunction G(~, K) admits the following integral representation:

G(~, K) = (21C~n-l f, e-i(mjfj+ ... +mnf.)S(t, K)dt (~, K E iln), (1.63)

D

where


Indeed, by applying the operator A~ to both sides of (1.63) and using (1.61), we get

1t

J exp{ - i[ (m] - mn - (kw(l) - kw(n))) tl + ... -1t

= L 0(11- roCK» An

(11, K E ~ ).

weSn

Consequently, the function G(Il, K) defined by (1.63) satisfies equation (1.62). Moreover, the fact that S (t, K) is skew-symmetric with respect to K implies that G (11, K) is also skew-symmetric with respect to K. To complete the proof it remains to use the fact that, as proved above, (1.62) is uniquely solvable in the class of functions skew-symmetric with respect to K.

• It follows from (1.63) that

S(t, K) = L G(Il, K)ei(mttt+···+mntn) (KE ~n, tE D). (1.64) Ile~n

By comparing (1.64) with (1.32), we establish that the function G (11, K) is equal to

the multiplicity of the weight 11 + p in the representation with dominant weight K + p. By applying the operator A~ to both sides of (1.63), we obtain

(A~G)(Il, K) = (_I)n(n-I)/2 L a(ro)o(ll- roCK»~. weSn

Hence, the operator A + has the inverse which acts in the class of symmetric and skew( _It(n-I)/2

symmetric' functions and is determined by the "Green function" G (11, K) as n!

follows:

( _It(n-I)/2 (A -lx )(ll) = L G(Il, K)X(K)

n! 1C

if X(K) is a skew-symmetric function, and ,


( It(n-I)/2 A -lx(lC):;;; - L G(~, lC)x(~)

n! 11

if x is a symmetric function. Following the reasoning of Subsection 1.7, we obtain

Consequently,

:;;; L (_l)a(ffi)G(~-ro(~I)'~2) (~I'~2,~E Q). ( 1.65) ffiESn

Relation (1.65) was obtained by Weyl. By using this relation, one can easily determine

C(~l' ~2; ~). The results of this subsection can also be used in the case of an arbitrary semisimple

Lie group. The corresponding relation for determining the multiplicity M(I"I, A2; A) of

the representation 1t A with dominant weight A in the decomposition of 1t A ® 1t A in-I 2

to irreducible representations coincides with the following Weyl formula:

M(AI,A,2;A,):;;; L (detro)n(A,+o-ro(A,2+ 0),A,1)' ffiE Ad t.

where n(~, AI) is the multiplicity of the weight ~ in the representation 1tAI with

dominant weight A I'

Section 2 Gelfand Pairs 223

2. Gelfand Pairs

In this section, we give an example of a hypercomplex system whose basis is the set of double cosets of a locally compact group with respect to a compact subgroup. Such a hypercomplex system is isomorphic to a subalgebra of the group algebra of the original

group G which consists of functions invariant under the right and left actions of a compact subgroup H. In this section, we mainly consider the case where this hypercomplex

system is commutative; in this case, the pair (G, H) is called a Gelfand pair. For

example, the pair (G, H) that generates a Riemannian symmetric space is a Gelfand pair; the hypercomplex systems considered in Section 1 are also related to Gelfand pairs.

In this section, we present basic theorems of harmonic analysis on Gelfand pairs. We omit the well-known facts from the theory of functions on symmetric spaces and the theory of discrete hypercomplex systems associated with the group of motions of a connected nonoriented graph. At the end of the section, we give an important example of a hypercomplex system associated with the Delsarte generalized translation operators; every hypercomplex system of this sort is isomorphic to a hypercomplex system associated with a certain Gelfand pair.

2.1. Definition of Gelfand Pairs. Consider a hypercomplex system that consists of functions defined on a locally compact group G and biinvariant under the action of its

compact subgroup H. If this hypercomplex system is commutative, then (G, H) is a called a Gelfand pair. Below, we give basic examples of Gelfand pairs and describe their properties.

Let G be a locally compact group, let H be its compact subgroup, let d/..,(g) = dg

be a left Haar measure on G, and let d~(h) = dh be the normalized (~(H) = 1) Haar

measure on H. The set G / H of left co sets of the group G with respect to the subgroup H is a homogeneous space with the group of motions G. Namely, an element

g E G transforms a class aH into the class gaH (a E G). Let n,~, 'Y, ... denote elements of the space G / H and let e be the element corresponding to the class H

(the fixed point). Let 'Y E G / H. Then the set {h'Yl hE H} is called an H -orbit of

the point 'Y (or a sphere centered at the fixed point), and G / H can be decomposed into

the set Q of H-orbits. Denote elements of Q by p, q, r, .... Obviously, Q is the set

of double co sets G / / H of the group G with respect to the subgroup H (in other words,

Q is the set of points of the form HgH={hlgh2Ihl,h2E H} (gE G)).

A function x E Ll (G, dg) is called orbital (or biinvariant) if it is constant on the

doublecosetsof G withrespectto H, i.e., if x(h 1gh 2)=x(g) for almost all gE G

and all hi, h 2 E H. It easy to see that the set K of all orbital functions is a closed

subalgebra of the group algebra Ll(G,dg) of the group G. Indeed, K is obviously


closed. If x, y E K, g E G, and hI' h2 E H, then

(x * y)(h 1gh 2) = J x(h 1gh 2a)y(a- 1 )da G

= f x(h 1ga)y(a- 1h 2)da = f x(ga)y(a-1)da = (x*y)(g). G G

• The algebra K can be reE{arded as a hypercomplex system whose basis is the space

Q = G / / H. Indeed, by virtue of the general construction presented in Subsection 1.1, it suffices to show that every function x (g) E K can be approximated in the norm of the space Ll (G, dg) by finite continuous functions taking constant values on the double cosets. For this purpose, we consider a sequence fn(g) E Co ( G) of functions such that

II x - fnlll ) O. Then, the functions n~oo

xn(g) = f f fnCh 1gh 2)dh 1dh 2 HH

are constant on the classes H g H and approximate x (g ).

• Denote the obtained hypercomplex system by Ll (G, H). Note that the multiplicative

measure m and the structure measure c (A, B, r) satisfy the following relations:

m(A)=A(21) (AE ~Q) and c(A,B,r)=A(g-l21nm-l)=(K~*K\B)(g) (A,BE

~(Q), gE r=HgHE Q), where A is the Haarmeasureofthe group G, and 21 and m are complete preimages of the sets A and B, respectively. The hypercomplex system will be normal if we introduce involution * in Q as follows: Q:3 H g H =

P H P * = H g-I H E Q. Obviously, the class H is the basis unity of the hypercomplex system Ll (G, H). Thus, by virtue of Corollary 1.2 in Chapter 1, the algebra K is semisimple.

Generalized translation operators associated with L 1 (G, H) have the form

Ls!(t) = J f(gl h g2)dh (gl E s, g2 E t, fE Ll(G,H». (2.1) H

Indeed, for any biinvariant functions x, y E Co(G), we have

f Lsx(t)y(t*)dt = (x*y)(s) = J (x*y)(glh)dh Q H


= f f x(glhg 2)y(g"21)dg 2dh = f f x(glhg2)dhy(g"21)dg2 HG GH

(gl E s, g2 E t, t,SE Q),

which implies (2.1).

• In particular, it follows from (2.1) that the hypercomplex system L 1 (G, H) satisfies

the condition of separate continuity. It is not difficult to show that Q = G / / H is a

hypergroup.

The modular function ~ (g) (g E G) of the group G is biinvariant and its pro

jection onto Q is the modular function of the hypercomplex system L 1 (G, H). In

deed, since H is a compact subgroup, the set {~( h) I h E H} is a compact subgroup of the multiplicative group (0, 00). On the other hand, the unique compact subgroup of

(0,00) is {I}. Thus, ~(h)=l (hE H) and,hence, ~(hlgh2)=~(hl)~(g)~(h2)= ~ (g) (g E G, hI' h 2 E H). The last statement follows from the equality

f (Rs·fXt)dt = f f f(g]hg";])dhdg] = f f f(glhg":/)dg]dh Q GH HG

= J ~(g2h-l)dhJf(gl)dgl = ~(s)Jf(t)dt H G Q

• Below, we study the case of commutative hypercomplex systems. The pair of groups

( G, H), where H is a compact subgroup of a locally compact group G, is called a Gelfand pair if the algebra K of orbital functions is commutative.

Note that if (G, H) is a Gelfand pair, then the group G is unimodular. Indeed, let ~(g) be the modular function of the group G. As shown above, ~(h) = 1 (h E H).

Hence, for any function f ELI ( G, d g), we have

J J J f(h 1gh 2)dh 1dh 2dg = J J J f(g)dg~(h2)dhldh2 = J f(g)dg. GHH HHG G

Therefore, to prove the unimodularity of G it suffices to show that

f x(g)dg = J x(g-l )dg G G


for any finite orbital function x E K. Denote A = supp x U supp x ~ w here x *( g) = x(g-l). Obviuosly, KA(g)E K. By virtue of commutativity of K, we obtain

f x(g)dg = (x * KA)(e) = (KA * x)(e) = f x(g-l )dg. G G

• Theorem 2.1. If there exists an involutive automorphism 0' of the group G such

that O'(g)E Hg-IH for all gE G, then (G,H) isaGelfandpair.

Proof. Let x, y E K. Since x( O'(g» = x(g-l), we have

x * y(g) = f x(ga)y(a- 1 )da = f x(O'«ga)-l»y(O'(a»da G G

= f x«O'(a»-1 O'(g-l »y(O'(a»da. G

Since the automorphism 0' is involutive, it preserves the Haar measure (see, e.g., [HeR]). Therefore, taking into account that y * x E K, we can extend the last equality as follows:

f x«(O'(a-1»O'(g-l»y(O'(a»da = f x(a-1O'(g-l»g(a)da G G

• Remark. It follows from the proof of Theorem 2.1 that it is not necessary that the

automorphism 0' be involutive; it is sufficient to require that it preserve the Haar

measure, i.e., A(O'(A» = A(A)(A E ~(G».

A pair (G, H) that generates a Riemannian symmetric space] is a Gelfand pair.

Indeed, since every element g E G can be represented in the form g = h . k, where

hE Hand O'(k) = k- 1 (see, e.g., [Hell]), we have O'(g) = hk-1 E Hk- 1 H = H g-l H.

• Riemannian symmetric spaces are thoroughly studied and the reader can find a rather

1 This is, in fact, the space G / H, where G is a connected Lie group and H is a compact subgroup of

G for which there exists an involutive automorphism cr of the group G such that (Ga)o C H eGa,

where Ga is the set of fixed points of the automorphism cr and (G a) 0 is the connected component of

Ga·


complete presentation of the theory of such spaces in many monographs; therefore, we do not present here the corresponding results and refer the reader to the well-known monographs of Helgason [Hell] and [HeI2]. Note only that, in this case, the algebra

tj)( G / H) of G-invariant differential operators on G / H is commutative and finitely generated; the number of algebraically independent generators of the algebra tj)( G / H)

is called the rank of the symmetric space G / H. Spherical functions (see Subsection 2.2) are determined by the following conditions:

(i) <p(e)=l,

(ii) <p E COO(G),

(iii) <p(hg h') = <p(g) for all g E G and h, h' E H,

(iv) D <p = AD <p (AD E <C) for every D E tj)( G I H) (note that, by virtue of condition

(iii), the function <p can be regarded as a function on G I H).

The converse statement is also true: If G is a Lie group, H is its compact sub

group, and the algebra tj)( G / H) is commutative, then (G, H) is a Gelfand pair (see [Th02l).

Another example of a Gelfand pair is the pair (G, H) such that, for any two points

ex, PEG / H, there exists a motion g E G which permutes these points, i .. e., gex = P

and g P = ex.2 Indeed, one can easily show that the classes H g Hand H g-I H coincide. Let a E G be such that aex = P anti a P = ex, where ex = gH and P = H. In

other words, there exist hl,h2E H such that ag=hl and a=gh 2. Therefore, g-I=

r I a-I a = (agr 1 a = hll gh2' i.e., g-l E H gH. Obviously, we can take the identical

automorphism as the automorphism 0' required in Theorem 2.1.

2.2. Spherical Functions. In this subsection, we show that characters of the hypercomplex system introduced in Subsection 1.1 coincide with bounded spherical functions and study general properties of functions of this sort.

A complex-valued function <p(g) (g E G) which is not identically equal to zero is called spherical if

f <p(glhg 2 )dh = <P(gl)<P(gl)

H

(2.2)

for any g l' g 2 E G. Note that spherical functions are constant on double cosets. In

deed; let <peg I) '::j:. O. Since

2 Gelfand pairs of this sort are called symmetric. As we know, they were considered for the first time

by M. Krein [Krell.


H H

we see that <peg) = <p(h' g) (g E G, h' E H). Similarly, we conclude that <p(gh') = <peg) (h' E H, g E G).

• Theorem 2.2. A biinvariant continuous function <p (g) (g E G) is spherical if and

only if the following conditions are satisfied:

(i) the equality <p * f = A ( <p,f) <p (A ( <p, f) E ([ I ) holds for any continuous finite biinvariant function f E K;

(ii) <p(e)=l.

Proof. Necessity. Since

<p(e)<p(g) = f <p(hg)dh = f <p(g)dh = <peg) (g E G) H H

and <p (g) "* 0, we have <p (e) = 1. Let f E K be an arbitrary continuous biinvariant function. Since

(<p * f)(g) = f <p(ga- I )f(a)da = f f <p(ga- I )f(a)dadh G H G

= f f <p(gha- I )f(ah- I )dadh = f f <p(gha- I )dhf(a)da HG GH

= f <p(g)<p(a- I )f(a)da = (<p * f)(e)<p(g), G

condition (i) is satisfied with A (<p,f) = (<p * f)( e).

Sufficiency. Since <p(e)= 1, we have A(<p,f) = A(<p,f)<p(e) = (<p * f)(e). Since

<p (g) "* 0, there exists a finite biinvariant continuous function f such that

A(<p,!) = (<p * f)(e) = f <p(a- 1 )f(a)da "* 0. G

In view ofthe unimodularity of the group G, we have

Section 2 Gelfand Pairs

where

(<p*f)(e)f <p(glhg2 )dh = H

f (<p * f)(g 1 hg 2)dh H

= f f <p(glhg 2a- 1)f(a)dadh He

= f f <p(gla- 1)f(ahg2)dadh He

f; (a) = f f(ahg 2)dh 2 H

229

is obviously a finite biinvariant continuous function. We compute A( <P,f;2). Indeed,

Thus,

A( <p,f;) = (<p * f;)(e) = J <p(a- 1 )f;2 (a )da e

= J J <p(a- 1 )f(ahg2)dadh = J f <p(ha- 1 )f(ag2)dadh He He

= f <p(a- 1 )f(ag2)da = (<p * f)(g2) = A(<p,f)<P(g2) e

(<p*f)(e)J <p(glhg2)dh = (<p*f)(e)<p(gl)<P(g2) H

for any g l' g 2 E G. Consequently, in view of the fact that <p * f(e) *- 0, we conclude

that the function <p (g) is spherical.

• Let us now present an important example of spherical functions. Assume that G =

P . H, where P is a closed subgroup and P x H ~ PH = G is a homeomorphism. Let

p: P ~ a:: 1 \ { o} be a continuous homomorphism of the group P. By assuming that p(ph)=p(p) (pE P,hE H), we extend it to a function on G. Obviuosly, p(gh)= P (g) (g E G, h E H). Moreover,


p(pg) = p(pp'h) = p(pp') = p(p)p(p')=p(p)p(g) (g=p'hE G, pE P).

The junction

<peg) = J p(hg)dh (2.3) H

is spherical with respect to the subgroup H. Indeed, since P x H --7 PH = G is a homeomorphism, for any h E H and g E G, there are elements p g( h) E P and u g( h) E

H such that hg = p g(h)u g(h) and the functions H 3 h H P g(h) E P and H 3 h H

ug(h) E H are continuous in h for any fixed g E G. In view of the invariance of the

Haar measure and the properties of p established above, we obtain

J <P(gl hgZ)dh = J J p(hg1h'gz)dh'dh H HH

= J J p(pgl(h)ug1(hh'gz)dh'dh HH

= J J P(pg/h)h'gz)dh'dh HH

= J J p(pgl(h»p(h'gz)dh'dh HH

= J P (p g 1 (h) U g 1 (h » d h J P (h ' g z ) d h' H H

= J P(hg 1)dhJ p(h'gz)dh' = <P(gl)<P(gZ)' H H

• Example. Let G = SLz(IR). Then G = PH, where H = SO(2) and the group P

consists of upper triangular unimodular matrices, i.e., matrices of the type

Let s E ([ 1. Obviuosly, the mapping pS: P --7 ([ 1 \ { O} defined by the formula

b )) = as -1 ' a


is a continuous homeomorphism of the group P in a: 1 \ { O} and, hence, the functions

<p/g) = J ps+l(hg)dh (s E a: 1\{O}) (2.4) H

are spherical. Note that the last formula exhausts the set of spherical functions (see, e.g., [LanD.

Let us now describe the space of characters of the hypercomplex system L] (G, H).

Theorem 2.3. The set of characters of the hypercomplex system L, (G, H) is in a

one-to-one correspondence with the set of bounded spherical functions. Moreover, a character X (p) and the relevant bounded spherical function <p (g) satisfy the equal

ity X(p)=<p(g) (gE pE G//H).

Proof. Let <p (g) (g E G) be a bounded spherical function. We set X (p ) = <p (g)

(g E P E Q) and show that X(p) is a character of the hypercomplex system L] (G, H).

It is obvious that the function X (p) is bounded and continuous. Let A, B E ~ (Q) and

let ~,Q3 E ~ ( G) be their complete preimages. Since the indicator K \8 is biinvariant,

we obtain

J c(A, B, r)x(r)dr = f (K~ * K\8)(g,)<p(g,)dg] Q G

= f K~(g2) f f K\8(g,)<p(g2 hg,)dg,dhdg2 G H G

= f f K~(g2)<P(g2)K\8(g,)<p(g,)dg,dg2 GG

= f X(p)dp J X(q)dq = X(A)X(B). A B

Conversely, let X (p) be a character of the hypercomplex system L, (G, H). We set

(gl'g2) = f <p(g,hg2)dh. H

In view ofthe unimodularity of the group H, the function (g l' g2) is biinvariant with

respect H in each variable, i.e., for any h" h2' h3' h4 E H, we have (h2g, h2'


h 3 g2h4 ) = <P(gI' g2)' Consequently, we can introduce the function <P(Pl'P2) = <P(g1'

g2) (gj E Pj; gj= 1, ~ definedon Q x Q. The function <P(gI' g2) is continuous by

the Lebesgue theorem (the function <peg) is continuous and bounded since I <peg) 1= Ix(p)I~I). We fix gj,g2E G and set PI = Hg1H,P2 = Hg2H. Let E~1) and E~2) be sequences of neighborhoods of the points P t and P 2' respectively, contracting to

these points and let ~(!) and ~(;) be their complete preimages. Then

<p(gt)<P(g2) = X(P!)X(P2)

= . 1 f ( (I) (2) ) hm (E(1») (E(2») c En ,Ek ,q X)q)dq

k,n--?oo m n m k Q

=

=

=

= <P(PI'P2) = <P(gl' g2) = f <peg! hg2)dh. H

• By virtue of the last theorem, it is important to know conditions of boundedness of

spherical functions. Thus, functions of the type (2.4) are bounded if and only if - 1 ~

Re s ~ 1 (see [LanD.

2.3. Representations of Class I. In this subsection, we establish a one-to-one cor

respondence between the set of spherical.positive definite functions on the group G and

the set of irreducible unitary representations of G with H-invariant vector.

A unitary representation 1t of the group G in a Hilbert space J{ is called repre-


sentation of Class I with respect to a subgroup H if it is irreducible and there exists a

vector ~:;t 0 in :J{ such that 1t (h ) ~ = ~ for all h E H.

Theorem 2.4. Let (G, H) be a Gelfand pair. Then, for any irreducible unitary

representation of the group G, the subspace of H-invariant vectors is at most one-dimensional. The set of equivalence classes of representations of Class I is in a one-toone correspondence with the set of positive definite spherical functions. This correspondence is given by the formula <p (g) = (1t (g)~, ~ ) J{ (1t is a representation of

Class I, ~ is a unit vector invariant with respect to all 1t (h), h E H, and <p is a

positive definite spherical function J.

Proof. Let us prove the first part of the theorem. Let G :3 g 1--7 1t (g) be an irre

ducible unitary representation of G in the Hilbert space :Jl. We consider a representation of the subalgebra K c L] (G, dg) ofbiinvariant functions

K :3 X 1--7 1t(x) = f x(g)1t(g)dg G

and set we = {~E :J{11t(h)~ = ~ for all hE H}. The subspace we is invariant with respect to 1t (x) (x E K) because, for all h E H,

1t(h)1t(x)~ = f 1t(h)1t(a)x(a)da~ = f 1t(ha)x(a)da~ G G

= f 1t(a)x(h-la)da~ = f 1t(a)x(a)da~ = 1t(x)~. G G

We set x*(g) = x(g-l) (x E K). Then x* E K and, since G is unimodular,

(1t(x)<p, 'IJ)H = f x(g)(1t(g)<p, 'V)Hdg G

= f x(g-l )(<p, 1t(g)'V)H d g = (<p,1t(x*)'V)J{' G

(2.5)

By Ax (x E K) we denote the restriction of the operator 1t (x) to we. By virtue of

(2.5) and the fact that K is commutative, the operators A x form a commutative family

of bounded normal operators in the Hilbert space we. By virtue of the spectral theorem (see, e.g., [BeKo]), every operator Ax can be represented in the form

Ax = f A(x)dE(A(-», [K


where dECA(')) is a joint resolution of the identity for the operators A X' Since E(~)

(~c (CK) commute with all operators Ax (x E K), the subspace E(~)'im is invariant

with respect to all A x (x E K).

Assume that dim'im > 1. Then either all operators A x are multiples of the identity

operator or E(~)'im differs from {O} and 'im for some ~. In both cases, 'im can be

represented in the form 'im = 91 1 Et1~, where 911 and 912 are mutually orthogonal nonzero closed subspaces of 'im invariant under the action of all operators A x and,

hence, 1t (x) (x E K). We take ~ E 91 1 and denote by n the closure of the set

{1t(X)~IXE LI (G,dg)}. Since

1t(g)1t(X)~1 = J x(a)1t(ga)~1 da = J x(g-la)1t(a)~2da = 1t(g_IX)~1 G G

(gE G,XE LI(G,dg); gx(h)=x(gh),g,hE G),

the space n is invariant under the action ofthe group G. Moreover, n -:t:. (0) because

~ 1 E Q. Let us show that n does not coincide with the entire :;{ (this contradicts the

irreducibility of 1t). To do this, it suffices to show that n.l 912 , Indeed, by virtue of the equality ~(h) = 1 (x E K), we have

(1t(x)~\, ~2)J{ = J x(g)(1t(g)~], ~2)J{dg G

= f J f x(g)(1t(h1gh2)~1' ~2)J{dgdh]dh2 GHH

= J J J X(hlgh2)dhldh2(1t(g)~\, ~2)J{dg HGH

for all x E L] (G, dg) and ~2 E 912 since

x*(g) = J J x(h l gh2)dh]dh2 E K HH

The first part of Theorem 2.4 is proved. Let us prove the second part of the theorem. Let with a cyclic vector ~o

such that <p (g) = (1t (g) ~o' ~o) J{fj) Let us show that 1t (g) is a representation of Class I.

By virtue of the invariance of the function <p with respect to H, we have


Since the vector ~o is cyclic, we have 1t(h)~o = ~o (h E H).

In order to prove the irreducibility of the representation 1t (g), we consider a bilinear

form in the space J{q>

B (~, 11) = J (1t ( h )~, 11) j{ II11IIj{q>' there exists abounded operator Pin J{q> such that

(P~ 11)j{q> = B(~, 11) ~,11 E J{q»' By virtue of (2.2), for all 11 = 1t(a)~o (a E G),

we have

(P1t(g)~o,11)j{q> = J (1t(h)1t(g)~o,1t(a)~o)j{q>dh H

= J (1t(a-lhg)~o'~o)j{q>dh = J <p(a-1hg)dh = <p(a-I)<p(g) H H

Since ~o is cyclic, we conclude that P1t(g) ~o = <p(g) ~o and P is a projector onto the

one-dimensional space generated by the vector ~o. Let J{' be the closure of the sum of

all closed subspaces V' c J{q> invariant under 1t and satisfying the condition P V' =

{O}. Then J{' and its orthogonal complement J{" in J{q> are invariant under 1t. Let

V be a closed subspace of j{" invariant under 1t (V:;:. {O}). Then P V:;:. {O} and, in

view of the fact that P j{q> = {A~O I A E a:}, we have ~o E V. But then 1t (g) ~o E V

for all g E G which gives V = j{q> because ~o is cyclic. Consequently, j{q> = j{"

and the representation 1t (g) is irreducible.

Conversely, let 1t(g) be a representation of Class I of the group G in a Hilbert

space Jf., let we be a (one-dimensional) subspace of vectors invariant under the action

of transformations from H, and let ~o E we be a unit vector (II ~o II j{= 1). Let us

show that the positive definite function <peg) = (1t(g)~o' ~O)j{ is spherical. Since we is invariant under the action of 1t (x) (x E K) and is one-dimensional, the vector

1t(x)~o is a scalar multiple of the vector ~o. Since

we have

(1t(x)~O' ~o)j{ = J x(g)<p(g)dg, G


(2.6)

Since the mapping x H 1t (x) is a representation of the sub algebra K of the group algebra L1 ( G, d g) formed by orbital functions and the latter is isometrically isomorphic to the hypercomplex system L1 ( G, H), the mapping

L1 (G, H) 3 x H J x(g)<p(g)dg

G

is a homomorphism of L1 (G, H) in a: 1. By virtue of Theorem 2.3, this implies that <p

is a spherical function on G. Since (1t (g ) ~, ~).'If does not depend on the choice of the unit vector ~ E we and on

the choice of the representation 1t in the corresponding equivalence class of irreducible representations, we conclude that the formula <p (g) = (1t (g)~, ~).'If defines a mapping

of the set n of equivalence classes of irreducible representations of Class I into the set S of positive definite spherical functions. This mapping is surjective since (x)~o' ~o)q>' where Cx) is the representation associated with E

00. The required mapping of the Hilbert space j{ (the space of the representation 1t)

onto j{q> is given by the formula

r r

L ai1tCg i)~ H L ai1tq>(g ;)~o' i=1 i=1

where g l' ... , g rEG and a1,'" , arE ([ 1 .

• It is well known (see, e.g., [HeI2]) that bounded spherical functions are not necessari

ly positive definite. In [Hull] (Theorem 4.2), it was proved that, for every bounded spherical function to be positive definite, it suffices that the group G be such that, for

every compact set A c G and every c > 0,

A(A n) = A(A. A· .... A) = o(cn) '------v--'

n

as n ---7 00 (here, A is the Haar measure of the unimodular group G). Clearly, compact groups satisfy this condition. Moreover, in [Hull], it was shown that locally compact


nilpotent groups (in particular, Abelian groups) and some other classes of groups also satisfy this condition. By using the fact that positive definite functions satisfy the equal

ity <p(g-I) = <peg) (g E G), Theorem 4.2 in [Hull], and Theorem 2.3, we conclude

that if (G, H) is a Gelfand pair, and the group G is compact or nilpotent, then the

hypercomplex system LI (G, H) is symmetric, i.e., every its character is Hermitian.

For real noncompact semisimple Lie groups, the description of positive definite spherical functions (with respect to the maximal compact subgroup) was given by Kostant [Kos] (see also [FlK2]).

2.4. Harmonic Analysis on Gelfand Pairs. In the present subsection, on the basis of the results of Section 3 in Chapter 1, we establish some theorems of harmonic analysis

for the hypercomplex system LI (G, H) and give three examples of hypercomplex systems associated with Gelfand pairs. These hypercomplex systems are related to simple examples of symmetric spaces, namely, to spaces of compact, noncompact, or Euclidean

type.

Lemma 2.1. The support of the Plancherel measure dm(<p) = d<p lies in the set S

of positive definite spherical functions.

Proof. Note that if a function y E La( G, dg) is orbital, then the following equality

is true for any function x E La,(G, dg) (l/a + l/a' = 1):

f x(g)y(g)dg = f x(g) f f y(h- I gh'-I )dhdh'dg

G G HH

= f f f x(hgh')dhdh'y{g)dg

GHH

= f xft (g)y{g)dg = f xft (p)y(P)dp, (2.7)

G Q

where

xft (g) = f f x(hgh')dhdh'.

HH

The function z{p) = {x** x)ft (p) (p E Q) is positive definite in the hypercomplex

system LI (G, H) for any function x(g) E Co(G). Indeed, for any function y E

LI (G, H), we have


f z(p)(y** y)(p)dp = f (x** x)(g)(y** y)(g)dg Q G

It is obvious that the function z (p) is bounded and continuous. Moreover, since x * * X E

L1 (G, dg), we have z(p) E L j (Q, dp).

By virtue of Corollary 3.4 in Chapter l, z(x) ~ 0 for all X E supp m c X h. At the

same time, according to Theorem 2.3, Xh belongs to the set of bounded spherical func

tions. Consequently, if <p E supp m, then, for any function x(g) E Co(G), we get

f <p(g)(x** x)(g)dg = f z(p)<p(P)dp = £(<p) ~ 0, G G

i.e., the function <peg) is positive definite.

• Thefollowing equalities take placeforallfunctions x E K and y E Lj (G, dg):

(2.8)

Indeed,

(x*y)lf(g) = f f (x * y)(hgh')dhdh' HH

= f f f x(hgh'a)y(a- l )dadhdh' HHG

= J J J x(hgh'a- j )y(a)dadhdh'

HHG

= J J J x(gh'a-1h)y(a)dadhdh'

HHG

= f f x(ga- I )y(h' ah)dadhdh'

HH

= J x(ga- l )ylf(a)da = (x * ylf)(g).

G


The second equality in (2.8) is proved similarly.

• By applying the results of Section 3 in Chapter 1 to the hypercomplex system L 1 (G,

H), we obtain the following theorem which actually contains the main results of harmonic analysis on the hypercomplex system LI (G, H):

Theorem 2.5. On the set S of positive definite spherical functions equipped with

the topology of the space of maximal ideals of the hypercomplex system LI (G, H),

there exists a regular Borel measure d<p such that the following statements are true:

(i) (Plancherel-Godement theorem, see [God]) the space L2 (Q, dp) of biinvari

ant functions square summable with respect to the Haar measure on G is isometrically isomorphic to the space L2 (S, d<p) and the Parseval equality takes

place

f x(g)y(g)dg = f x(<p)y(<p)d<p (X,YE L2(Q,dp», (2.9) G S

where x( <p) is the spherical Fourier transform of the function x given by

x(<p) = f x(g)<p(g)dg (2.10)

G

for x E L] (G, dg).

(ii) (inversetheorem,see[BerJ). Let W be the linear span of p(Q)nL1(G,dg),

where P (Q) is the set of continuous biinvariant functions on G positive definite with respect to the hypercomplex system LI (G, H) (i.e., the set of func

tions satisfying relation (3.1) of Chapter 1 for all x(g) E L] (G, H»). For any

function x E ~n, its spherical Fourier transform x(<p) belongs to L] (S, d<p)

and the following inversion formula holds:

x(g) = f x(<p)<p(g)d<p. (2.11 ) S

(iii) (an analog of the Bochner theorem, see [God]). For every biinvariant function

f (g) (g E G) to be representable in the form

f(g) = f <p(g)dJ.1(<p) (g E G) (2.12) Xh


where f.l is a nonnegative finite regular measure on the space X h of bounded

spherical functions, it is necessary and sufficient that it be continuous and positive definite with respect to the hypercomplex system L I (G, H). If f (g) is

positive definite on the group G, then the measure df.l in (2.12) is concentrated on the set S of positive definite spherical functions.

(iv) (an analog of the Pontryagin duality theorem). The product of any two positive

definite spherical functions is positive definite; therefore, if G is amenable, then duality takes place for the hypercomplex system Ll (G, H).

(v) every continuous biinvariant function on G can be uniformly approximated by

linear combinations of spherical functions on every compact set.

Proof. The statements (i), (ii), (v), and the first part of (iii) follow from Theorems

3.3,3.2,3.13, and 3.1 in Chapter 1 and the fact that supp m c S (Lemma 2.1). It is well known that the product of any two continuous positive definite functions on a group is positive definite (see, e.g., [Dix2], Subsection 13.14.9). Moreover, Voit proved that

1 E supp m if and only if G is amenable [Voi7]. In this case, supp m = Sand (iv) follows from Theorem 3.5. Therefore, it remains to prove the second part of (iii).

Let f(g) be a continuous biinvariant positive definite function. It can be represented

in the form (2.12). Assume that there exists <p(g) E X h such that <p E (X h\S) n supp f.l.

It is well known ([Dix2], Section 13.5) that S is closed in X h and the topology of S

coincides with the topology of convergence on compact sets. Therefore, there exists a

neighborhood U of the point O. Since the func

tion < 0 G

for all <p E U (here, we have used equality (2.7». Therefore,

f (x* * x)" (<p)df.l(<p) = f (x* * x)#"(<p)Ku(<p)df.l(<p) < O. u x h

On the other hand, since the set £) (Q, d p) is dense in Coo (X h) in the uniform norm,

Coo (X h) is dense in L2 (X h' df.l), and the measure f.l is finite, we can find a sequence of

functions XnE LI(G,H) suchthat II.in -KuIIL2(Xh,dJl)~0 as n~oo. By using the


fact that the function f(g) is positive definite, equalities (2.7) and (2.8), representation (2.12), and the Lebesgue theorem, we obtain

0:::; Jf(g)[(x*xn)**(X*xn)](g)dg G

= J f(g)(x~ * (x** X)9 * xn)(g)dg G

= J J q>(g)#L(q»(X~ * (x** X)9 * xn)(g)dg G xh

= J (x~ * (X**X)9 *xn)-(q»dA(q» xh

= J I xn(q»12(x* * x)#''(q»dll(q» xh

J Ku(q»(X* * x)#I\(q»dq>. xh

At the same time, somewhat earlier it was shown that the last integral is negative .

• Remark. Let (G, H) be a Gelfand pair and let G be a noncompact group. Then

the conditions of Theorem 3.5 in Chapter 1 are, generally speaking, not satisfied. Thus,

let G = SL2 (1R) and H = SO(2). The spherical functions q> s(g) can be found from

(2.4), X = {s 1-1:::; Re s:::; I}, and the Plancherel measure

h S (1tS) ds dm (s) = -;- tanh -. -. I I 2m

(Res= 0)

(see, e.g., [Lan]). Thus, Q = supp m = {s E a:: I Res = o} and 1 (g) = q>_1 (g) $. Q. Hence, in the general case where the group G is not amenable, we fail to prove the

duality theorem (iv). Nevertheless, we can construct a hypercomplex system on Xh (generally speaking without basis unity). HypercompJex systems of this sort (in terms of

the operation of generalized convolution of measures on X h) were studied in [May] (see also [Miz1] and [Miz2]).

In conclusion, we present several examples of hypercomplex systems associated with Gelfand pairs.


Example 1. Let G = SO(3) and H = SO(2). For definiteness, we assume that SO(2) is realized in SO(3) as the group of rotations about the axis Oz. As is khown,

every element g of the group SO(3) is parametrized by the Euler angles g = g 1' q>2::;; 2n, 0::;; 8::;; n), where

sinq>

cosq>

o

o cosq>

-sinq>

this parametrization is unique if 0 < 8 < n. The homogeneous space SO(3) 1 SO(2) is

homeomorphic to the unit sphere in 1R 3 , and S 0(2) is the stationary subgroup of the

point Z 0 = (0, 0, 1). Obviously, SO(3) /1 SO(2) is homeomorphic to [0, n]. Since

dg = (8n2)-\ sin8dq>\dq>2d8 is the Haar measure on SO(3) (see, e.g., [Vii]), the

measure (1/2) sin q> d q> is the multiplicative measure of the hypercomplex system

L \ (G, H). For convenience, we consider a hypercomplex system isomorphic to L 1 (G,

H). Consider the homeomorphism [0, n] 3 8 H t = cos 8 E [-1, 1] = Q. The measure

(l/2)sinq>dq> tums into the Lebesgue measure (1/2)dt on Q. By virtue of (2.1), we have

By representing g=g9 1g <pg92 in the form gif>\ g9 gif>2' we get cos8=cos81cos8 2-

sin 8 1 sin 8 2cOS q> and, hence, the generalized translation operators in L \ (Q, ( 1/2) d t) have the following form:

1 21t (Lsf)(t) = - ff(cos8\cos8 2-sin8\sin8 2cosq»dq>

2n o

1 21t = - f f ( t s - .J1=t2 ·,h - s2 cos q> ) d q> (f E C (Q)).

2n 0

This enables us to establish the multiplication law in L \ (Q, (1/2 )dt):

(x*y)(t) =

(X,yE L\(Q,(l/2)dt».


As is khown (see, e.g., [Hell]), in this case, the spherical functions have the form

<Pn(g6) = Pn(cos 8)(n = 0, 1, ... ; 8 E [0, nD, where Pn are the Legendre polynomials

normalized by the condition P n( 1 ) = 1. Therefore, the set of Legendre polynomials

P n( t) (t E [- 1, 1], n = 0, 1, ... ) forms the system of characters of the hypercomplex

system L 1 (Q, ( 1/2) d t), and relation (2.2) turns into the multiplication formula for the Legendre polynomials, namely,

The set Q = {O, 1, ... } is the basis of the dual hypercomplex system, and the Plancherel measure is given by the relation

= 2k+ 1

(see, e.g., [Sze]). Thus, harmonic analysis on the hypercomplex system Ll (Q, (1/2 )dt) reduces to the Fourier analysis on the Legendre polynomials.

Remark. Similarly, for the Gelfand pair (SO( n), S O( n - 1», we establish that the characters of the hypercomplex system L 1 (SO(n), SO(n - 1» have the form

(k=O, 1, ... ; tE [-1,1]),

where U1n- Z)IZ(t) are ultraspherical polynomials (see, e.g., [Vil]).

Example 2. Let G = SLz(lR), let H = SO(2), and let A be the group of diagonal unimodular matrices

Denote by A + the set of matrices a E A such that a ~ 0. By using the polar decompo

sition, one can uniquely represent any matrix g E S L Z (lR) in the form g = s h, where h E SO(2) and s is a positive symmetric matrix. Moreover, by orthogonal transformation, the matrix s can be diagonalized. If the obtained diagonal matrix a belongs to

A \A +, then the adjoint matrix mam- 1 belongs to A + (here,


ill = (~1 ~) E SO(2)

is the Weyl matrix). If a*" 1, then the matrix hi in the expression s = hll ah I (a E A +,

h 1 E H) is uniquely defined to within multiplication by ± 1. Thus, we obtain the de

composition G = HA +H (the Cartan decomposition), which is uniquely defined for

a*"1 to within the multiplication of the matrices hi, h 2 by ± 1. Consequently, Q = Gil H = A +. Below, we represent matrices from A + in the form

(A, ~ 0).

Thus, Q can be identified with the semiaxis [0,00). Denote the elements of H = SO(2)

by h = g (0 ~ <p ~ 21t). Relation (1.31) takes the form <p

1 21t 21t 00

f f(g)dg = 81t f J f f(g<PlaA,g<p)sinhA,dA,d<pjd<P2 GOO 0

and, hence, the measure (1t/2) sinh A,dA, is the multiplicative measure on Q. It is convenient to consider a hypercomplex system isomorphic to L j (G, H). Namely, consider

the homeomorphism [0, 00 ) :3 A, H t = cosh A, E [1, 00). The measure (1t /2) sinh A, d A,

tumsintotheLebesguemeasure (1t/2)dt on [1,00). Byvirtueof(2.1),

Rewriting a A,lg <paA,2 in the form g 6 laA,g 6 2, we obtain cosh A, = cosh A, I cosh 11,2 +

sinh A, 1 sinh 11,2 cos 2<p. Consequently,

By using the explicit form of the generalized translation operators, one can easily es

tablish the multiplication law in L I (Q, (1t1 2 ) d t). The spherical functions in this case are expressed via the Legendre functions (see, e.g., [ViI]):

Section 2 Gelfand Pairs

The integral representation of the Legendre functions

1 2It Pp(COShA) = -f(coshA+sinhAcosu)Pdu

2It o

and the multiplication formula

follow from (2.4) and (2.2), respectively.

245

The relations obtained in Example 2 are similar to the corresponding relations in Example 1. This is explained by the fact that the corresponding Riemann spaces are in Cartan duality (see, e.g., [Hell]).

Example 3. Let G = M 0 (2), where M 0 (2) is the identity component of the group

M (2) of all motions of plane, and let H = SO(2). The group M 0 (2) = JR 2 Xl SO(2) is isomorphic to the group of matrices of the form

( cosa

g(a, b, a) = l-s~na sina

cosa

° ~l (a, b E JR 1, a E [0, 2It]).

It is easy to see that G / H = JR 2, and Q = G / / H is parametrized by the semiaxis [0, 00 )

(every orbit Hg(a,b,cp)H is a circle of radius r= ~a2+b2 E [0,00) in JR2). Since

(l/4rc 2)dadbda is the Haar measure of the group Mo(JR 2), by passing to the polar coordinates, we establish that rdr is the multiplicative measure of the hypercomplex system L 1 ( G, H). Let us find the explicit form of the generalized translation operators

associated with L 1 (G, H). Rewriting g(a l' b l' O)g(O, 0, a )g(a2' b 2, 0) in the form g(a,l},a), we obtain

r =

246

and, therefore, .

Examples oj Hypercomplex Systems

1 21t

(L r J)(r2) = 21t J J( ~ r? + ri + 2'V2 cosa )da. o

Chapter 2

It is khown (see, e.g., [Vii]) that, in this case, spherical functions are expressed in terms of the Bessel functions

. I 21t <Pip(g(a, b, a)) = Jo(pr) = - J eiprsin<Pd<p (r = ~a2 +b2 , p? 0).

21t 0

It is easy to derive the multiplication formula for Bessel functions from the explicit form of the generalized translation operators Ls. The Fourier transformation associated with

the hypercomplex system LI (G, H) reduces to the Hankel transformation. As is khown,

in this case, the Plancherel measure is concentrated on Q = [0, 00) and has the form

pdp.

Remark. If G is the group M o(n)= lRnXl SO(n) of motions of the Euclidean

space lR n preserving orientation and the subgroup H is equal to S O(n), then Q = G / / H = [0,00) and the characters of the hypercomplex system LI (G, H) are expressed

in terms of Bessel functions J(I)-2)12 as follows:

(r = [0, 00), p? 0)

(see [ViI]).

2.5. Hypercomplex Systems Associated with the Delsarte Generalized Translation Operators. Let G be a locally compact group and let B be a compact subgroup of the group of automorphisms of G. Then the set of B-orbits equipped with the quotient topology is a hypergroup with respect to natural convolution. This important class of hypergroups was introduced by Delsarte [DeI4]. In the present subsection, we show that hypergroups of this sort are generated by Gelfand pairs.

Let G be a locally compact unimodular group and let B be a subgroup of the group

Aut G containing the group of inner automorphisms I ( G). The group G is called a

[FIA]B-grouP if the closure B- of B in Aut G is compact. The set B-(g) = {~(g) I ~ E B-} is called a B--orbit of the element g E G. Consider the space GB of B-orbits

equipped with the quotient topology.

The subalgebra


of the group algebra of the group G is commutative. Moreover, it is a hypercomplex

system and its basis GB consists of B--orbits of the group G.

Obviously, fE ZBLI (G) if and only if it is constant on B--orbits. For all g E G

and x, y E ZBLI (G), we have

(x*y)(g) = J x(a)y(a-1g)da G

= J x(a)y(a- 1 ga-1a)da G

= J x(a)y(ga- 1 )da = (y * x)(g). G

By the general construction presented in Subsection 1.1 of Chapter 2, it suffices to

show that any function x(g) E ZBLj (G) can be approximated in the norm of the space

L, (G, dg) by continuous finite functions taking constant values on B- -orbits. This can be done in exactly the same way as in Subsections 1.2 or 2.1 of Chapter 2.

• If B does not contain leG), then ZBLj (G) is also a hypercomplex system (gener

ally speaking, noncommutative).

Generalized translation operators associated with the hypercomplex system

Z B L, (G) are called Delsarte generalized translation operators and have the form

(2.13)

where y is the Haar measure on the group B- normalized by the condition y(B-) = 1.

Indeed, by introducing necessary (obvious) modifications in the proof of relation (2.1), we arrive at the first formula in (2.13). The second formula follows from the in-

variance of the function f and the fact that the compact group B- is unimodular.

• In particular, it follows from (2.13) that the hypercomplex system ZBL, (G) satisfies

the condition of separate continuity. Clearly, the orbit {e} is the basis unity of this hy

percomplex system. It is also easy to check that the convolution of measures on GB


generated by generalized translation operators (2.13) satisfies all axioms of hypergroup.

Let us show that the space G B can be identified with the space Q = Gil if of

double cosets of the group G with respect to if for a properly chosen Gelfand

pair (G,if).

Indeed, we set G = G ><I B- and if = {e} ® B- = B-. Let (g,~) E G, then

This implies that Gil if is homeomorphic to G B. In order to show that (G, if ) is a

Gelfand pair, we define an involutive automorphism cr as follows: cr «g, ~)) = (r I , P g 0 ~), where p g is the inner automorphism of the group G generated by an element

g. Let us show that cr is indeed an automorphism. Since ~ 0 Pg = P 13(g) o~, we have

Obviously, cr is an involutive automorphism. Since

by virtue of Theorem 2.1, we conclude that (G,if) is a Gelfand pair.

• Note that if G is commutative, then we can take a simpler automorphism cr«g, ~)) =

(g-I,~) instead of cr. Thus, the sub algebra 2BLI (G) of the group algebra LI (G, dg) coincides with the

hypercomplex system LI ( G, if). It is easy to see that characters of the hypercomp1ex

system 2BLI (G) are continuous functions not identically equal to zero on G taking


constant values on B--orbits, and such that

<P(gl)<P(g2) = f <p(gl~(g2»d~. (2.14) B-

As shown in [Mos], the space of characters of the hypercomplex system Z B LI (G)

can be identified with the set of extreme points of the space of B- -invariant positive definite functions <p on G normalized by the condition <p (e) = 1 and equipped with the topology of uniform convergence on compact sets. Harmonic analysis on the hypercompie x system Z B LI ( G) was constructed in [Mos] and [LiM]. Obviously, these results can also be derived from Theorem 2.5. In [HHL], it was also shown that the space of Hermitian characters E (G, B) of the hypergroup G B is a commutative hypergroup with

respect to the convolution of measures on E (G, B) which is defined as follows: Since the product of positive definite functions <p, \If E E( G, B) is positive definite, by virtue of Theorem 3.1 in Chapter 1, we have

(<p\lf)(p) = f e (P) dllcp, lJI(e) (p E GB), E(G,B)

where IlCP,1JI is a probability measure on E(G, B). Further, for any <p, \If E E(G, B),

we set Ocp * 0'1' = Ilcp,'V" According to [Hart], on the hypergroup E(G, B), there exists

the left-invariant measure m whose support coincides with E( G, B). The measure m normalized in a proper way turns into the Plancherel measure. By applying Theorem 3.5 to the hypercomplex system Z B LI (G), we conclude that duality is true for the hyper

complex system ZBLI (G) (this result was first established in [Hart]). It is worth not

ing that the subclass of [FIA]B-groups composed by so-called Z-groups, i.e., locally

compact groups G with center Z such that the group G / Z is compact, is fairly well

studied (see [GrM], [GMM], [Rosl], [Ros2], and [HKKD. In this case, B- = leG).

2.6. Center of Group Algebra as a Gelfand Pair. If G is a compact group and B

is the group of all inner automorphisms of the group G, then the reasoning used above can be significantly simplified. In this case, the conjugacy classes of the group G are

B --orbits and the hypercomplex system Z B LI ( G) coincides with the center Z LI (G)

of the group algebra of the group G. This hypercomplex system was studied in detail in

the previous subsection. In this case, we take the direct product G ® G as G and the

subgroupofallpairs (g,g) (gE G) as iI. Clearly, relation (2.12) for characters of the hypercomplex system ZL1 (G) can be reduced to the Weyl formula (1.20).


3. Orthogonal Polynomials

As shown in Section 2, some known systems of orthogonal polynomials form sets of spherical functions of a Gelfand pair (G, H), where G is a compact group and H is its subgroup. In particular, spherical functions of Riemannian symmetric spaces of compact type are orthogonal polynomials of, generally speaking, several variables (the number of variables is equal to the rank of the symmetric space, [Vret]). Thus, these families of orthogonal polynomials form the set of characters of a normal hypercomplex system

with basis unity and compact basis G / H. It is natural to construct examples of hyper

complex systems with compact basis E C IR n (n ~ 1) whose sets of characters coincide

with given systems of orthogonal polynomials on E. This is done in Subsection 3.5. The main attention in this section is focused on the analysis of the dual case, namely, for

fairly general families of orthonormal polynomials P n(t) (t E 1R 1; n E No = {O,

1, ... n, we construct discrete hypercomplex systems with basis No whose characters

are X (n) = Arf n( z) (An E IR, Z E <t) and study harmonic analysis for these systems.

3.1. Discrete Hypercomplex Systems Associated with Orthogonal Systems of Polynomials. In this subsection, we construct a discrete Hermitian hypercomplex system with the basis Q = {O, 1, ... }, the nonnegative structure measure, and the basis unity

e = 0 associated with a given family of orthogonal polynomials defined on a bounded

closed subset E C IR 1. We describe the set of characters and give a sufficient condition for this hypercomplex system to be symmetric.

Let E be a closed bounded subset of the real axis and let dcr be a nonnegative measure on E with infinitely many points of growth satisfying the conditions cr(E) = 1,

f I tlndcr(t) < 00 (n=I,2, ... ), and suppcr=E. Let poet), PI(t), ... be a sequence

of polynomials orthonormal with respect to the measure dcr (t) with positive leading coefficients and let a = inf E and b = sup E. As is known (see [Sze]), all zeros of the family of orthogonal polynomials P n( t) are located inside the interval (a, b). Hence,

Pn(b»O and (-ltPn(a»O (nE N). We set

Theorem 3.1. If f Pp(t)PqCt)Pr(t) dcr (t) ~ 0 for all p, q, r = 0, 1, ... , then

c (p, q, r) are structure constants and m (p) is the multiplicative weight of a certain

Section 3 Orthogonal Polynomials 251

Hermitian hypercomplex system with the discrete basis No = {O, 1,2, ... } and basis

unity O. We denote this hypercomplex system by I] (dcr, m). The space X of char

acters of the hypercomplex system I] (dcr, m) consists of the set of points Z of the

complex plane such that the sequence R n(z) is bounded. The space of Hermitian

characters is Xh = X n IR.

Proof. Orthogonal polynomials possess the property that f Q (t)P n(t )dcr (t) = 0

for any polynomial Q(t) of degree less than n. Hence, f Pp(t)Pq(t)Pr(t)dcr(t) = 0 for

all r > p + q and r < 1 p - q I. This obviously implies that the functions R pC t) (p E

No) form an A-system. By applying Theorem 3.8 from Chapter 1, we establish the validity of the first part of the theorem.

Note that, for all z E a:], orthogonal polynomials satisfy the equality

00

Pp(z)Piz ) = L Pr(z) f Pp{t)Pit)Pr(t)dcr(t) (p, q E No)· (3.2) r=O

By dividing both sides of (3.2) by P p( b )P q( b), we obtain relation (1.27) from Chap

ter 1, whence it follows that if the sequence Rp(z) is bounded, then X (p) = RpCz) is a

character of the hypercomplex system I] (dcr, m).

Conversely, assume that X (p) (p E No) is a character of the hypercomplex system

I] (dcr, m). Then it follows from (3.2) that

p+q = L X(r)Pr(b) f Pp(t)Pit)Pr(t)dcr(t) (p, q E No)·

r=lp-ql

This implies that all values of the character X (p) are uniquely determined by the values

X(O)=I and x(1)· We set z = X(1)(b-cr]) +0"], where cr] = f tdcr(t) is the first

moment of the measure dcr. Since

we have X(O)=Po(z) and X(l)P 1(b)=P1(z). This and (3.2) imply that X(r)Pr(b)=

P r(z) or X (r) = R r(z) (r E No). Since the character X (p) is bounded, the sequence

RpCz) is also bounded. The equality X h = X n IR follows from the fact that Hermitian

characters of an Hermitian hypercomplex system are real-valued.

•


Remark 1. It may happen that (-1l+ q+r J PpCt)P qCt)Pr(t)dcr(t) ~ 0 for all

p, q, rENo. In this case, it is necessary to set

S(t)- Pp(t) []2 p - Pp(a) , m(p) = Pp(a) ,

c(p, q, r) = m(p)m(q) J Sp(t)Sit)Sr(t)dcr(t).

For the numbers c(p, q, r) and m (r) thus defined, all statements of Theorem 3.1 are true.

Remark 2. Note that X ~ {z E ([111 z - crl I ~ b - crl} and, in particular, X h ~ {2crI-b,b}.

Remark 3. By virtue of Theorems 3.2 and 3.3 in Chapter 1, there exists a regular Borel measure drc on X h such that the polynomials P r(t) are orthogonal with respect to it. For a system of polynomials orthonormal on a bounded subset of the real axis, such

a measure is unique (see, e.g., [BerlO]). Consequently, we have dcr = drc. Since Q =

E=suppcr=suPPrc~Xh' we conclude that IRr(t)1~1 for all tE Q or Ipr(t)I~

Pr(b) for all tE E.

Remark 4. One can easily check that if conditions of Theorem 3.1 are satisfied, the

functions R~(t) = Pp(t)/ Pp(c) form an A-system for any c> b. Consequently, they

also can be used for the construction of an Hermitian hypercomplex system l{(dcr, m')

with basis unity. This hypercomplex system has the set of characters larger than E

because, on the one hand, E ~ X h and, on the other hand, the point c li!: E generates a character of the hypercomplex system identically equal to one.

By applying Theorem 3.1 from Chapter 1 to the constructed hypercomplex system, we establish the following statement: In order that a bounded sequence s(p) of real

numbers be representable as

s (p) = f R pC t )d~ (t ), Xh

where ~ is a nonnegative finite measure on X h, it is necessary and sufficient that

this sequence be positive definite in the sense of (3.1/) in Chapter 1.

Theorem 3.2. Suppose that the conditions of Theorem 3.1 (or the conditions of Re

mark 1) are satisfied. If a sequence P r (b) is bounded, then the hypercomplex system

II (dcr, m) is symmetric, i.e., X = X h .


Proof. It is necessary to verify that x(r) = x(r) or Pr(z) = Pr(z) (z E X). If

Z E Xh, then this equality is obvious. Therefore, assume that Z ~ 1R I. Let us fix r. For

sufficiently large N, we have

1 N

L f Pp(t)PqCt)Pr(t)dcr(t) Pp(z) PqCz) p,q=o

N N+r

L L J Pr(t)Pp(t)PqCt)dcr(t) Pp(z)PqCz) p=N-r+1 q=N+I

(3.3)

(here, we have used the fact that J Pp(t)PqCt)Pr(t)dcr(t) = 0 for all r > p + q and r <

Ip-ql). It follows from Remark 3 that Ipr(z)I~Pr(b) foraH ZE X and,hence, the

numbers Pr(z) are uniformly bounded. But then the sum

N N+r

L L f Pr(t)PpCt)Pq(t)dcr(t) Pp(z)PqCz) p=N-r+l q=N+l

is bounded in N. It is known (see, e.g., [BerlO]) that the series L;=o I PpCz) \2 is di

vergent for real z. Therefore, it follows from (3.3) that

N

L f Pr(t)Pp(t)Pq(t)dcr(t) Pp(z) PqCz). p,q=O

On the other hand, it is easy to see that the numbers

are real.

N

L f Pp(t)PqCt)Pr(t)dcr(t)Pp(z)Pp(z) p,q=O

•


3.2. Jacobian Matrices and Generalized Translation Operators. Here, we outline another approach to the construction of hypercomplex systems in terms of orthonormal polynomials Pr(t). First, we briefly recall several facts from the spectral theory of difference expressions (Jacobi matrices) (see [BerlO]).

For the difference expression with coefficients Clj and ~j

in the space l2 of sequences U = (u)'j=o, we construct an operator A as the closure

of the mapping 103 u ~ Lu (to is the linear subspace of l2 consisting of finite se

quences). To calculate (Lu)o we assume that the boundary condition u_l = ° is satis

fied. The operator A is Hermitian. We assume that Clj and ~j are such that this oper

ator is self-adjoint (we can even take bounded coefficients Cl j and ~j; in this case, the

operator A is bounded and, hence, self-adjoint). The Parseval equality for the decomposition of the operator A in the eigenfunctions has the form

(u, v)12 = J u(t)v(t)da(t) (u, v E 10)· (3.5) JRl

Here, da(t) is the spectral measure (equal to d(E(t)oo, (0)' E is the resolution of the

identity for the operator A, 00 = (1, 0, ... ) E 12 , and

00

fI(t) = L upPp(t) p=o

is the Fourier transform associated with the sequence (Pp(t));=o of so-called polynomials of the first kind, which can be recursively calculated as solutions of the Cauchy problem for the difference equation

As follows from (3.5), the polynomials Pp(t) form an orthonormal basis in ~(1R 1, da).

The inverse spectral problem now takes a fairly simple form, namely, for an arbitrary nonnegative probability measure da (t) defined on the axis (- 00, 00), having infinitely

many growth points, and such that all powers tP (p E No) are integrable, by applying

the orthogonalization procedure to the powers 1, t, t 2, ... , we construct polynomials Pp(t) with a positive leading coefficient. These polynomials are polynomials of the first

kind for the difference expression L whose coefficients are uniquely determined by the relations


00

up = f tPpCt)Pp+ I (t)dcr(t), ~p = f tP;(t) dcr(t).

Consider the partial difference equation

where (up,q);,q=o is an unknown infinite matrix. It follows from (3.7) that the ele

ments of the (q + 1 )th column of the matrix (up,q);,q=o are uniquely determined by

the elements of the qth and (q - 1 )th columns. This implies that the matrix Up,q is

completely determined by its "initial" values-the column with index q equal to zero.

On the linear span of the sequences x = (x(p));=o, we define a family of operators

(T~x )(p) = up, q' where the matrix (up,q );,q=o is determined by the "initial" value

Up, 0 = x (p) (p E Q). Let us show that T~x can be directly calculated by using the relation

(T~x)(p) = Up,q = L x(r) f Pp(t)Pit)Pr(t)dcr(t) (p, q E Q) (3.8) r

(clearly, the summation is carried out over finitely many indices 1 p - q I::;; r::;; p + q).

Indeed, since an element Up,q is completely determined by finitely many "initial" values

ur,o (I p - q I ::;; r ::;; p + q), it is sufficient to prove equality (3.8) for sequences x with finitely many nonzero terms. We set

Then, by the Parseval equality,

r

x(t) = L x(P)Pp(t). p

The matrix (vp,q);,q=o is a solution of equation (3.7). Indeed, by using (3.6), we get


Since vp,o = J Pp(t)x(t)dcr(t) = x(P) = up,o (p E ~+), we have Vp,q= Up,q' and re

lation (3.8) is thus proved. Also note that (T;P.(z»(p) = Pp(z)Piz ) for any p, q E Q, I

Z E [ .

We set

. [T;xP'(b)](p)

(Tqx )(p) = [T;P'(b)](p)

We introduce a convolution in II (m) as follows:

= [T;xP'(b)](q)

Pp (b)Pq (b)

(~*T\)(r) = L (Tp~)(r)T\(P)m(p) (~,T\E II(m), rE No)· p

(3.9)

It is easy to see that ifthe conditions of Theorem 3.1 are satisfied, then II (m) endowed with such a convolution is a commutative hypercomplex system with the structure measure

c(p, q, r) = L TrKp(l)KqCl)m(l) = T,~(q)m(q) I

P (b)P (b) J = p q Pq(t)Pp(t)Pr(t)dcr(t).

P,(b)

The operator Tp defined by (3.9) is a generalized translation operator in the hypercom

plex system II (dcr, m). Conversely, since Up,q are calculated in terms of the coeffi

cients fi.p' ~p of the difference expression L by the recurrence relation (3.7), formula

(3.8) allows us to calculate the integrals J Pp(t)Pit)Pr(t)dcr(t) (and, hence, the struc

ture constants c(p, q, r) of the hypercomplex system II (dcr, m) for a given difference expression.

3.3. Characterization of Hypercomplex Systems Associated with Orthogonal Polynomials. Below, we give a criterion for a particular hypercomplex system to be generated by a system of orthogonal polynomials.


Theorem 3.3. For an Hermitian hypercomplex system II (m) with basis Q = No to be generated by a system of orthogonal polynomials, it is necessary and sufficient that the following conditions be satisfied:

(i) m(O) = 1;

(ii) suppc(p,I,r)C[lp-ll,p+l] and c(p,l,p+l»O. This condition implies r

that suppc(p,q,r)C [lp-ql,p+q] forany p,qE Q. r

Proof. The necessity obviously follows from Theorem 3.1. Let us establish the suf

ficiency. Consider the mapping a: Xh ---7 [-1, 1] given by the relation X h 3 X ~

a(x) = X(I) E [-1, 1] (since the hypercomplex system is Hermitian, we have X (1) E

1R 1). Since

J X(l)X(O)dX = 0, Xh

we get X (1) ;F. X (0) = 1. Consequently, a(Xh):# {I}. Let us show that a is injective.

Let X (1) = '1'(1) for some X, 'I' E X h. By using equality (1.27) from Chapter 1 with

q = 1 and condition (ii), we obtain

p+I

X(p)X(l)m(p)m(l) = L c(p, I, r)x(r)m(r) (p ~ 1) (3.10) r=p-I

or

c(p,I,p+ I)X(p+ l)m(p+ 1)

= X(p)X(l)m(l)m(p)-c(p, 1,p)x(p)m(p)-c(p, 1,p-l)x(p-l)m(p-l).

Sincr c (p, 1, P + 1) > 0, any value of the character X (p) is uniquely determined by the

values X (0) = 1 and X (1). Consequently, X (p) = 'I'(p) for all p E Q, i.e., X = 'I' and a is injective. Let E = a (Xh) ~ [1, 1]. It is clear that a is continuous and, since

X h is compact, the set E is closed.

By writing (3.10) for the character Xt E Xh, setting Pp(t) = Xt(P) -Jm(p) , where

Xt = a-I (t) (t E E), and using the equality c(p, q, r)m(r) = c(r, q,p)m(p), we obtain


c(p -1, 1, p)m(p)Pp_l(t) ~ = I + c(p, l,p)'\jm(p)Pp(t)

'\jm(p - 1)

+ c(p, l,p + 1)..Jm(p + 1) Pp+1 (t)

or

P () c(p - 1, 1, p) m(p) P -I(t) t p t = m(1) m(p -1) p

+ c(p, 1, p) P (t) + c(p, 1, P + 1) m(p + 1) P () m(1) p mel) m(p) p+1 t .

If we set

a = c(p, 1, p + 1) ~ m(p + 1) p mel) m(p) ,

Ap = c(p, 1, p) I-' m(l) ,

the last equality can be rewritten in the form

(LP. (t») p = Up_I Pp_1 (t) + ~pPpCt) + upPp+ I (t) = tPp(t) (p E Q, tEE)

(3.11)

(by virtue of (i), Po (t) = X t (0) -J m(O) = 1; it is also easy to see that (3.11) holds for

p = 0). By virtue of condition (ii) and the fact that the structure constants are nonnegative, we have up> 0 and ~p;::: O.

Let us show that up and ~p are such that the operator A is essentially self-adjoint.

It is sufficient to show that up and ~p are bounded. Since the hypercomplex system is normal,

U = p c(p, p + 1, 1)

-Jm(p)m(p + 1) (p E Q).

But by virtue of (1.13) in Chapter 1, c (p, P + I, I) $; min { m (p ), m (p + I)} and

o < < min {m(p), m(p + I)} < 1 - up - ..Jm(p)m(p + 1) -.

Similarly, we get 0 $; ~p $; 1. Consequently, the polynomials of the first kind P pC t)


form an orthonormal basis in ~ (E, dcr), where dcr (t) is the spectral measure of the operator A.

Let us show that the relationship between c (p, q, r), m (r) and the polynomials

P p(t) is described by formulas (3.1). It is clear that P;(l) = Xf(p) m(p) = m(p) be

cause X 1 (p) = 8 -1 ( 1) == 1. The measure dX is the image of the Plancherel measure dcr under the mapping 8. Indeed, denoting by dp the image of dX under the mapping 8, by virtue of (3.14) in Chapter 1, we get

whence it follows that the polynomials Pit) are orthogonal with respect to the measure

dp. Since such a measure is unique for a system of polynomials orthonormal on a bounded subset of the real axis" we get dcr = dp. By (3.15) in Chapter 1, we have

c(p, q, r) = m(p)m(q) f X(p)x(q)x(r)dx

xh

• Remark. As follows from the example of the join of countably many hypergroups

constructed in Subsection 4.4 in Chapter 1, the condition c(p, 1, p + 1) is essential.

3.4. Another Method for the Construction of a Hypercomplex System Associated with Orthogonal Polynomials. Examples. In this subsection, we give another method, which is due to Lasser [Las2], for the construction of a discrete hypercomplex system associated with a system of orthogonal polynomials. First, let us prove an auxiliary result, which, nevertheless, is of independent interest.

Theorem 3.4. If a hypercomplex system II (m) is associated with a system of

orthogonal polynomials, then the structure measure c(p, q, r) is completely deter-

mined by its values c(p,p,O)=m(p), c(p, l,p), and c(p, l,p+ 1) (pE No).

Proof. By Theorem 3.3, supp c(p, q, r) ~ [lp - q I, p + q]. Let us write the asso-r

ciativity relation (1.25) in Chapter 1 for p=l, q=n-l, l=k, s=n+l (n,kE No).

Taking into account that supp c(l, n - 1, r) ~ [I n - 21, n], supp c(r, k, n + k) = r

suppc(k,n+k,r) ~ [n,n+2k], suppc(n-l,k,r) ~ [In-k-lln+k-l], and r


supp c(r, 1, n + k) = supp c(n + k, 1, r) ~ [n + k - 1, n + k + 1], we see that relation r r

(1.25) in Chapter 1 takes the form

c(n-I, I,n)c(n,k,n+k) = c(n-I,k,n+k-I)c(n+k-I, I,n+k),

whence

c(n + k - I, I, n + k)c(n - I, k, n + k - I) c(n, k, n + k) =

c(n - I, I, n) (3.12)

because c(p,I,p+l»O (pE No). Since all numbers c(k,l,k+l) areknown,re

lation (3.12) enables us to recursively determine from c(p, l,p+ I) all numbers

c(p, q,p + q). By writing the associativity relation (1.25) from Chapter 1 for p = I, q =

n - 1, I = k, s = n + k - 1 and repeating the same argument, we obtain the equality

c(n, k, n +k-l)

= c(n - 1, k, n + k - I)[c(n + k - 1, 1, n + k - 1) - c(n - 1, I, n - 1)]

c(n-l,l,n)

+ c(n - I, k, n + k - 2)c(n + k - 2, I, n + k - 1) , (3.13) c(n - I, I, n)

which allows us to determine all numbers c (p, q, p + q - I) by recursion in n. Since the hypercomplex system is commutative, without loss of generality, we can assume that

p~q. Thus, by setting p=l, q=n-l, l=k, s=n+k-r (r=2,3, ... ,2n-2) in relation (1.25) in Chapter I, we obtain the following equality (recall that n ~ k):

c(n,k,n+k-r)

= c(n - I, k, n + k - r - I)c(n + k - r - I, I, n + k - r)

c(n -I, I, n)

c(n - I, k, n + k - r)[c(n + k - r, I, n + k - r) - c(n - 1, k, n + k - r)] +~------------~------------------------------

c(n-l,l,n)

c(n - 1, k, n + k - r + I)c(n + k - r + 1, k, n + k - r) +~--~--------~~--------------~

c(n -1, 1, n)

c(n - 1, 1, n - 2)c(n - 2, k, n + k - r)

c(n -1,1, n) (3.14)

Section 3

Note that

and

Orthogonal Polynomials

c(n - 2, 1, n - l)m(n - 1) c(n-l,l,n-2)=

men - 2)

c(n + k _ r + 1, k, n + k _ r) = c(n + k - r, k, n + k - r + l)m(n + k - r + 1) men + k - r)

261

By using (3.14), we determine by recursion (first, in n and then in r) the structure con

stants c(p,q,p+q-r) (r=2,3, ... ,2p-2). Finally, the remaining structure constants c (p, q, q - p + 1) and c (p, q, q - p) can

be found as follows:

c(p,q,q-p) = c(q,p,q-p) = c(q-p,p,q) ~(q)) (p~q), mq-p

c(p,q,q-p+ 1) = c(q,p,q-p+ 1) = c(q-p+ l,p,q) m(q) (3.15) m(q - p + 1)

(note that c(q-p,p,q) and c(q-p+l,p,q) are numbers of the form c(n,m,n+m)

and c(n, m, n + m - 1), respectively).

• Theorem 3.4 allows one to introduce the following method for the construction of a

discrete hypercomplex system: Let three sequences (an);=o, (bn);=o, and (cn);=o

(an>O, cn>O, bn~O, n~ 1, co=O), an+bn+cn= 1, be given. We set

Cp +l c(P,O,p) = 1, c(p, l,p+ 1) =

bp c(p, l,p) =

cl

c(p,l,p-l) = ap _l (p=I,2, ... ) cl

define c(p, q, r) = c(q, p, r) for r E [Ip - q I, p + q] by using formulas (3.12)-(3.15),

and set c(p, q, r) = ° for other r.

Theorem 3.5. If c(p,q,r)~O (p,q,rE No), then c(p,q,r) is the structure

measure for the hypercomplex system II (m) with basis unity ° associated with a certain system of orthogonal polynomials.


Proof. Consider the difference expression (3.4) with the coefficients determined by

the formulas ao = ao jC;, ~o = bo, an = ao ~cn+lan' and ~n = aobn+ bo (n ~ 1). Let Pn(t) be the corresponding polynomials of the first kind. Since

and ° ~ ~n = ao bn + bo < ao (an + bn + cn) + bo = 1, the operator A, that corresponds

to the difference expression L is bounded and, hence, self-adjoint. Thus, P n(t) (n E

No) is a sequence of polynomials orthogonal on a bounded subset of the real axis with

respect to a certain measure dcr. It is not difficult to show by induction that

> 0.

Consider the numbers

P (1)P (1) C'(p, q, r) = p q f Pp(t)Pit)Pr(t)dcr(t).

Pr (1)

By repeating the argument used in the proof of Theorem 3.4, we see that C'(p, q, r) satisfy the associativity relation and, hence, are completely determined, according to relations (3.12)-(3.15), by their values c'(p,O,p), C'(p, 1,p+ 1), c'(p,l,p), and C'(p, 1,p - 1). But it is not difficult to check that

I I cp+1 = c (p,O,p) = 1 = c(p,O,p), c (p, 1,p+ 1) = c(p, 1,p+ 1), cI

bp I ap_1 C'(p, l,p) = = c(p, 1,p), and c (p, 1,p-l) = = c(p, l,p-l).

Thus, C'(p, q, r) = c(p, q, r) and, since c(p, q, r) ~ ° by the condition of the theorem, it remains to use Theorem 3.1 to complete the proof.

• As was noticed by Bloom and Selvanathan [BlS], this method for the construction of

a hypercomplex system is universal in the sense that, for any hypercomplex system as

sociated with a system of orthogonal polynomials, it is possible to find sequences

(an);;'=o, (bn);;'=o, and (cn);;'=o which generate this hypercomplex system as indi

cated above. More precisely, let II (dcr, m) be associated with a system of orthogonal


polynomials P n( t), which are polynomials of the first kind for the difference expression

(Lu)p = Up_1 up-I + ~pup+ Upup+I' We set

a = p

b = _1_ f PI (t) i;(t)dcr(t) ~ 0, p ~(1)

Co = 0, cp = Pp- I (1) f PI (t)Pp I (t)Pp(t)dcr(t) = Pp- I (1) ap_1 > O. ~(l)Pp(l) - ~(l)Pp(l) up

Since

we have

The reader can easily verify that c(p, q, r) are related to ap' bp' and cp as required .

• Below, we give sufficient conditions for the integrals f P p (t)P qC t)P r (t )dcr (t) to be

nonnegative and present several examples of hypercomplex systems associated with orthogonal polynomials and having nonnegative structure measure.

Theorem 3.6. Let up and ~p be coefficients of the difference expression (3.4)

associated with a system of orthogonal polynomials P p(t). If the sequences a p and

~p are not decreasing, then f Pp(t)PqCt)Pr(t)dcr(t) ~ 0 (p, q, rENo)·

Proof. The integrals f Pp(t)Pit)Pr(t)dcr(t) are, in fact, the Fourier coefficients

in the decomposition of the product Pp(t)Pq(t) in the polynomials Pr(t). Without loss

of generality, we can assume that p $; q. To prove the theorem, we use induction on p.

Since PI(t) = (t-~o)/a, wehave


for all q~l. Since C1.p >O and ~q-~O~O, the theorem is true for p=l, q~l. Let

us assume that the statement of theorem is true for p = 1, ... , I and prove it for p = I + 1, q > I. We have

By using the induction hypothesis and the inequalities C1.q-l - C1.l-1 ~ 0, ~q - ~l ~ 0, and

C1.q > 0 (q > I), we see that the theorem will be proved if we show that the last term is nonnegative. We have

= (C1.q_l Pit) + ~q-l Pq- 1 (t) + C1.q_l Pq- 2 (t)PI- 1 (t)

Continuing in this manner, we get terms 'which have nonnegative Fourier coefficients in

decomposition (3.2) except, possibly, the last term C1.0(P1(t)Pq-l(t)-Pq-l+l(t). But


for this term, we have ao (PI (t)P q-l(t) - P q-l+ I (t» = (t - ~o)P q-l(t) - aoP q-l+ I (t) =

(aq-I-aO)Pq-I+I(t)+(~q-I-~O)Pq-I(t)+aq-I-IPq-I-I(t), which completes the proof.

• The second condition for a structure measure to be nonnegative was established by

Koornwinder [Ko02] and involves an additional formula. It was formulated in Section 2 in Chapter 1 for an arbitrary A-system of functions.

Examples of hypercomplex systems associated with orthogonal polynomials are described in detail in [Las2]-[Las4]. Here, we just give a list of systems of orthogonal polynomials which lead to hypercomplex systems with nonnegative structure constants.

These systems include the Jacobi polynomials P~o.·~) (t) (a ~ ~ > - 1, a (a + 5) (a +

3)2~(a2-7a-24)b2), where a=a+~+1 and b=a-~, more general (see

[Rahl], [Rah2D q-Jacobi polynomials p~o.,~)(t; q) (a ~ ~ > -1, a + ~ + 1 ~ 0, 0 < q < 1), which turn into the Jacobi polynomials as q ~ 1, q-ultraspherical polynomials Cn( t; ~ I q) (- 1 < ~ < 1, 0 < q < 1) (see [BreD, associated Legendre polynomials

Cn( t, v) (v ~ 0) (see [BaD]), associated q-ultraspherical polynomials C~(t; ~ I q)

(0 < ~ ~ q < 1, 0 < a < 1) (see [Bul]), polynomials related to homogeneous trees

P n(t, a) (a ~ 2) (see [Letl]), Pollacheck polynomials P n(t; a, c, a) (a ~ 0, c ~ 0,

a ~ - 1/2, 2a + 1 + c ~ 0, 1 ~ 4aa + 4a2 + 4ac + 6a) (see [Sze]), generalized Che

byshev polynomials Tno.'~(t) (a ~ ~ + 1, ~ > -1) (see [LaiD, and polynomials orthog

onal with respect to the weight functions P / (1- ~Lt2) (Jl = a - (a/2 )2, a ~ 2)

and l/[I-vt2]P (v=2a-a2, a~2) (see [GerD. As follows from Theorem 3.7 and Corollary 3.1 in Chapter 1, for an object dual to a

hypercomplex system associated with orthogonal polynomials to be a hypercomplex system, it is necessary and sufficient that, for any t, sEE, the following integral representation hold:

Rp(t)Rp(s) = J R/r)dJlt,s(t) (p E No), E

(3.16)

where Jlt,s(t) is a nonnegative Radon measure on the space X ir In this case, we have duality. Necessary and sufficient conditions for representation (3.16) to exist are established in [Gas3] for the case of Jacobi polynomials, and a sufficient condition for generalized Chebyshev polynomials is found in [Lai].

3.5. Compact Hypercomplex Systems Associated with Orthogonal Polynomials. In this subsection, we consider hypercomplex systems with compact basis associated with a system of orthogonal polynomials. We preserve the notation of the Subsection 3.4. Let

Pn = max I P net) I and set Rn(t) = Pn(t)/Pn' Denote by P (F) the set of all poly-tEE


nomials on E. By the Weierstrass theorem, peE) is dense in C(E) and, hence, in

L2 (E, cr). Suppose that the following conditions are satisfied:

(i) there exists a point e E E such that Pn = P n( e) for all e E No;

(ii) for any t, sEE, there exists a measure d~ (t, s; z) E Mt (E) such that

(3.17)

for all n E No.

Remark 1. It follows from Remark 3 to Theorem 3.1 that if it is possible to con

struct a hypercomplex system with discrete basis associated with polynomials P n( t),

then condition (i) is automatically satisfied.

Remark 2. As a rule, d~(t, s; z) should be sought in the form d~(t, s; z) = K(t, s, z)dcr(z), where

K(t, s, z) = L Pn(t)Pn(s)Rn(z) n=O

(we are not concerned with the convergence of the series). If K(t, s, z) ~ 0, then, after formal calculations, we obtain

f Rn(z)K(t, s, z)dcr(z)

We do not prove these formulas here and refer the reader to [Gas2], [Gas3], and [Osil].

In the theory of special functions, relations of the form (3.17) are called multiplication

formulas. If a multiplication formula exists for a certain system of functions, then, as a rule, it is possible to construct a hypercomplex system associated with these functions. The necessity to have a multiplication formula for the construction of a hypercomplex system associated with a certain system of functions and having a nonnegative structure measure follows from relation (3.10) in Chapter l.

Let us define operators Ls (s E E) on pee) by the equality


(Lsf)(t) = J f(z)dJl(t, s; z) (fE pee), t, sEE). (3.18)

Let

N

f(t) = L fkPk(t)· k=l

It is easy to see that

N

II(Lsf)(t)ll; = IlfkI2IRk(S)12 Ilfl@. k=l

Consequently, the operators Ls (s E E) can be continuously extended to the entire

L2 (E,cr) with IILsll~ 1 (SE E). Approximating an arbitrary function fE C(E) unifonnlyon E by polynomials and passing to the limit in (3.18), we conclude that relation (3.18) is true for all functions continuous on E.

By setting p = 0 in (3.16), we establish that, for all t, sEE, the measure dJl (t, s; z)

belongs to M 1 (E) and Ls 1 = 1. Since d/l(t, s; z) E Ml (IR), it follows from (3.18)

that inequality (2.7) in Chapter 1 holds for the operators Ls (s E E).

Let us show that the operators Ls (s E E) form an Hermitian commutative fam

ily of generalized translation operators in L 2 (E, cr) which preserves positiveness, is

weakly continuous with respect to the invariant measure cr, and satisfies the condition of separate continuity. Indeed, let us show that Ls are, in fact, generalized translation

operators. It is clear that (Lsf)(t) = (Lsf)(s) (t,sE E; fE C(E». This implies that

(L.f) (t) = (Ltf) ( . ) E L2 (E, cr) for almost all tEE with respect to the measure cr.

Let us show that Le = I. We have Rn(t) = J Rn(z)d/l(t, e; z) for every n E No. Since

the polynomials Rn(t) form a basis in pee), we have tn = J zndJl(t, e; z). Thus, we

have obtained the sequence of moments sn = tn. Clearly, this moment problem is

solvable (i.e., the measure dJl(t, e; z) can be uniquely constructed from its moments t~

see, e.g., [BerlO]). Consequently, d/l(t,e;z) = Ot and (Lef)(t)=f(t) (fE C(E».

Hence, Le = I. Since P (E) is dense in L2 (E, cr), it is sufficient to verify the associa

tivity axiom only for the polynomials Rn(t). But for Rn(t), the corresponding equality

is obvious. Thus, Ls are generalized translation operators. It follows from (3.18) that

Ls preserve positiveness; the commutativity of the generalized translation operators Ls

is obvious. Since involution on E is trivial, the generalized translation operators are

involutory if and only if they are real, which again follows from (3.18): (Ls 1) (t) = Lsf(t) (f E L2 (E, cr» for almost all (t, s) E Ex E. Let us show that the measure cr

is invariant under the action of Ls (s E E). For every n, m E No, we have


and it remains to recall that the family of the functions R n (t) is total in L2 (E, cr). Let

us show that the condition of separate continuity is satisfied. Let sn -7 s, n -7 00 (s,

sn E E). Since, for every n E No, (LSkR J(t) = Rn(t)Rn(sk) n-too) R n(t)Rn(s) =

(Ls R ;J (t), the function Ls Q( t) is separately continuous for an arbitrary polynomial

Q(t). Let f E C(E) be an arbitrary function. We fix € > O. By the Weierstrass the

orem, there exists a polynomial Q (t) such that II f - Q 1100 < € / 3. Then

for sufficiently large n. Finally, let us prove that the generalized translation operators

Ls (s E E) are weakly continuous. Let f E C( E) and g E L2 (E, cr). By using se

parate continuity, the inequality II Ls II ~ 1, and the Lebesgue theorem, we establish that

the function (L s,f, g h is continuous. Now let f, g E L2 (E, cr ). Approximating the

function f by continuous functions fn and using the inequality I (Ls (f - f J, g) 2 I ~ IIfn - fll211 g 11 2, we find that the sequence of functions (Lsffl' g) 2 uniformly converges

to (Lsf, g). This implies that the generalized translation operators Ls (s E E) are weakly continuous.

• If we apply Theorem 2.1 in Chapter 1 to the generalized translation operators Ls

(s E E), we obtain the Hermitian hypercomplex system Ll (E, cr) with basis unity satisfying the condition of separate continuity. We summarize this discussion by the following statement:

Theorem 3.7. Let P n(t) (n E No, tEE) be polynomials orthonormal with

respect to a measure cr with compact support E. Suppose that the following condi

tions are satisfied:

(i) There exists a point e E E such that P n( e) = max I P n( t) I (n E No); tEE

(ii) for any t, SEE, there exists a measure d Il (t, s; z) E Mt(E) such that

Rn(t)Rn(s) = J Rn(z)dll(t, s; z) (n E No), where Rn(t) = Pn(t)! Pn(e).


Then the operators (Lsf)(t) = J f(z)dll(t, s; z) (fE C(E); t, sEE) can be

extended to Hermitian generalized translation operators in L 2 (E, cr) which preserve

the identity and positiveness, are weakly continuous with respect to the invariant

measure, and satisfy the condition of separate continuity. The space L1 (E, cr) with

convolution f* g(t) = J (Ldhr)g(r)dcr(r) (f, g E L2 (E, cr); tEE) generated by

these generalized translation operators becomes an Hermitian hypercomplex system with compact basis E and basis unity e satisfying the condition of separate con

tinuity. The set of polynomials R n(t) (n E No) forms the set of characters of the

hypercomplex system L1 (E, cr).

Proof. It is necessary to prove only the last statement of the theorem. By virtue of

(3.17), (LsRJ(t) = Rit)Rn(s) and, consequently, by using the statement proved at

the end of Section 2.1 in Chapter 1, we conclude that R n( t) is a generalized character.

But, by condition (i) of the theorem, I R n( t) I ~ 1 (t E E) and, hence, R n (t) is a char

acter. Let us show that the hypercomplex system L1 (E, cr) does not have other charac

ters. Indeed, let X (t) (t E E) be a character of the hypercomplex system L1 (E, cr)

different form Rn(t) (n E No). By Theorem 1.4 in Chapter 1, X(t) is orthogonal to

all Rn(t) with respect to the measure cr. But Rn(t) (n E No) is an orthonormal basis

in L2 (E, cr), whence X (t) == O.

• Let p~(l,~\t) (t E [-1, 1]; a, ~ > -1) be Jacobi polynomials orthonormal with

respect to the measure cr = (1- t)(l(l- t)~dt. Since p~(l,~)(_t) = (_I)n p~(l,~)(t), we

can assume that a ~~. But then we have (see, e.g., [Sze]) I p~a,~)(t) I ~ p~a'~\I) if

max{a,~}>-1/2 (if ~>a, wehave Ip~a,~)(t)I~(-ltP~(l'~)(-I». For a~~

and max {a, ~} > - 1/2, denote Rn(t) = p~a,~)(t) / p~(l'~)(l). It was shown in [Gas3] that the integral representation

1

Rn(t)RnCs) = J Rn(z)dll(t, s; z), -1

where the measure Il(t, s; . ) E Mb( [-1, 1]) is uniquely determined, holds if and only if

a ~~, a + ~ + 1 ~ O. The measure dll (t, s; z) is positive if and only if ~ ~ - 1/2 or

a + ~ ~ O. Hence, if a ~ ~ and a + ~ ~ 0 or ~ ~ - 1/2, then, by Theorem 3.7, one

can construct the Hermitian hypercomplex system L2 ( [ - 1, 1], (l - t)a (1 + t) ~ dt) associated with Jacobi polynomials and having a nonnegative structure measure. Clearly, this hypercomplex system is dual to the discrete hypercomplex system associated with


Jacobi polynomials constructed in Subsection 3.4. However, it should be noted that the set of values (a, ~) for which a discrete hypergroup with nonnegative structure measure can be constructed is larger than the corresponding set for a compact hypercomplex system with nonnegative structure measure. More precise, let us introduce the following

sets: U = {( a, ~) I a ~ ~, a + ~ ~ o} U {( a, ~) I a ~ ~, ~ ~ -1/2}, V = {( a, ~) I a ~ ~,

a+~+I~O}, and W={(a,~)la~~, a(a+5)(a+5)2~(a2-7a-24)b2, where

a = a + ~ + 1, b = a - ~}. It is clear that U C V. It is also easy to show that V C W and V \ U :f; 0, W \ V:f; 0. If (a, ~ ) E U, the structure measures of the compact and discrete hypercomplex systems are nonnegative and the duality theorem (Theorem 3.7 in

Chapter 1) is true. If (a, ~) E V\ U, then the structure measure of the discrete hypercomplex system is nonnegative, and the structure measure of the hypercomplex system

L1 ([ - 1, 1], (1 - t)Ct ( 1 + t)~ dt) is real. Nevertheless, in this case, duality holds in the sense of quantized hypercomplex systems (see Section 5 in Chapter 1). If (a, ~) E

W \ V, the structure measure of the discrete hypercomplex system is still nonnegative, but equality (3.17) does not hold. This is why the generalized translation operators Ls

(s E E) should be introduced in a different way, namely, if f(t) = 'Lf,ftit) is the

expansion of a function f E L2(E, cr) in the basis R k(t), then Lsf(t) = 'Lf,ft k(s)R k(t).

Generally speaking, the object generated by these generalized translation operators is not

a quantized hypercomplex system but, since I R k( t) I ::; 1 for (a, ~ ) E W (see Remark 3 to Theorem 3.1), it is a real hypercomplex system with compact basis in the sense of Vainerman (see Subsection 5.4 in Chapter O. In this case, duality also takes place. As shown in [Vai9], for an arbitrary system of orthogonal polynomials satisfying the estimate

with some K> 0 and /l E Mb(E), it is possible to construct compact and discrete hypercomplex systems (in the sense of Vainerman) dual to each other. In the case where ( a, ~ ) ~ W (a, ~ > - 1), we did not succeed to verify this estimate. And although we succeeded in constructing a discrete hypercomplex system in the sense of Subsection 2.4 in Chapter 1 (see below), the case (a, ~) ~ W remains unclear from the duality point of view.

Formulas of type (3.17) are also established for generalized Chebyshev polynomials [Lai] and q-Jacobi polynomials [Rah2]. Moreover, in [CoS2], the following result was announced: Suppose that, for some family of orthogonal polynomials on an interval

Ie 1R 1, the multiplication formula (3.17) holds and the measure d/l(t, s; z) possesses

the following properties:

(i) There exists a point eEl such that d/l (t, e; . ) = 0(;

(ii) lim diamsupp /l(t, s; .) = o. s~e


Then, to within a linear change of variables, these polynomials coincide with the Jacobi

polynomials R~a,~)(t) with (a, ~) C U. This theorem is converse to the result of Gasper [Gas3]. The existence of a multiplication formula for generalized Chebyshev

polynomials and q-Jacobi polynomials does not contradict this theorem because, in the first case, condition (ii) is not satisfied and, in the second case, condition (i) is not satisfied. Thus, generalized Chebyshev polynomials produce an example of a hypercomplex system which is not a hypergroup.

In [Osil], a multiplication formula of type (3.17) was obtained for a wider class of

orthogonal polynomials with measure 11 (t, s; .) not necessarily positive, and the corresponding generalized translation operators were studied.

It should also be noted that similar constructions of compact and discrete hypercomplex systems can be carried out for polynomials in several variables. However, we do not consider this case in detail but give an example (see, e.g., [AnT] and [Ko02]). Let

D = {z = x + i y I x2 + l ~ I} be the unit disk. For arbitrary a > 0, we define a probability measure on D as follows:

a + 1 ( 2 2)a = -- I-x -y dxdy. 1t

Consider the following family of polynomials in two variables x and y: R~a~(z) =

R(a) (rei<p) = rl m-nlei(m-n)<p R(O:,lm-nl)(2r2 -1) where m n E N and R(a,~)(t) are m,n mm(m,n) , , 0 k

Jacobi polynomials normalized by the condition Rka,~)(t) = 1. These polynomials form

an orthogonal basis in L2 (D, ma) and satisfy the relation

(L R(a))(v) = ~f Rm(an)(vw+~I-lvI2~1-lwI2z)dma(Z) w m,n a + 1 '

where

D

= f R~~~(z)Ea(v, w, z)dma(z) = D

R(a) (v)R(U) (w) m,n m,n (v, wED),

Eu(v, w, z) = a (1-lvI2 -lwl2 -Ize + 2Revwzt- 1

a + 1 (1 - 1 v 12)U (1 - 1 w 12 t (1- 1 Z 12 t

We set

(Lvf)(w) = f f(z)Eu(v, w, z)dma(z).

D

Annabi and Trimeche [AnT] established that Lv are commutative generalized transla-


tion operators in the space L2 CD, ma) which preserve the identity and positiveness and

satisfy the condition of separate continuity; furthermore, ~ = 4; and C Lv!) (w) = 4/*(15), where r(v) = f(v). Thus, the space LI CD, ma) with convolution induced

by these generalized translation operators and with involution D:3 Z ~ ZED is a normal hypercomp1ex system with the basis unity e = (1,0) satisfying the condition of separate continuity. It is obvious that he characters of this hypercomplex system are the

polynomials R~~~(z). By virtue of the equality R~~(z) = R~~~(z), all characters of

the hypercomplex system L1 CD, ma) are Hermitian. Hypercomplex systems dual to

Ll CD, ma) (a > 0) were, in fact, studied by Kanjin [Kan]. It should be noted that, in the discrete case, the nonnegativity of the structure measure was proved by Koornwinder

[Ko02]. The polynomials R~~~(z) with a E No are the characters of the hypergroup

S U (n ) / S U (n - 1) and were first studied by Shapiro [Sha I], [Sha2].

3.6. The Case of Not Necessarily Nonnegative Structure Constants. In this subsection, we consider discrete hypercomplex systems associated with orthogonal polynomials with not necessarily positive structure constants. Let Pit) be a sequence of

polynomials defined on a bounded subset E of the real line and orthogonal with respect to the measure dcr. Suppose that there exists a sequence m (p ) (p = 0, 1, ... ) of positive numbers such that

(3.19) r

We set

Pp(t) f Rp(t) = -Jm{ij)' c(p, q, r) = Rp(t)RqCt)Rr(t)dcr(t) (p, q, r E Q = No).

(3.20)

By replacing Pp(b) with .Jm(p) and repeating the argument of Theorem 3.1, we can

show that c(p, q, r) are structure constants (generally speaking, nonpositive) and m (p)

is the multiplicative weight of some Hermitian hypercomplex system II Cdcr, m) with basis Q. The space of characters of this hypercomplex system consists of the set of

points Z E ([: I such that the sequence R pC z) is bounded; X h = X n IR I. Note that

these statements remain valid if we set R p (t) = (- l)fp P P)/ .Jm(p) in (3.20), where

Ep is an arbitrary sequence of natural numbers.

Let us describe a method for choosing the numbers m (p ). Suppose that

If Pp(t)Pq(t)Pr(t)dcr(t)I~C forallp,q,rE Q. Note that if the numbers vp>O sat

isfy the inequality

Section 3 Orthogonal Polynomials

p+q L vr :::; VpVq'

r=lp-ql

we can set .Jm(p) = CVp in (3.19). Indeed,

r

p+q L If Pp(t)Pp(t)Pp(t)dcr(t)ICvr

r=lp-ql

p+q :::; c2 L Vp:::; C2VpVq = .Jm(p)m(q).

r=1 p-ql

273

The numbers v p can be chosen as v p = (2p + 1) A or V p = sinh (2p + 1) A / sinh A.

Indeed, by using the formulas for the sum of sines, one can easily check that

p+q

L sin(2r + l)z sin(2p + l)z sin(2q + l)z =

r=1 p-ql sinz sinz sinz

By setting here z = 0 and z = iA (A> 0), we get

p+q

L r=1 p-ql

Since

p+q L (2r + 1) = (2p + 1)( 2q + 1),

r=1 p-ql

sinh(2r + I)A sinh(2p + I)A sinh(2q + I)A =

sinhA sinhA sinhA

it follows from the first relation that

A A ( 2p + 1) (2q + 1) (A ~ 1).

In particular, if the polynomials Pr(t) are uniformly bounded in r E Q and tEE,

we can set .Jm(r) = K (2r + l)A or


= K sinh(2r + I)B sinhB

(A;::: I, B>O, K = [sup IPr (t)1]3). rEQ

Indeed, it is necessary to check inequality (3.19). We have

p+q

L I J Pp (t)Pq (t)Pr(t)dcr(t) IK(2r + I)A r=lp-ql

For the multiplicative measure

p+q

L (2r + I)A ~ -Jm(p)m(q) r=lp-ql

= K sinh(2r + I)B sinhB '

inequality (3.19) can be verified similarly.

• For hypercomplex systems with real structure constants, Theorem 3.3 remains true.

To prove this it suffices to repeat the proof of Theorem 3.3, changing the expression for Pp(t) as follows:

p

P p (t) = X t (t ) -J m(p) sign n c( k - 1, 1, k). k=!

In this case, in the recurrence relation (3.11), we have

ic(p, I, p + 1) I ex =

p m(l) m(p + 1)

;::: 0, m(p)

~ = c(p, 1, p) E IRI p m(l) ,

and the other arguments remain the same. We only note that the structure constants

c(p, q, r) are calculated by formulas (3.20) with R pCt) = Xt(P)/ .,Jm(p). Thus, a hypercomplex system with real structure constants which satisfies the conditions of The

orem 3.3 can be uniquely recovered from a sequence of orthogonal polynomials Pp(t),

multiplicative weight m(p), and a given sequence of numbers cp , where cp = ± 1.

• Theorem 3.4 also remains true for hypercomplex systems with real structure con

stants because, in the proof of an analog of Theorem 3.3 for such hypercomplex systems,

it was shown that c(p, l,p + 1):;t: 0 (p E No).


3.7. Examples of Hypercomplex Systems with Real Structure Constants. Let us give examples of hypercomplex systems with real structure measure associated with orthogonal polynomials. According to Subsection 3.6, it is sufficient to prove that the orthogonal polynomials Pr(t) are uniformly bounded with respect to r E Q and tEE.

First, let us show that the Jacobi polynomials ~a,I3)(t) (-1 < ex, ~::; -1/2) are

uniformly bounded on [- 1, 1]. Indeed, for the Jacobi polynomials p~a,I3)(t) (ex, ~ >

- 1) normalized by the condition p~a,I3)(1) = (n ~ ex), the following asymptotic esti

mate holds (see [Sze, p. 75]):

max I p~a,I3)(t) I - n- 1f2 ( - 1 < ex, ~ ::; - 1/2). -1::::t::::1

(3.21)

Since

(see [Sze]), by using the Stirling formula for the 'Y-function, we get

(3.22)

It follows from (3.21) and (3.22) that

max Ip~a,I3)(t)I_ 1 (-I<ex,~::;-1/2). -1::::t::::1

(3.23)

• Consider the hypercomplex systems 11 ( 1 - t)a( 1 + t) 13 dt, K2(2p + 1 )2A) and

Let us show that the spaces of characters of these hypercomplex systems are the

segment [- 1, 1] and the region of the complex plane given by I z + (Z2 - 1 ) If 2 I ::; e2B ,

respectively. Indeed, for the Jacobi polynomials p~a,I3)(t), the following asymptotic

relation is true (see [Sze]) for z lying outside the closed interval [- 1, 1]:

276 Examples of Hypercomplex Systems

It follows from (3.22) and (3.24) that, for fixed Z E [- 1, 1],

p~a,~)(z)

K(2n + I)A

[z + (z2 -IF2t+1/2

K(2n + I)A

Chapter 2

(3.25)

Since I z + (Z2 - 1) 1/21' > 1 for Z E [-1, 1], it follows from (3.25) that, in the first case,

the space of maximal ideals is the segment [- 1, l]. The second case is considered by analogy.

• We can now show that it is always possible to construct a hypercomplex system with

discrete basis associated with the Jacobi polynomials ~a,~)(t) (a, ~ > - l) (in fact, this result was first obtained by Askey and Gasper [AsG 1 D. Indeed, let us first consider

the case a ~~. Denote a = a + ~ + 1 and b = a -~. As follows from the results of Gasper [Gasl], for the structure constants c(p, q, r) to be nonnegative, it is necessary

and sufficient that the inequality a(a + 5)(a + 3)2 ~ (a2 -7a - 24 )b2 hold. If this inequality does not hold, then by virtue of Theorem 1 in Gasper [Gas3], we get

r

for a ~ ~ > - 1 and a ~ - 1/2. Consequently, inequality (3.19) holds in this case, and

we can choose the sequence m (r) = [~(a'~)(1)] 2 to be a multiplicative measure. The

case a, ~ < - 1/2 has just been considered. It should be noted that the results of Askey and Gasper [Gasl] imply that, in this case, one can also consider the hypercomplex system with the structure constant

and the multiplicative weight mer) = [Pr(a'~)(c)]2, where c>1.

Now let ~ ~ a. By virtue of the well-known identity for Jacobi polynomials

Pr(a,~)(_t) = (-1 {Pr(~,a)(t), it is easy to see that we have the situation described in Remark 1 to Theorem 3.1. Consequently, in this case, it is also always possible to con-

struct a hypercomplex system by replacing p:a'~)(I) with Pr(a'~)(-l) in all reasonings .

• In order to consider polynomials more general than Jacobi polynomials we use an

extension of the Koraus theorem on bounded orthonormal polynomials. Before formulating the required result, we establish an identity necessary for what follows.


Let Pr(t) and Pr(t) (r E No) be systems of polynomials orthonormal with respect

to measures dcr(t) and h(t)dcr(t), respectively, where h(t) is a nonnegative function. We set

n

Kn(z, s) = L Pk(Z)Pk(s) k=O

Since every polynomial Pn(t) is expressed in terms of the first n + 1 polynomials of

the system {Pr(t)}, for any complex Z, we have

n

Pn(z) = L J Pn(s)Pk(s)dcr(s)Pk(z) = J Kn(z, s)Pn(s) dcr(s). (3.26) k=O

It is clear that Kn(z, s) = P n(z)P n(s) + Kn- 1 (z, s). Hence, it follows from (3.26) that

h(t)Pn(Z) = h(t)Pn(z) J Pn(s)i:z(s)dcr(s) + h(t) J Kn-l(z, s)i:z(s)dcr(s)

for any tEE. This equality is true because, for fixed z, Kn_ I (z, s) regarded as a poly

nomial of degree n - 1 in s is orthogonal to the polynomial Pn (s) with respect to the

measure h(s)dcr(s). The Cristoffel-Darboux identity (see [Sze, p. 56]) yields

an_I Pn(Z)Pn-l(s) - Pn-I(Z)Pn(s) Kn_l(z,s) = ------------

an Z - s

(aj is the leading coefficient of Pj } For any Z E a: and tEE such that h (t) * 0, we

conclude from (3.27) that

where

Bn(z, t) = - an-l _1_ J h(t) - h(s) Pn(s)Pn(s)dcr(s) (n E No; P- 1 (s) == 0). an h(t) Z - s

This is the just the required identity.


Lemma 3.1. If h (t) > 0 (t E E) satisfies the Lipschitz condition, then An (t, t),

Bn(t, t) and An(zo, t), Bn(zo, t) are uniformly bounded with respect to nand tEE

for fixed Zo lying outside E.

Proof. Let us estimate an_I / an and J nk = J I P n(s) Pk(s) I do(s). It is known

[Sze,p.55] that an_I/?~ = J sPn_l(s)Pn(s)do(s), whence, by using the Cauchy-Bu

niakowski inequality, we establish that an-l / an are bounded by a certain constant a. To estimate Jnk we can also use the Cauchy-Buniakowski inequality and get

1 [J 2 ] 112 [J 1- 12 ] 112 ~.fh Ipn(s)1 dcr(s) Pk(s) h(s)dcr(s) ~ ~

By using these estimates and the inequalities

[ h(t) - h(s) 1 < - "{,

t-s h ~ h(s) ~ H,

we obtain

The boundedness of An(zo, t) and Bn(zo, t) can be established similarly if we note that

[-1-1 ~K (SE E). Zo - s

This and (3.28) imply the following statement:

•

Corollary (Koraus theorem). If the polynomials {p n(t)} are uniformly bounded,

then the polynomials {P'z(f)} are also uniformly bounded.


Consider the system of polynomials p~a,~)(t) (-1 < a, ~ :::; - 1!2) orthonormal on

the segment [-1,1] with respect to the measure (l-t)a(l+t)~h(t)dt, where h(t»O satisfies the Lipschitz condition. By using the Koraus theorem, we conclude that these polynomials are uniformly bounded and, hence, it is reasonable to consider the hyper-

complex systems I) « 1 - t)a( 1 + t)~ dt, K2( 2p + 1) 2A) and

The spaces of characters of these hypercomplex systems are the segment [-1, 1]

and the region in the complex plane defined by the inequality I Z + (Z2 - 1) 112 I :::; e2B .

Indeed, we use Lemma 3.1, taking p~a,~,h)(t) and p~a,~)(t) as the systems Pn(t) and

Pn(t), respectively (this is possible because 1/ h(t) > 0 satisfies the Lipschitz condition). Let Zo be a fixed point of the complex plane. By virtue of (3.28), we have

Pn(a·~)(zo) 1 p(a,~,h)(z ) 1 1 p(a,~,h)(z ) I :::;IA(z,t)1 n 0 +IB(z,t)1 n-) 0

(2n + I)A n 0 (2n + l)A n 0 (2n + l)A

Thus, if

. I p(a,~,h) (z ) I hm sup n 1 < 00,

n~oo (2n + 1)

i.e., if Zo generates a character of the hypercomplex system I) « 1- t)a( 1 + t)~h(t)dt,

K2(2p + 1)2A), then

. 1 p(a,~)(z )1 hm sup n ~ < 00,

n-t OO (2n + 1)

1.e., Zo generates a character of the hypercomplex system I) « 1 - t)a( 1 + t)~dt,

K2 (2p + 1) 2A) and, hence, Zo E [- I, 1]. The first part of the statement is proved. The second part can be proved similarly.

•


Since the series

00

L I Xp IC2p + I)A, p=O

f Ix I sinh(2p + I)B p=o p sinhB

converge or diverge together with the series

we obtain the following analog of the Wiener-Levi theorem:

Theorem 3.8. If the function

is such that the series

00

xCt) = L xpp;a,p,h)Ct) (t E [-1,1]) p=o

00

L Ixp IpA, A ~ 1 p=o

Chapter 2

is convergent, then the function f{x(t)), where fez) is regular on the range of

values of xC t), t E [ -1, 1] ,

also possesses this property.

3.8. Transmutation Operators. Here, we consider transmutation operators for hypercomplex systems associated with orthogonal polynomials. Consider two hyper

complex systems II{dcr, m) and II (da ,m) with the same set E. An operator U

thatactsfrom II{dcr,m) into ll(da,m) and is defined by the matrix (upq);,q=o=

<f PqCt)Pq(t)da(t) according to the formula


(Ux)(p) (PE Q , X(P) E 11 (da, m) )

is called a transmutation operator. It is not difficult to show that the operator U is continuous if and only if the inequalities

L IU pq IJm(p) ~ AJm(q) (qE Q) P

hold. Denote by V a transmutation operator acting from 11 (dcr, m) into 11 (da, m ) .

Clearly,ifbothoperators U and V are continuous, then 11(da,m) and 11(dcr,m) are topologically equivalent.

Before formulating the main theorem about transmutation operators, we recall that it

was shown in Section 3.3 in Chapter 1 that the discrete hypercomplex system 11(m) is

isometrically isomorphic to the algebra of continuous functions on Xh which can be ex

panded into absolutely convergent series in the complete orthonormal system of func

tions p(X) (XE X h' pE Q) with ordinary algebraic operations. Therefore, the nota

tion xU) E I) (da. m) makes sense and means that

x(t) = L x(p) Pp(t) = L x(p) .Jm(p) Q/t) pEQ pEQ

with

IIx(p) II '1 (dcr,m) = L x(p) .Jm(p) < 00.

PEQ

Consider the hypercomplex system 11 (g( t) dt, m) associated with polynomials or

thogonal with respect to the measure g(t) dt (here, g(t) ~ is an integrable function ).

Theorem 3.9. Suppose that the hypercomplex system 11 (g( t) dt, m ), h ( t) >0

(tE E) hasaboundedfirst-orderderivative, l/h(t)E 11 (g(t)dt,m), and

(3.29)

uniformly in sEE, where Lp kp< 00. Then the transmutation operators U and

V for the spaces l) (g(t)dt, m) and I) (h(t)g(t)dt, m) are continuous (it readily

follows from (3.28) and Lemma 3.1 that Pp(t)/ .Jm(p) are uniformly bounded and,


hence, the space ll(h( t)g( t)dt, m) exists). If the polynomials P p( t) are uniformly

bounded, then condition (3.29) can be replaced by the following weaker condition:

L I J h(t~ = :(S) Pp(t)g(t)dt I "",m(p) ~ K < 00 (s E E) . p

A proof of this theorem can be found in [Ber4].

Let us now consider the ultraspherical polynomials U~A) (t) (A= 1, 2, ... , -1 ~

~ 1) orthonormal with respect to the measure (1 - tZ ) 1...12-1. It is known (see, e.g., [ViI]) that they coincide to within a factor with zonal spherical functions for the Gelfand

pair (SO(n) , SO(n-l) ). As indicated in Subsection 3.4, we can consider the hyper-

complex system 11«1-? )A/2-l dt ,m), where m(p)= [ U;/Z-l(l)f. By using the

known relations for Jacobi polynomials (see [Sze, pp.70, 80]), one can easily find that

m(p) = [U~A)(l)]Z = (A 1)/; [~(2P+A-l)r(P+A-l)] -l-Z 2 - ['(A/2) p!

and, hence, the space 11«(1-? )A/2-1 dt ,m) is topologically equivalent to the space

11«I-? )Al2-l dt ,pA-I). By using Theorem 3.9, one can prove the following statement:

Theorem 3.10. Let h(t) > 0 and let U and V be the transmutation operators

constructed for the spaces 11«(1- rZ)A/Z-ldt,pA-l) and 11(h(t)(I-? )A/Z-l dt ,

pA-l), where A=l, 2, .... For A=2, 3, ... , the operators U and V are con

tinuous if h(t) has a bounded derivative of order [(A+5 )12]. For A=I, these

operators are continuous if h(t) has a bounded derivative of the second order.

A proof of this theorem can also be found in [Ber4].

Let U~A,hJct) (-1 ~ t ~ 1) be polynomials orthonormal with respect to the measure

h( t) ( 1 - r2 ) 1...1 2 - 1 dt. The results obtained above imply the following statement:

Theorem 3.11. Let A=I,2, ... and let h(t»O (-1:::; t :::; 1) have a bounded

derivative of the second order (for A =1 ) or of order [(A+5 )/2] (if A = 2,

3, ... ). If


x(t) = L x(r)U~t..)(t) = L x(h\r)U~A,h)(t), rEQ rEQ

then the series

L I x(r)I/t..-1)/2 and L Ix(h)(r)lr(t..-1)/2

r r

are either simultaneously convergent or simultaneously divergent.

This theorem, in particular, yields the Wiener-Levi theorem for expansions in the

orthononnal polynomials U~t..,h)(t).


4. Hypercomplex Systems Constructed for the Sturm-Liouville Equation

Chapter 2

Many hypercomplex systems considered above (e.g., hypercomplex systems associated with symmetric Riemann spaces of rank one, compact hypercomplex systems constructed from Jacobi polynomials, and others) have the following property: characters of these hypercomplex systems are eigenfunctions of a certain differential equation of the Sturm - Liouville type. In the present section, we describe hypercomplex systems constructed for the Sturm-Liouville equation on a semiaxis by the method of Levitan and Povzner and present a survey of the related results.

4.1. Riemann Function. First, we present preliminary results necessary for our subsequent discussion. Consider the Sturm-Liouville equation on a semiaxis

d2y L [ y] = -2 - q (t) y = - AY (0 ~ t < 00 ),

dt (4.1)

where q (t) is a continuous function. Denote by <p (t, A) a solution of equation (4.1) satisfying the initial conditions <p(0, A) = I and <p;(0, A) = 0. We extend the function q(t) to the entire axis by evenness and consider the following partial differential equation:

(4.2)

The solution u(t, s) of equation (4.2) is completely determined by the initial conditions of the form

u(t,O) = f(t) and dU(t, O) = 0,

dS (4.3)

where J(t) is a twice continuously differentiable function which can be found by using the Riemann formula. For convenience, we now deduce this formula as applied to equation (4.2) with initial conditions (4.3). We set

For any two differentiable functions u and v, it is easy to see that

Section 4 Hypercomplex Systems Constructed/or Sturm-Liouville Equation 285

v1;[u]-u1;[v] = - v--u- -- v--u-. a (au dv) a (au dv) at at at as as as

By integrating this equality over the domain G bounded by a piecewise smooth curve r and using the Green formula, we obtain

s

N

M

t B"

Q (t, s)

Fig. 1.

If {v1;[u]-u1;[vndtds= f(v au _uav)dt + (v au -udv)dS. (4.4) as as at at G r

Let P (to, so) (so:;t: 0) be a fixed point in the plane. We construct a triangle PAB as shown in Fig. 1 and apply relation (4.4) to this triangle by assuming that u is a solu

tion of the equation 1; [u] = 0 with the indicated initial conditions and v is, for the present, an arbitrary function. After simple calculations, we arrive at the identity (for the

sake of definiteness, we assume that so> 0):

If u1;[vJdtds = 2u(P)v(p) - u(A)v(A) - u(B)v(B)

G

+ 2 f u(av - aV)dt + 2 f u(av + aV)dt + toro /(t) av I dt. (4.5)

r as at r as at as 1=0 I 3 ~-~

Suppose that the function v (t, s) = v (to, so; t, s) (- 00 :5; t, s < 00) is chosen so that


(b) av = aVon PA and av = at as at

(c) x; [v] = 0.

av on PB;

as

It follows from (b) that v is constant on PA and P B and, hence, it is equal on

these lines to 1. For this choice of the function v, relation (4.5) implies the formula

to +so ( 1 [f( ) ( )] 1 f av to, so; t, 0) (4.6) = - to - So + f to + So - - ---'~---'!..:.--'---....:.f(t)dt, 2 2 ~

to -so

for the solution of equation (4.2) satisfying the initial conditions. The function v (to, so;

t, s) = v (P; Q) is called a Riemann function. Its existence and uniqueness can be

proved as follows: Since :c [v] = 0, we can write

a2v a2v -2 - -2 = [q(t)-q(s)]v (-oo:::;t,s<oo). at as

By applying the Green formula to this relation with PM Q N = G (see Fig. 1), we easily obtain

If [ q ( t ) - q ( s )] v dt ds = f av dt + av ds as at

G r

= v(P; M) + v(p; N) - 2v(P; Q) = 2 - 2v(P; Q),

whence we conclude that, for fixed P, the Riemann function satisfies the integral equation

v(p; Q) = 1 + ~ Sf [q(s)-q(t)]vdtds. (4.7)

G

Conversely, it is clear that each solution of (4.7) is a Riemann function. At the same

time, in the coordinate system A" P B", (4.7) is a two-dimensional Volterra equation.

Therefore, its solution v (p; Q) always exists is unique and defined for all Q. This solution can be obtained by using the Neumann series

v(P;Q) = 1 + ~Jf [q(s)-q(t)]dtds+ ... , G

(4.8)

Section 4 Hypercomplex Systems Constructedfor Sturm-Liouville Equation 287

where dots stand for the sum of iterations of the kernel of (4.7) applied to 1.

We need a formula for the derivative dv (to, so; t, 0)1 ds with to - so::; t::; to + So (so> 0). By using (4.7), we obtain (in the notation of Fig. 2)

= - 21h If [q(s) - q(t)]vdtds - 21h If [q(s)-q(t)]vdtds Gj G2

- 21h If [q(s)-q(t)]vdtds G3

= - 21h If [q(s)-q(t)]vdtds G2

- !(to+so-t) 2 If [q(s)-q(t)]vdtds 4 (to+so-t)h

G1

- !(t-to+so) 2 If [q(s)-q(t)]vdtds. 4 (t - to + so)h

G3

By passing to the limit as h -7 0, we get

dv(to, so; t, 0) ds

= - ± {(to + So - t) I [q(s) - q(t)]vdl

+ (t - to + so) I [q(s) - q(t)]Vdl}. (4.9)

where dl is a differential of the length of the arc. Relations (4.8) and (4.9) yield some important properties of the Riemann function.

First, if q (t) does not increase as t -7 +00, then the Riemann function v (to, so; t, s) is nonnegative for 0 < So ::; to and 0::; s ::; t. Indeed, if equation (4.7) is considered in the region 0::; s::; t, then its kernel is nonnegative. Hence, all iterations of this kernel

are also nonnegative but, in this case, relation (4.8) implies that v(P; Q) is nonnegative.

•


s

t

(t ,-h)

Fig. 2

Let 0 < So ~ to' If q (t) is a function nonincreasing as t ~ + 00, then the de

rivative dv (to, so; t, 0)1 ds is nonpositive for to - So ~ t ~ to + so.

This assertion directly follows from (4.9) because the Riemann function is nonnegative in the triangle PAB.

• Up to now, we did not impose any restrictions on the function f(t). Let us now

prove that if the function f(t) is even, then the solution u(t,s) of equation (4.2) sat

isfying the initial conditions (4.3) is symmetric about the coordinate axes and the bisectrices of the coordinate angles.

It is easy to see that the fact that the solution is symmetric about the coordinate axes follows from the uniqueness of the solution of problem (4.2) -(4.3) and the fact that the functions q(t) and f(t) are even. Let us prove that u(t, s) = u(s, t). For this purpose,

we consider equation (4.1) in the entire axis (q (t) is extended by evenness) and denote

by <p(t, A.) its solution satisfying the initial conditions <pet, 0) = 1 and d<p(O, A.)ldt = O. It is clear that <p (t, A.) = <p ( - t, A.). The function <p (t, A.) <p (s, A.) is a solution of problem (4.2)-(4.3) under the condition f(t) = <p(t, A.) and hence, if

n

f(t) = I. c) <p(t, J.,), )=1

then the corresponding solution has the form

Section 4 Hypercomplex Systems Constructed for Sturm-Liouville Equation

n

u(t, s) = L Cj<P(t, ,:;)<p(s, ').). j=l

289

This implies that, under the indicated initial conditions, the function u (t, s) is symmetric.

We fix t and s. It follows from formula (4.6) that, first, the value of u(t, s) de

pends only on the values of the initial function f (r) on the interval [t - s, t + s] and,

second, that if fn ('!) ~ f('!) uniformly on [t - s, t + s], then un(t, s) ~ u(t, s). Thus, to prove the required assertion in the general case, it suffices to show that any twice continuously differentiable even function can be approximated by functions of the form

L~=l Cj<P('!, Aj) uniformly on [t- s, t + s]. But this is an immediate consequence of

the theory of the Sturm-Liouville equation. Indeed, we choose a> 0 so large that

[ t - s, t + s] C (- a, a) and consider the Sturm-Liouville problem on the interval (0, a) with boundary conditions y (0) = 1, y' (0) = 0, and y (a) = O. For some values of the parameter A, this problem has eigenfunctions <Pet, A). By using linear combinations of these functions, one can construct uniform (on (0, a)) approximations of any twice continuously differentiable function satisfying the boundary conditions. Thus, to complete the proof, it remains to perform an appropriate change of values of the function f('!)

outside the segment [t - s, t + s] and use the fact that the functions f ('!) and <p (t, A) are even. This proves that the solution u (t, s) is symmetric about the bisectrix of the first quadrant. The fact that it is symmetric about the second bisectrix follows this fact and from the symmetry about the coordinate axes.

• 4.2. Hypercomplex Systems Constructed/or the Sturm-Liouville Equation. In the

present subsection, we construct a hypercomplex system associated with the SturmLiouville equation. The potential q (t) satisfies the conditions guaranteeing that the structure measure of the relevant hypercomplex system is nonnegative.

Suppose that q(t) is a nonnegative nonincreasing function. Denote by Il(t) (0::;; t < 00) the solution of equation (4.1) with A = 0 satisfying the initial conditions 11(0) = 1 and 11'(0)=0. We always have Il'(t) ~ 0 and, therefore, the function Il(t) is non

increasing and bounded from below by one. Indeed, equation (4.1) implies the following integral Volterra equation for the func

tion 11' (t):

y'(t) = f q(t)y(s)ds = J q(S)[1 + J y'(r)drjds 000

t t t t

= f q(t)ds + f y'(r) f q(s)dsdr = p(t) + f [p(t)-p(r)]y'(r)dr, (4.10) o 0 r 0

290

where

Examples of Hypercomplex Systems

t

p(t) = f q(s)ds.

o

Chapter 2

The solution Il'(t) of equation (4.10) can by obtained by using the Neumann series

t

Il' (t) = p(t) + f [pet) - p(r)]p(r)dr + ... , o

(4.11)

and, hence, the nonnegativity of Il' (t) follows from the nonnegativity of the kernel of the integral equation (4.10) and its iterations.

• For a twice continuously differentiable even function x(t), we introduce generalized

translation operators by setting (Rsx )(t) = u(t, s)/Il(s) Il(t), where u (t, s) is a solution of equation (4.2) satisfying the initial conditions

u(t,O) = x(t)Il(t) and du(t,O) = 0 (-oo<t,s<oo, Il(-t) = Il(t)). ds

Let us now rewrite this solution in terms of the Riemann formula. We get

1 (Rsx)(t) = [x(t-s)Il(t-s) + x(t+s)Il(t+s)]

21l(t)ll(s)

1 t+s

+ f w(t, s, 't)x('t)Il('t)dt, 21l(t)ll(s)

t-s

(4.12)

where

( ) " _ dv(t, s; 't, cr) I w t, s, 't - - -:'I •

ocr cr=o

By using relation (4.12), we can now define generalized translation operators Rs for any

locally integrable functions x(t) defined for t ~ 0 and extended as even functions to the entire axis.

Theorem 4.1. Let q (t) be a nonnegative non increasing function and let Il (t) be

a solution of equation (4.1) with A = 0 satisfying the initial conditions y (0) = 1 and

Section 4 Hypercomplex Systems Constructed for Sturm-Liouville Equation 291

y' (0) = O. The space L 1 ([ 0, 00 ), J.t 2 (t), dt) of functions summable with respect to the

measure J.t 2 (t) on the semiaxis [0, 00) with the operation of convolution defined by the formula

(x*y)(t) = f (Rsx)(t)y(s)J.t2 (s)ds o

1 00

= -- J [x(t-s)J.t(t-s)+x(t+s)J.t(t+s)]y(s)J.t(s)ds 2J.t(t) 0

1 00 t+s

+ -- J J w(t,s,'t)x('t)J.t('t)d'ty(s)J.t(s)ds 2J.t(t) o t-s

(4.13)

is a commutative Hermitian hypercomplex system with basis unit e = 0 and nonnegative structure measure vanishing for large t. Moreover, this hypercomplex system satisfies the condition of separate continuity.

We first prove several useful lemmas. The only assumption imposed on the function

q (t) (0 ~ t < (0) in these lemmas is its continuity.

Lemma 4.1. Let u(t, s) and v (t, s) be, respectively, the solutions of problem

(4.2) with even twice continuously differentiable and compactly supported initial functions f ( t) and g (t ). Then

co 00

J u(t,s)g(t)dt = J f(t)v(t,s)dt (-oo<s<oo). (4.14)

Proof. First, we note that the conditions imposed on the function f(t) and relation

(4.6) imply that u(t, s) vanishes for large 1 t I. Hence, the function

00

yes) = J u(t, s)cp(t, 'A)dt

is well defined and satisfies the initial conditions

00

yeO) = J f(t)cp(t, 'A)dt and dy(O) = O. ds

(4.15)


The function y (s) satisfies the differential equation (4.1). Indeed, by using the Green formula, we get

Ls[y] = J LJu(t, s)] <p(t, A)dt = J Lt[u(t, s)] <p(t, A)dt -00

= J u(t, s)Lt[<p(t, A)] dt = -Ay.

On the other hand, the function J~oo f(t)<p(t, A) <p(s, A)dt also satisfies equation

(4.1) with the initial conditions (4.15) and, hence, by the uniqueness theorem,

00

J u(t, s)<p(t, A)dt = J fU)<p(t,A)<p(s,A)dt (-oo<s<oo). (4.16)

It is clear that equality (4.16) also holds for linear combinations of the functions <p(t, A). By constructing uniform approximations of the twice continuously differentiable even function g (t) on its compact support supp g by sums of the form

n

gn(t) = L ck<P(t, Ak) k=1

and passing to the limit, we arrive at relation (4.14).

• Lemma 4.2. Any three functions x, y, and z from the collection 9( of functions

finite and bounded on [0, 00) satisfy the relations of commutativity and associativity

(4.17)

and, moreover, x * y E 9(.

Proof. If fU), g(t) E 9( are twice continuously differentiable functions, then it follows from (4.14) that

00 1 00

(j* g)(t) = J (Rsf)(t)g(s)J.l2 (s)ds = - J u(t, s)g(s)J.l(s)ds o J.lW 0


1 00

= - f f(s)ll(s)v(t,s)ds = g*f(t), Il(t) 0

293

where u(t, s) and v (t, s) denote the solutions of differential equation (4.2) satisfying

theinitialconditions u(t,O) =f(t)Il(t) and du(t,O)/ds=O and v(t,O) = g(t)Il(t)

and dv (t, O)j ds = 0, respectively. By approximating (almost everywhere) arbitrary

functions x, y E 9( by twice continuously differentiable functions fn and gn bounded

uniformly in n and passing to the limit in (4.13), we conclude that x * y = y * x. It also follows from relation (4.13) that x * y E 9( for any x, y E 9(.

Relation (4.13) implies the existence of a convolution of the form x * 'V, where x E

9( and 'V(t) is a function summable on every bounded interval. For a twice continuously differentiable function x E 9(, we can write

( X * <p(', A»)(t) = _1_ J u(t, s)<p(s, A)ds = <pet, A) J x(s)<p(s, A)Il(s)ds. (4.18) <p(.) Il(t) Il(t)

By approximating (almost everywhere) an arbitrary function x E 9( by twice continu

ously differentiable finite functions bounded uniformly in n, we prove that (4.18) holds for any x E 9(. By using (4.18), we obtain

( x * y) * <p(., A»)(t) = J (x * y)(s) <pet, A) <pes, A)Il(s)ds <p(.) Il(t) o

Consequently,

A 0000

= <p(t, ) J J (Rrx)(s)y(r)1l2 (r)dr<p(s, A)Il(s)ds Il(t) 0 0

A 00 00

= <p(t,) fy(r)f x(s)<p(s, A)<p(r, A)dsll(r)dr Il(t) 0 0

A 00 00

= <p(t, ) J x(s)<p(s, A)Il(s)ds J y(r) <p(r, A)Il(r)dr. Il(t) 0 0

( x * y) * <p(-, A») Il(· )

= x * ( * <p(., A») y 110 '


whence we arrive at the relation of associativity for

By passing to the limit, we establish this relation first for functions z E 91 whose second derivatives are continuous and then for arbitrary z E 91.

• Corollary 4.1. The function

c(A,B,r) = (KA*KB)(r) (A,BE 'B([0,00)), t~O)

is a structure measure. The operation of convolution with respect to this measure has

the form (4.13).

Indeed, this assertion is a direct consequence of Lemma 4.2 and relation (4.13) from which it follows that the function x * y(t) is finite and continuous for x, y E 91.

• Proof of Theorem 4.1. Since q(t) is nonincreasing as t -7 + 00, the function w(t,

s, 't) is nonnegative for It - s I ~ 't ~ t + s. Let us check relation (1.2) in Chapter 1 for the measure dm(t) = Jl2(t)dt. It follows from (4.13) that there exists a convolution x * \jf, where x E 91 and \jf is a locally integrable function. For this convolution, relation (4.14) remains valid. Thus,

00

f c(A, B, t)Jl2 (t)dt

o

00 00

= f KB(S)Jl2(s) f (RsKA)(t)Jl2(t)dtds

o 0

00 00

= f KB(s) f [KA(S) Jl(t)] Jl(t)Jl(s)dtds

o 0

00

= J KA (t)Jl2 (t)dt J KB(S)Jl2(s)ds = m(A)m(B).

o 0

To show that the hypercomplex system LJ ([0, 00), Jl2 (t), dt) is Hermitian, we write

00 00

c(A,B, C) = f KB(s) f (RsKA)(t)KC(t)Jl2(t)dq.l2(s)ds

o 0

Section 4

Since

Hypercomplex Systems Constructed for Sturm-Liouville Equation

00

= J KB(s) J (Rt~)(S)KC(t)1l2(t)dq.L2(s)ds o 0

00 00

= f KB(s) f KA(s)(RtKC)(s)1l2(t)dtIl2(s)ds o 0

= f KA(t) J KB(s)(RsKC)(t)1l2(s)dsIl2(t)dt = c(C,B,A) o 0

(A, B, C E~([O, 00 ).

00

c(A, B, 0) = J (Rs~)(O)KB(s) 1l2(s)ds

o

00

= f (R OKA)(S)KB(S)1l2(s)ds o

00

= f KA(S) KB(s) 1l2(s)ds = m(AnB),

o

we conclude that 0 is the basis unit of this hypercomplex system.

295

4.3. Structure Measure which Is Not Necessarily Nonnegative. In the present subsection, we construct a hypercomplex system whose structure measure is not necessarily nonnegative (see Subsection 2.4 in Chapter 1). Assume that q (t) is a continuous function of bounded variation on the semiaxis [0, 00). Let V(t) be the variation of the

00

function q(t) on the interval (t, (0), i.e., Vet) = Var q. The function Vet) is cont

tinuous and nonnegative, moreover, it approaches 0 as t -t 00. Since

b OQ 00

Var q = Var q - Var q = V(a) - V(b) (a $, b), a a b

we conclude that

I q(t') - q(t") I $, V(t') - V(t") (0 $, t' $, t" < 00).


Let ij(t) (0:::; t < 00) be a continuous nonnegative nonincreasing function such that

I q(t') - q(t") I :::; ij(t') - ij(t") (0:::; t':::; t" < 00). (4.19)

(in particular, we can assume that ij(t) is equal to V(t); we always have ijU);::: V(t». Parallel with (4.1), we consider the equation

y" - ij(t)y = -t..y. (4.20)

According to Theorem 4.1, equation (4.20) can be associated with a hypercomplex

system Ll ([ 0, 00), jl2 (t), dt) with the nonnegative measure c(A, B, t), where jlU) is

a solution of (4.20) with t.. = 0 satisfying the initial conditions yeO) = 1 and y'(O) = O.

Theorem 4.2. Le t q (t) be a function continuous of bounded variation on the

semiaxis [0,00), let ijU) (0:::; t < 00) be a fixed continuous nonnegative nonin

creasing function satisfying inequality (4.19), and let jl(t) be a solution of equation (4.20) with t.. = 0 satisfying the initial conditions yeO) = 1 and y' (0) = O. Then

the space L1([0,00), jl2(t),dt) with the operation of multiplication given by (4.13)

is a commutative hypercomplex system whose structure measure is not, generally speaking, positive (it is assumed that the function II in relation (4.13) is replaced by jl).

Proof. Let c(A, B, t) = (KA * KB)(t). By virtue of Lemmas 4.1 and 4.2, c(A, B, t)

is a structure measure which induces convolution (4.13). To prove the theorem, it suffices to check inequality (2.14) in Chapter 1. This inequality follows from the inequality

I c(A, B, t)1 :::; c(A, B, t) (A, B E $([0,00», 0:::; t < 00). (4.21)

Indeed, since c(A, B, t) ;::: 0, it follows from (4.21) that V(A, B, t):::; c(A, B, t). Thus,

00 00

f V(A, B, t)jl2(t)dt :::; f c(A, B, t)jl2(t)dt = f jl2(t)dt f jl2(s)ds,

o o A B

as required. Hence, it remains to prove inequality (4.21). Since (Rs ~ )( t) = (Rt ~ )( s),

we get

00

I c(A, B, t)1 = f (RsKA)(t)KB(s)1l2(s)ds

o

t 00

:::; f (RsKA)(t)KB(s)it2(s)ds + f (RtKA)(s)KB(s)jl2(s)ds

o

Section 4 Hypercomplex Systems ConstructedJor Sturm-Liouville Equation

1 t t+s

+ -_- J J I w(t, s, "C) IKA ("C)P,("C)d"CK8(s)P,(s)ds 2~(t) a t-s

1 00 t+s

+ --- f f Iw(t, s, 't)IKA('t)P,('t)d'tKB(S)P,(s)ds. ~(t) t s-t

297

Therefore, it suffices to check that 1 w(t, s, r) 1 ::; wet, s, r) (0::; s::; t < 00, t - s ::; r ::;

t + s), where w(t, s, r) is a function constructed by using equation (4.20) in exactly the

same way as the function w(t, s, r) was constructed by using equation (4.1).

Let to ~ So > O. Riemann functions satisfy the inequality

Indeed, we fix a point (to, so) (0 < so::; to) and denote by K(t, s; 't, 0') and K(t ,s; 't, 0'),

respectively, the kernels of the integral equation (4.7) considered in the region 0::; s ::; t

and the same integral equation with q replaced by ij. Clearly,

{.!.[q(O') - q('t)],

K(t, s; 't, 0') = 2

0,

('t, 0') E G,

('t, 0') ~ G,

and a similar relation holds for K(t ,s; 't, 0'). This and relation (4.19) imply the inequal

ity 1 K(t, s; 't, 0')1 ::; K(t,s; 't, 0'). But then the same relations hold for the iterated kernels and, thus, inequality (4.22) follows from the expansion of (4.8) of the Riemann function in the Neumann series.

By using inequalities (4.19) and (4.22) and relation (4.9), for 0::; So ::; to and to -

So ::; t::; to + So (see Fig. 2), we obtain

M

I av(to ,so; t, 0) I - 1 ( ) J [ () ( )] dl - - to + So - t q s - q t v as 4 Q

N

+ (t - to + so) f [q(s) - q(t)]vdl

Q


os; ± { (to + So - t) I [ij(s) - ij(t)]vdl + (t - to + so) I [ij(s) - q(t)]Vdl}

= av(to ,so; t, 0)

as

This inequality actually means that I w(t, s, r) I ~ w(t, s, r) (0 ~ s ~ t < 00, t - s ~ r ~

t + s).

• It should be noted that the function q(t) can be nonincreasing and nonnegative it

self; in particular, if (j(t) = q(t), we arrive at the case considered above.

Theorem 4.3. Characters of the hypercomplex system L1 ([ 0,00), fi 2(t), dt) have

theform X(t,A) = <p(t,A)/fi(t), where <p(t,A) is a solution of equation (4.1) satisfying the initial conditions <p(0, A) = 1 and <p/(0, A) = 0 and the complex number

A is such that <pet, A) = O(fi(t».

Proof. In one direction the proof is evident. Indeed, if A is such that <p (t, A) = O(fi(t), then X (t, A) is bounded. Moreover,

00

J c(A, B, t)X(t, A)fi2(t)dt

o

00 00

= J KB(s) J (RtKA)(s)<P(t, A)fi(t)dtfi2(s)ds

o 0

= f X(t, A)fi2(t)dt f Xes, A)fi2(S)ds

A B

= X(A,A)X(B,A) (A,BE ~([O,oo))).

Let us prove the converse assertion. First, we show that X (t) is twice continuously differentiable. Let f(t) be a twice continuously differentiable even function with com

pact support. Then (Rsf)(t) = u(t, s)/ fi(s)fi(t), where u(t, s) is a solution of equation

(4.2) with the initial conditions u(t, 0) = f(t) fi(t) and au(t, O)/as = 0, whence it fol-


lows that (Rsf)(t) is twice continuously differentiable with respect to s and, for s

varying within a finite interval, (Rsf)(t) regarded as a function of t is finite. By passing to the limit in the equality

00 00 00

J (f * gn h(t)ft 2 (t)dt = J f(t)X(t)ft2(t)dt J gn(t)X(s)ft2(s)ds,

o o o

where gn is a sequence of functions which converge to (is, we get

o o

We choose f(t) such that the integral on the right-hand side is nonzero. Since integration on the left-hand side is actually carried out over a finite interval, this integral is twice continuously differentiable with respect to s, which implies the differentiability of

the function X (s). This means that the function

ft(t)ft(s) (Rs X)(t) = u(t, s) = ft(t)ft(s) X(t)X(s) = <p(t)<p(s) (<p(t) = ft(t) x(t)

is a solution of equation (2) with the initial conditions u(t, 0) = <Pet) and au(t, O)/as = O. Since <p(t);t:0, these initial conditions imply that <p(0) = 1 and cp'(O)=O. Further, by substituting u(t, s) = <p(t)<p(s) in (4.2), we find that

[ <p" (t) - q (t) cp (t)] <p (s) = [cp" (s) - q (s ) <p (s)] <p (t),

whence it follows that cp" (t) - q(t)cp(t) = - Acp(t), where A is a constant, i.e. cp(t)

<p(t, A). Finally, <pet, A) = O(ft(t) because! X(t)!::; 1.

• Thus, the space of characters coincides with a certain set X of values of the parame

ter A lying in the complex plane.

Theorem 4.4. The topology of the space of characters coincide with the ordinary

topology in the complex plane. The point 00 corresponds to the maximal ideal Moo.

Lemma 4.3. If ! An I ~ 00 (An E X), then, for any function x (t) ELI ([ 0, 00),

ft 2 (t), dt), we have

00

J x(t)X(t, An) Jl2(t)dt ~ o. o


Proof. Let a < band ° < h < b - a be arbitrary numbers. By integrating the

equality <p(t,An) = -[<p"(t,An)-q(t)<p(t,An)]/An, we obtain

1 b Hh

h f d't f <p(t, An)dt a 't

b

::; h'~n' f [<p'('t + h, An) - <p'('t, An)]d't a

1 b

+ -IA 1 sup 'q(t)1 f jl('t+h)d't n O:O::t<oo a

I b + -, -, sup Iq(t)1 J jl('t + h)d't

An O<t<oo - a

= _12 1 [jl(a+h) + jl(b+h)] hAn

This implies that the integral on the left-hand side of this inequality tends to zero as

I An I ~ 00. By changing the order of integration, we easily see that this integral is equal

to

oof x(t; a, b, h) ( 'l.)- 2( )d _ X t, I'.n Il t t, Il(t) o

where x(t; a, b h) is a function depicted in Fig. 3. Thus, Lemma 4.3 is proved for the

functions x(t; a, b, h )/Il(t) E L J ([0, 00), jl2(t), dt) and their linear combinations y (t).

Now let x(t) be an arbitrary function from LJ([O, 00), jl2(t),dt). We choose yet) such that

00

J I x (t) - y (t) I jl2 (t) dt < E.

o

Section 4 Hypercomplex Systems ConstructedJor Sturm-Liouville Equation

Then

s

1

o a a+h

Fig. 3

00

J x(t)x(t, An)Jl2(t)dt o

00 00

::; J [x(t) - y(t)]x(t, An)Jl2(t)dt + J y(t)X(t, An)Jl2(t)dt o 0

00

< E + J y(t)X(t, An)Jl2(t)dt --? E.

o

301

• To prove the theorem, it is necessary to show that the space me of maximal ideals of

the algebra ~1 ([0, 00),112 (t), dt) is homeomorphic to XU {oo}. Since the relevant mapping is, clearly, bijective and me and XU { oo} are compact, it suffices to prove that convergence in XU {oo} implies convergence in we Let An --? A'* 00. Since

X(t,'An) --?X(t,A) pointwise and X(t)E L1([0,00), Jl2(t),dt), wehave

00 00

J x(t)X(t, An)Jl2(t)dt --? J x(t)X(t, An)Jl2(t)dt. o o

If An --? 00, then, by virtue of Lemma 4.3,

00

J x(t)X(t, An )112 (t)dt --? O. o

•


Since X (t, 'A) is real if and only if 'A is real, X (t, 'A) is Hermitian if and only if

'AEXnIR1=xh· Since ii(t) is monotone, there exists a finite limit c = lim ii(t) ;::: O. Inequality

(4.19) remains valid with ii(t) replaced by ii(t) - c and, in addition, ii(t) ) o. t~oo

Hence, without loss of generality, we can always assume that this condition is satisfied.

4.4. Set of Characters of the Hypercomplex System Associated with the SturmLiouville Equation. Let us now describe the structure of the space of characters. In fact, we want to show that this space coincides with the spectrum of the operator acting

in the space L2 ([0,00), dt) and generated by the differential expression L with the boundary condition y'(O) = 0 (for details, see [BerlO], [Gla], and [LeS]). First, we formulate several auxiliary lemmas concerning the Sturm-Liouville equation (14.1) with

continuous real potential q(t) approaching zero as t ~ 00. Denote 'A = s2, where s =

(J + i't varies in the upper half plane.

Lemma 4.4. The solution <P (t, 'A) of equation (4.1) admits the following estimate:

I <pet, 'A) I ~ exp (t't + _1 J I q(a) Ida] (0 ~ t < 00). (4.23) lsi 0

Proof For <pet, 'A), we obtain

1 t

<P (t, 'A) = cos st + - J sin s (t - a) q ( a) <P (a, 'A) da. s

o

By setting <PI (t, 'A) = <pet, 'A) e- tt, in view of relation (4.24), we get

whence

t

<PI(t,'A) = cosste-tt + .!. f sins(t-a)e-'t(t-O:)q(a)<PI(a,'A)da, s o

(4.24)

By using the well-known lemma (see [LeS, Chapter 4, Section3]), we conclude that


which implies (4.23).

• Lemma 4.5. Assume that q (t) ----7 O. Le t p > 0 and £ > 0 be given num-

1-7 00

bers such that £ < -lP. Then, for any A from the interval (- 00, - p) there are two

linearly independent solutions of equation (4.1) YA. (t) and z). (t) such that I y). (t) I ~ C,(A)iM-c)t and I z).(t) I ::0; C2 (A)e(MH)1 (O::O;t::O;oo, -oo<A<-p). More

over, for any fixed t, the functions y).(t), dYA.(t)ldt, z).(t) and dz).(t)ldt are an

alytic functions of A in the interval (- 00, - p ).

Proof. We now choose sufficiently large to such that, for t ~ to, we have I q(t) I <

2£ -lP - £2 (this is possible since, by the condition, 2£ -lP - £2 = £ (2 -lP - £) > 0). Denote by h (t) the solution of (4.1) (0::0; t < 00, - 00 < A < - p) satisfying the initial

conditions YA. (to) = 1 and y~ (to) = O. Let us show that this is the first required solu

tion. For t ~ to and A < - p, we have I q (t) I < 2£ -lP - £2 < 2£..JfII - £2 and, hence,

- A + q (t) ~ I A I - 2£..JfII + £2 = (..JfII - £ )2. By using this inequality, we obtain

t h(t) = 1 + f (t - a) [- A + q ( a) ] h ( a) da

10

1

= 1 + f (t - a) [ - A + q ( a ) ] da

10

t a

+ f (t - a) [ - A + q ( a) ] da f (a - ~ ) [ - A + q ( ~ ) ] d~ ...

t

~ 1 + f (t - a) (..JfII - £ ) 2 da

10

1 a

+ f (t - a) (..JfII - £) 2 da f (a - ~ ) (..JfII - £) 2 d~ +... = 11 (t)


where 11 (t) is a solution of the equation y" == (ffiT - £)2 Y satisfying the initial condi

tions y(to) == 1 and y'(to) == 0, i.e., 11 (t) == cosh«t- to)( ffiT - E)). This implies that there exists a constant C 1 which, generally speaking, depends on 'A such that

I yA,(t) I ~ C 1 /vfAl-E)t (0 ~ t < 00). Since yA, (t) satisfies the initial conditions

YA,(to) == 1 and y~(to) == 0, for any fixed t, it is an analytic function of 'A in the inter

val (-00; -p).

To find the second partial solution, it is necessary to establish an estimate for

dYA,(t)/dt == y~(t). By using the same notation as above, we obtain

y~(t) == f [-'A+q(a)]YA,(a)da == f [-'A+q(a)] [1 + 1 y~(~)d~]da to to to

t t t ~

== f [-'A+q(a)]da+ J f [-'A+q(a)]da J [-'A+q(y)]dyd~+ ... to

t t t ~

~ J (-JiTf - £ )2 da + J J (ffiT - £ )2 da J (-JiTf - £? dy d~ + ...

== 11' (t) == (-JiTf - £) sinh (t - to)( -JiTf - £),

whence it follows that there is a constant C3 which, generally speaking, depends on 'A

such that I yHt) I ~ c3 i M -E)t (O~t). The second partial solution ZA,(t) defined

on the interval (to, 00) is now sought in the form ZA(t) == uA,(t)YA,(t). By substituting

this relation in (4.1), we find that uA(t) is a solution the equation u~yA, + 2u~y~ == 0, whence

~ da uA(t) == A J -2- + B

to yA, (a)

100 da (for t ~ to, YA (t) > 0). We set A == 1 and B == - -2-. Then

to YA,(a)

By using the Cauchy formula, we find a point rt > t such that


This gives the estimate

It follows from (4.24) that the solution z'" (t) satisfies the initial conditions

(4.25)

We use these initial conditions to extend z'" (t) to the interval (0, to) and to con

struct a solution z'" (t) such that I z'" (t) I ~ C2 (t) i --JII1+e)t (0::;; t < 00, - 00 < 'A < - p). The function z",Cto) analytically depends on 'A (-00 < 'A < - p) because the inte

grand 1/ y£(t) possesses the indicated property; moreover, we have

Thus, 1/ y£(t) < 2 e-C.JP-E)Ct-to) and the integral roo e-C.JP-E)(t-tO)dt is convergent. Jto

Hence, the initial conditions (4.25) are analytic functions of 'A (- 00 < 'A < - p) and,

consequently, the solution z'" (t) and its derivative are analytic in 'A for all fixed t .

• We say that a function f (t) (0::;; t < (0) grows slower than any exponential function

if f(t)= O(eEt ) as t-7 00 for any £>0.

Lemma 4.6. Let q(t) ~ O. There exists an (at most countable) family of t~oo

negative values of 'A bounded from below and having at most one limiting point

(whose role may be played only by zero) for which <p(t, 'A) grows slower than any ex-

ponential function. Then O.

Proof. Let ~ > 0 be given and £ > 0 be such that £ < -ft;/2. It suffices to show

that the number of negative 'A < - ~ such that <p (t, 'A) grows slower than any exponen-

tial function is finite and <pet, 'A) = O(ec-..JfI!+e)t) for each 'A of this sort. In Lemma


4.5, we now set p = 8/2. The solutions YA,(t) and zA(t) and their derivatives with re

spectto t are analytic in A in the interval (-00,-8/2). Because YA,(t) and zA(t) are

linearly independent for all A, we have

By differentiating (4.26) with respect to t, we get

<p'(t,A) = A(A)yi(t) +B(A)zi(t) (O::;t<oo; -OO<A< %} (4.27)

By setting t = 0 in (4.26) and (4.27), we obtain the following system of equations:

(4.28)

A(A)Y~(O) + B(A)zi(O) = <p'(O,A) (-oo<A< %)

For all A, the determinant of this system is nonzero because the solutions YA (t) and

z'" (t) are linearly independent. Since all coefficients of system (4.28) are analytic func

tions in the interval (- 00, - 8/2), its solutions A (A) and B (A) are also analytic. It is easy to see that A (A) cannot have zeros with arbitrarily large absolute values.

Indeed, if A(AO)=O (-00<Ao<8/2), then

and, hence, <p(t, 'Ao) E L, (0,00) n L2 (0,00). Therefore, by integrating the equality

we conclude that <p' (t, Ao) = 0 ( I), whence, by integrating by parts, we obtain

00 = f <p"(t, Ao)<P(t, Ao)dt = - f (<p,)2(t, Ao)dt ::; o. o o

By multiplying (4.29) by <pet, Ao), integrating, and using the relation established above, we get

00

f [ q (t) - AO] <p2 (t, Ao) dt ::; 0, o


for arbitrarily large - 11,,0' which is absurd. We can now complete the proof of the lemma as follows: It is clear that <p(t, A)

grows slower than any exponential function if and only if A is a zero of the function

A(A) analytic in the interval (-00, -8/2). Since zeros of the function A(A) are bounded from below, they cannot have limiting points other than - 8/2.

• Corollary 4.2. The following statements are equivalent:

(i) a point A < 0 belongs to the spectrum of the differential operator (4.1) with the boundary condition y'(O) = 0;

(ii) the point A < 0 is such that <p (t, A) grows slower than any exponential function as t ~ 00;

(iii) the point A < 0 is such that, for any 0 < £ < -JfI1,

(4.29)

Proof. The equivalence of (ii) and (iii) has already been proved. Now let A < 0 be

a point of the spectrum. Since the spectrum is discrete for negative values of A (see,

e.g., Glazman [Gla, Chap. 2, Sec. 27]), we conclude that A is necessarily an eigenvalue.

By virtue of (4.29) and Lemma 4.5, A(A) = 0, which yields (ii) and (iii). Conversely, if

(ii) and (iii) hold, then A (A) = 0, i.e., A is an eigenvalue because <p(t, A) E L2 (0, 00) .

• Lemma4.7. Let q(t) (O::;t<oo) be bounded. If,forsome A, <p(t,A) = O(eEt )

for any £ > 0, then A is real.

Proof. We set

1 1 r+s 00

2" [J(r-s)+f(r+s)] + 2" J w(r,s,t)f(t)dt = J f(t)d(W(r, s, t).

r-s o

For N> r, we have

N N

J <p(s,A)ds J <p(t,A)d(W(r,s,t)

o o


NooN 00

= f <p(s,A)ds f <p(t,A)drW(r,s,t) - f <p(s,A)ds f <p(t,A)drW(r,s,t) o 0 0 N

N N N+r

= <p(r, A) f 1<p(S,A)1 2 ds - f <p(s,A)ds f <p(t,A)drW(r,s,t). (4.30)

o N-r N

The left-hand side of this equality is real because it is a composition of a function

equal to N with the conjugate function (for

Il(t)= 1). Thus, by dividing (4.30) by ioN I <p(s, A)1 2ds and passing to the limit as

N -7 00, we conclude that <per, A) is a real function provided that a sequence Nk -700

is chosen so that

k ) O. ~oo

(4.31)

We consider Nk so large that Nk > 2r. Then 0 < s - r < Nk and r + s? Nk for Nk-

r S; s S; Nk. Thus, in view of the fact that I W( r, s, t) I S; C (0 S; t, s < 00), we obtain

Nk Nk+r

f <pes, A)ds J <pet, A)dt W(r, s, t)

Nk-r Nk

1 S;

2

Nk

f <pes, A)<p(r + s, A)ds

Nk-r

1 + -

2

Nk Nk+r

f <pes, A) f w(r, s, t)<p(t, A)dtds

Nk -r Nk

(4.32)


Since

Nk Nk

J 1 <p(s, A)12ds :;; J 1 <p(s, A)12ds, o

it follows from (4.32) that, in order to prove (4.31), it suffices to show that

We set

Nk+r

f 1 <pes, A) 12 ds

Nk

kr

ak = J 1<p(s,A)12ds. (k-l)r

k ) O. ---too

309

Let us show that there exists a sequence kn such that akn+1

----- ---t 0 as n ---t 00.

al + ... + akn

Assume the contrary. Then there exists ~ > 0 such that ak+l ~ ~ (k = 1, al + ... + ak

2, ... ). These inequalities easily imply that a k> al ~(1 + ~)k-l = C l (l + ~)k=

Cl ek1n ( 1 +~) (k = 1,2, ... ). At the same time,

and, thus, by choosing £ > 0 so small that 2£ r < In (1 + ~), we arrive at a contradiction. Hence, (4.32) is true and, therefore, <per, A) is real, i.e., A is real.

• Theorem 4.5. If q(t) ~ 0, then the space of characters coincides with the

t---too

spectrum of equation (4.1) with the boundary condition y'(O) = 0, i.e., this space consists of all nonnegative A and of at most countably many negative A bounded from below and having at most one limiting point whose role may be played only by zero.

Proof. First, we note that if q(t) ~ 0, then /l(t) grows slower than any ex-t---too

ponential function. Indeed, /let) is a solution of equation y" - q(t) y = 0 or y" +


(J2y = [(j(t) + (J2] y, where (J> ° is an arbitrary number satisfying the boundary con

ditions yeO) = 1 and y'(O) = 0. It follows from the last equation and Lemma 4.4 that

We choose to so large that (j(a) < (J2 for a ~ to. Then fi(t) < C) iu(t-to) = C2 e2Ut

(t ~ 0). Since (J is arbitrary, the assertion is proved. The collection of characters consists of those and only those points Iv for which

<pet, Iv) = O(fi(t». By virtue of the already proved result, this means that <p(t, Iv) also grows slower than any exponential function but, in this case, in view of Lemma 4.7, this

Iv is real. Lemma 4.6 and its corollary describe the structure of the negative part of the set X. It remains to prove that every nonnegative Iv lies in X.

It is known ([Gla, Chap. 5, Sec. 55] and [Bed 0, Chap. 6, Subsection 2.10)) that, for

any £ > ° and almost all Iv ~ ° (with respect to the spectral measure), the following in-

equality is true 1<p(t,Iv)I::;C(Iv,£)ltI I/2 +E (tE [0,00». On the other hand, if 0::;

(j(t) ¢ 0, then it follows from (4.11) that W(t) ~ a> ° and, therefore, fi(t) ~ at (t E

[0, 00 ». Thus, for the indicated values of Iv and sufficiently small £ > 0, we have <p(t, Iv) = O(il(t» , i.e., these 'A lie in X. At the same time, the collection of these A is dense in [0,00) and X, as the space of maximal ideals, is closed in the ordinary topol-

ogy of the complex plane. Hence, [0, 00) C X. In the case where (j(t) == 0, the last inclusion is obvious.

• Corollary 4.3. For all functions <p (t, 'A), where 'A belongs to the spectrum of

equation (4.1) with the boundary condition y'(O) = 0, the following inequality holds:

I <p(t, 'A) I ::; ilU),

00

where il(t) is a solution of the equation y" [Var q] y = ° satisfying the initial

conditions fiCO) = 1 and W(O) = 0.

Proof. This statement follows from the fact that each character of a hypercomplex system is bounded by one.

•


By applying Theorems 3.2 and 3.3 from Chapter 1 to the hypercomplex system

L J ([ 0, 00 ), fl2 (t), dt), we obtain the well-known Plancherel theorem and the inversion formula for the Sturm-Liouville equation (although for a fairly poor set of potentials; this fact serves as an additional explanation of the necessity of generalization of the concept of hypercomplex systems; in this connection, see [Vai6], [Vai7]). The duality

Theorem 3.5, generally speaking, is not true for the hypercomplex system L J ([0, 00),

fl2(t), dt).

4.5. Survey of Related Results. Chebli [Chel]-[Che3] and Achour and Trimeche [AcT2] studied hypercomplex systems associated with the operator

1 d ( dy ) Ly = --- ACt)-A(t) dt dt

(4.33)

in the space H = L2 ([ 0,00), A (t)dt), where the function A (t) satisfies the following conditions:

1) A(O)=O and A(t»O for t>O;

2) in a neighborhood of zero, A'(t)/ A (t) = (a/t) + B(t), where a> 0, BE cOR);

3) the function A (t) is increasing and A I (t) / A (t) is decreasing as t approaches 00.

Chebli [Chell showed that the operator L is self-adjoint in the region .is) L = {y E

HI Ly E H and lim A(t)y'(t) = O} and has a simple spectrum whose continuous part 1--,)00

is [p2, 00), where

p = lim.!. A'(t) . Hoo 2 ACt)

Theeigenfunctions <p(t,A) of the problem Ly=Ay, y(O)=I, and y'(O)=O are

equibounded, Le., I <p(t, Iv) I:::; 1, and the generalized translation operators Rs associ

ated with the operator L by the method of Levitan and Povzner preserve positivity (this result follows from the maximum principle established in [Wei]). It is not difficult to show that the generalized translation operators R s satisfy all conditions of Theorem 2.1

in Chapter 1. Thus, L J ([ 0, 00 ), A (t )dt) is a commutative Hermitian hypercomplex sys

tem with basis unity e = 0 satisfying the condition of separate continuity. In [Che2l. it was proved that the operator L is unitarily equivalent to the operator of multiplication

by an independent variable in the space ~ (Q, m), where m is a Plancherel measure, and an analog of the Paley- Wiener theorem for the Fourier transformation over the


characters of the hypercomplex system LI ([0, 00), A(t)dt) was established (see also [Fle]). For the further development of harmonic analysis for operator (4.33), see [AcT2].

Note that the radial part of the Laplace-Beltrami operator acting in noncompact Riemannian symmetric spaces of rank 1 (i.e., in the Euclidean spaces and Riemannian noncompact-type symmetric spaces of rank I) has the form (4.33) with the function A (t)

satisfying conditions 1) - 3). In particular, the case A (t) = t 2 a + I leads to the Hankel transformation (this case was already analyzed in [DeI2] and [Del3]).

The operator L given by (4.33) on the compact set [0, nI2], where A (t) =

(sin 2t)2a+ I B(t) and B(t) E e([ 0, n12]) is a strictly positive function satisfying the conditions: 1) B'(O)=O, B(t)=B(nI2)-t); 2) A(t) is an increasing function in the

interval (0, lt/2]; 3) A'(t)1 A(t) decreases in the interval (0, nI2), was studied in [AcT2]. The relevant generalized translation operators satisfy all conditions of Theorem

2.1 in Chapter 1 and L 1([ 0, nI2], A (t)dt) is an Hermitian hypercomplex system with basis unity e = ° satisfying the condition of separate continuity. If B (t) = 2a-~+ 1 sin2~-2at, then the Jacobi polynomials Rra,~)(x) normalized by the condition

Rka'~)(l) = 1 are the eigenfunctions of the problem under consideration (the change of variables x = cos 2t transforms the corresponding equation into the ordinary differential equation for the Jacobi polynomials).

If, in (4.33), we set A (t) = (2a + 1)1 t (a> - 1/2) then the eigenfunctions of the boundary-value problem

Ly(t) = A2y(t), yeO) = 1, y(l) = 0, y'(O) = ° (tE [0,1]) (4.34)

are the Fourier-Besselfunctions J1a)(t) = ja(Akt), where ° < Al < ... < Ak <... is

the set of positive zeros ofthe normalized Bessel function ja(t) = 2 a r(a + 1 )t-a Ja(t), written in the order of increasing. It is known that the Fourier - Bessel functions form a

complete orthonormalized system in L2 ([ 0, 1], t 2a + I dt). Consider the functions

J1a)(t) = J1a)(t)/ Jba)(t). The functions J1a)(t) are solutions of the boundary-value

problem (4.34) with A(t)= [Jba)(t)ft2a+1 and the relevant eigenvalues are A2= AlAt. It is not difficult to calculate that J;;1I2(t) = Rk-1/ 2,-1I2\cOSltt) and Jf2)(t) =

Rrll 2, 11 2) (cos ltt) (here Rra'~)(t) are normalized Jacobi polynomials with Rka'~)(I) =

1). In other cases, Jra)(t) are not reduced to orthogonal polynomials. Generalized

translation operators associated with problem (4.34) for A(t) = [J6a)(t)]2 t2a+ I were studied in Markett [Mar4]. These generalized translation operators have the form

I

Rcf(s) = f f(r)K(a)(t, s, r)[Jba)(r)f?a+ldr.

o


For a = n + 1/2 (n = -1,0, 1, ... ), the explicit form of the kernel K(a)(t, s, r) was

found. If a E {1/2, 3/2, 5/2}, the kernel K(a)(t, s, r) is expressed in terms of elementary functions, and the corresponding generalized translation operators preserve po-

sitiveness. This implies that, for a E {1/2, 3/2, 5/2}, Ll ([0, 1], [J~a)(t)]2t2a+ldt) is an Hermitian hypercomplex system with basis unity e = 0 satisfying the condition of

separate continuity, and the functions Jka)(t) are characters of this hypercomplex system.

In [CoS3] (see also [CoS4]), Connett and A. Schwartz studied the properties of the characters and the Plancherel measure of hypergroups generated by equation (4.33) on

the segment [O,n;] with the boundary conditions y'(O)=O, lim [C1y(t)+ y-Ht-

C2A(t)y'(t)] = 0 (Cf + ci "* 0). Here, the function A(t) satisfies the following con

ditions: (i) A(t»O; (ii) A(t)E CI(O,It); (iii) for some a,~E IR, thereexistsposi

tive limits lim r 2a - I A(t) and lim (It_t)-2~-lA(t). This class of hypergroups, t---7O+ t---7O+

called by the authors "Jacobi-type hypergroups," includes hypergroups associated with Jacobi polynomials and Fourier - Bessel functions; it also includes hypergroups obtained for ultraspherical polynomials by perturbation of a differential equation ([CoS5]).

Generalized translation operators associated with the regular Sturm - Liouville prob

lem

Ly = -y"+q(t)y = Ay (O$t$n;), (4.35)

y'(O) = hyCO), y'(n;) = hly(It), (4.36)

were studied by Levitan [Lev9] and Vainerman [Vai2], [Vai3] (see also Section 3 in Chapter 3). Vainerman [Vai2], [Vai3] showed that these generalized translation oper

ators can be associated with a real hypercomplex system with compact basis [0, n;] defined in the sense of Subsection 5.5 in Chapter 1. The hypercomplex system with discrete basis dual to this system is determined by the structure constants

where i,j, kENo and <per, Ak ) is an eigenfunction of problem (4.35), (4.36), and by

the Plancherel measure II <pC Ak ) 11;2 . By virtue of the theorem on approximation, linear combinations of eigenfunctions are

uniformly dense in C([O, n;]). The application of the duality principle leads to the following version of the inverse problem for the Sturm-Liouville operator on a segment:

The parameters q (t), h, and hI of problem (4.35), (4.36) are uniquely recovered from

the "spectral data" ct and J..Lk [Vai3]. It is interesting to clarify under what conditions

on the potential these generalized translation operators preserve positiveness.


Trimeche [Tri1], [Tri2] and Markett [Mar2] considered the operator

1 d ( d) L = -- A(t)- + q(t) A(t) dt dt

on the semiaxis and on a finite interval and studied generalized translation operators

associated with it; here, A(t)=t2U + 1B(t) (a>-l/2), B(t)E C"'(lR) is an even

strictly positive function, and q(t) E C"'(lR). A particular case where B(t) == 1 was investigated by Markett [Marl], [Mar2], Braaksma [Bra], and Braaksma and de Snoo [BrS]. In particular, Braaksma and de Snoo [BrS] established sufficient conditions for the corresponding generalized translation operators to preserve positiveness, and Markett [Mar2] obtained estimates for the norms of these generalized translation operators.

If one takes B(t)== 1 and ql (t)= t 2 or

then the eigenfunctions of the problem Ly = Ay, yeO) = 1, y'(O) = 0, are, respectively,

the normalized Wittaker functions <PI (t, A) = t-U- 1 M').)14,U/4(t 2) and normalized Jacobi functions

( Sinht)(U+l)/2 R <P2(t, A) = -t- (coshd~+1)/2<p~U'f')(t).

Generally speaking, unbounded generalized translation operators associated with the equation

!!.- (P(t) dy ) + q(t)y = Ar(t)y, dt dt

where p (t) and r (t) are continuous positive functions on the semiaxis, were studied by Vainerman [Vai6], [Vai7]. For these generalized translation operators, an unbounded Hilbert algebra was constructed (see the cited works), and the Plancherel theorem and inversion formula were established for the generalized Fourier transformation related to these generalized translation operators.

3. ELEmEnTS OF LIE THEORY FOR GEnERALIZED TRAnSLATIon OPERATORS

As in the case of topological groups, there are two approaches to the investigation of generalized translation operators-global and infinitesimal. The first one includes the theory of representations, harmonic analysis, the theory of almost periodic functions, etc. For groups, the second approach reduces to the construction of Lie theory, i.e., to the analysis of, generally speaking, nonlinear law of multiplication in a certain neighborhood of the identity element of the Lie group in terms of a linear object, i.e., in terms of the Lie algebra. In the present chapter, we construct elements of Lie theory for generalized translation operators. It is worth noting that the Lie theory of generalized translation operators is far from being complete.

In the first section we show that, as in the case of Lie groups, generalized translation operators Rs (s E Q, Q is a smooth manifold) are associated with infinitesimal opera

tors (generators)

The algebra generated by all generators of generalized translation operators is an analog of the universal enveloping algebra of the Lie algebra. However, unlike the case of Lie groups, for generalized translation operators, there is no general concept similar to the concept of Lie algebras. This is explained by the fact that generators corresponding to generalized translation operators may be degenerate and nondegenerate generators satisfy much more general relations than in the case of Lie algebras. Therefore, it is interesting to analyze the problem of selection of various special classes of generalized translation operators for which it is possible to construct meaningful Lie theory.

In the second section, we describe three classes of generalized translation operators of the indicated type associated with Lie groups: Delsarte generalized translation operators, Delsarte-type generalized translation operators, and generalized translation opera

tors associated with the hypercomplex system L) (G, H) with basis Q = G / / H, where G is a Lie group and H is its compact subgroup.

315

316 Elements of Lie Theory for Operators of Generalized Translation Chapter 3

In the third section, we present another point of view on the generators of one-dimensional commutative hypercomplex systems. Namely, the first Nth-order nonzero gener

ator X N of the one-dimensional hypercomplex system L1 (Q, m) (Q = [a, b], -00:::;

a < b:::; 00) is associated with a self-adjoint operator X acting in L2 (Q, m) and such

that (Xf)(t) = (XNf)(t) for all fE Coo(Q) n V(X):t:. 0. By using the decomposition

of the operator X in generalized eigenfunctions X(t,'t) (tE Q; 'tE Q), we determine

the relevant collection of generalized translation operators by the formula (R sf X t) =

(X (s, X)f) (t) (t, SEX), where X (t, 't) regarded as functions of t are characters of the hypercomplex system L1 (Q, m) and, as functions of 't, they are characters of the dual hypercomplex system. In the third section, we also establish an "infinitesimal analog ofPontryagin's duality," i.e., for a given generator X of the hypercomplex system

L 1 (Q, m), we construct the "dual" generator of the hypercomplex system L 1 ( Q, fit), and vice versa.

1. Basic Concepts

It is well known that the Lie agebra g of a Lie group G, i.e., the tangent space at the unit element of the group endowed with the operation of commutation generated by the operation of multiplication in the group G, is its infinitesimal object. The Lie algebra g is also isomorphic to the algebra of the first-order group generators which are differential operators of the first order (with respect to the ordinary operation of commutation of operators). The notion of generator is one of the most important notions in the infinitesimal theory of generalized translation operators (the precise definition of generators of generalized translation operators is given in what follows).

For arbitrary generalized translation operators R s (s E Q), the situation is more

complicated than in the case of Lie groups: First, in many interesting cases (e.g., Q = G / / H, where G is a Lie group and H is its compact subgroup), the set Q is, generally speaking, not a manifold. Second, all generators of generalized translation operators under consideration may be degenerate up to a certain fixed order and nondegenerate generators are, generally speaking, not differential operators. Finally, the relationships between nondegenerate generators are, in general, more complicated than in the case of Lie algebras. This explains the necessity of the investigation of nonreduced generalized translation operators on smooth manifolds.

Let us show that the algebra generated by all generators is the infinitesimal object for generalized translation operators of this type and the sub algebra generated by all right generators is the infinitesimal object for reduced generalized translation operators (in the case of Lie groups, this means that, instead of Lie algebras, the role of the infinitesimal object is played by universal enveloping algebras of the corresponding Lie algebras).

We also study general properties of the generators of generalized translation operators and develop a pure algebraic approach to the infinitesimal theory of formal general-

Section 1 Basic Concepts 317

ized translation operators. Subsection 3.2 deals with the theory of topological bialgebras. It is included in the book because constructions presented there are helpful in understanding the algebraic nature of the objects studied in Section 3 (thus, it is shown that generalized translation operators can be regarded as coassociative comultiplication with right counit). At the end of the section we present necessary facts from the theory of topological vector spaces. In our presentation, we mainly follow the survey [Lit3].

1.1. Hypergroup Algebra of Infinitely Differentiable Generalized Translation Operators. In this subsection, we show that every smooth family of (nonreduced; see

Section 1 in Chapter 2) generalized translation operators in COO (Q) (Q is a smooth

manifold) admits a unique description by endowing 'D(Q) = (Coo(Q»)' with the struc

ture of associative topological algebra with right identity 8e (e E Q, R e = I). This re

sult is similar to Theorem 2.1 in Chapter 1 (also see remarks to this theorem). In this case, reduced generalized translation operators are characterized with the help of a subal

gebra of the algebra 'D( Q). We recall that the space 'D( Q) is equipped with the Mackey topology.

Consider a family of generalized translation operators Coo(Q) 3JH Rsf(t)E

Coo(Q) given in the space Coo(Q), (s E Q; we do not assume that the generalized translation operators R s are reduced). The family of generalized translation operators

Rs is called infinitely differentiable if the correspondence J(t) H (RsI)(t) is a conti-

nuous mapping from Coo(Q) into Coo(Q x Q). This implies that, for any FE 'D(Q)

and JE Coo(Q), the functions Q:3 S H (F, Rsf) and Q:3 S H (F, Lsi) belong to the

space Coo(Q).

For any F E 'D( Q), we introduce operators R (F) and L (F) acting in the space

COO(Q) by setting

L(F)J(t) = (F, Rtf) and R(F)J(t) = (F, Ltf) (t E Q). (1.1)

for all functions J E COO (Q). The operators L (F) and R (F) are continuous in view of

the continuity of R t and Lt as operators in COO (Q).

We intFoduce the operation of convolution in 'D(Q) as follows: For any F, G E

'D( Q) and J E COO (Q), we set (F * G, J) = (F, R (G )J). Since the operator R (G) is

continuous, we conclude that F * G E 'D( Q).

Since elements of the space 'D(Q) satisfy an analog of the Fubini theorem (see [Rha]), we can write


(as in Section 2 in Chapter 1, the superscript of the functional denotes the variable upon which it acts). As a result, we obtain

(F*G,f) = (F,R(F)f) = (G, L(G)f) (F,GE 1J(Q);fE Coo(Q». (1.2)

Since the operator F H F * G in 1J(Q) is adjoint to the continuous operator R(G) in

Coo (Q), the operation of convolution is separately continuous in 1J (Q). In view of the fact that the Mackey topology is stronger than the weak topology, this operation is also

separately continuous in the topology 0'( 1J( Q), COO (Q». By virtue of the result pre

sented at the end of Subsection 1.6, the convolution F * G is continuous in 1J( Q).

The mapping 1J(Q) 3 F H R(F) is a representation of the algebra 1J(Q) and

the mapping 1J (Q) 3 F H L (F) is its anti representation. Indeed, for any F, G E

1J(Q), fE Coo(Q) we have

(R(F)R(G)f)(t) = (F, L1R(G)f) = (L;F, R(G)f)

where t E Q and L; is the operator adjoint to the operator Lt.

Similarly,

(L(F)L(G)f)(t) = (L(G * F)f)(t).

• Theorem 1.1 (Litvinov [Litl], [Lit3]). The space 1J(Q) with operation * is an

associative topological algebra with continuous multiplication and right identity be

which is a multiplicative functional on Coo (Q).

Conversely, suppose that 1J(Q) is endowed with the structure of an associative

algebra and there is a multiplicative functional which is the right identity in 1J(Q). If

the multiplication in 1J(Q) is separately continuous in some locally convex topology

compatible with the duality (1J( Q), COO (Q », then this multiplication is continuous

in the Mackey topology on 1J( Q). In this case, there exists a unique infinitely differ

entiable family of generalized translation operators R s (s E Q) such that the multi

plication in 1J( Q) coincides with the relevant operation of convolution.


Proof. Let us prove the first assertion of the theorem. The fact that the algebra

f})( Q) is associative is a consequence of the following chain of equalities which is true

for all F, G, H E f})(Q) and f E COO (Q):

«F * G) * H, f) = (F * G, R(H)f) = (F, R(G)R(H)f)

The continuity of multiplication in f})( Q) has already been proved. Since

= (F, (Lrf)(e» = (F, RJ) = (F,J),

the multiplicative functional 0 e is the right identity of the algebra f})( Q).

Let us prove the converse assertion. Suppose that f})(Q) is an associative algebra with multiplication separately continuous in some topology compatible with the duality

( f})( Q), COO (Q» and right identity which is a multiplicative functional on Coo (Q).

Since multiplication in f})( Q) is separately continuous in a topology compatible with

the duality (f})( Q), COO (Q», it is separately continuous in the weak topology on

f})( Q). Since Coo (Q) is a Frechet space, this multiplication is also continuous in the

strong topology P(f})(Q), COO(Q») on f})(Q), which coincides with the Mackey topo

logy in the case under consideration. Thus there exists a continuous mapping f})( Q) ® f})(Q) ~ f})(Q). Let

be the dual mapping. Since Coo (Q) and COO (Q x Q) are reflexive, the mapping fl. is

continuous. For any SEQ, we set (F, (RsfXt» = (F * Os.!), i.e., we define the

operator R s in COO (Q) as an operator adjoint to the operator of right multiplication by

Os in f})(Q). Since the mapping f(t) H u(t, s) = (Rsf)(t) coincides with fl., the fam

ily of operators R s is infinitely differentiable.

Let us show that the operators R s satisfy the axioms of generalized translation oper

ators. First, we demonstrate that there exists a point e E Q such that the right identity in

f})(Q) is equal to De. Indeed, let <P E f})(Q) be a multiplicative functional and let K

be its support. We show that there exists a point to E Q such that <P = Oro- Suppose

that K contains at least two different points t, S E K. Since Q is the Hausdorff space,

there are disjoint neighborhoods Ut and Us of the points t and s, respectively, and


functions f, g E Coo (Q) such that supp f c U f , supp g C Us and (f) (g) * 0.

However, in this case, by virtue of the multiplicativity of <1>, we get

° = (Cf>,fg) = Cf>(f)(g) = ° because f g == 0. Consequently K = {to}. In a neighborhood U of the point to, we introduce coordinates <j)u such that <j)u(to) = 0. By virtue of the well-known theorem, in

these coordinates, the functional can be rewritten as

(cfJ,f) = L Ca.('66a.),f 0 <j)i}) lal:5k

where k < 00, a = (a l' .. , ,an) is the multiindex, n is the dimensionality of the mani

fold Q, and ca. E ([.

A functional of the indicated form is multiplicative only in the case where

{O,

ca. = 1,

a * (0, ... ,0),

a = (0, ... ,0).

Since the right identity cfJe ofthe algebra 'iJ(Q) is a multiplicative functional, we con

clude that cfJ e = '6 e' where e is a point from Q. This implies that R e is the identity operator.

Let L t be an operator in Coo (Q) adjoint to the operator of left multiplication by '6 f' i.e., (F, Ltf) = ('6 t * F,f). Let us show that (Ltf)(s) = (Rsf)(t). Indeed,

whence it follows that, for any t, s, r E Q, we have

which yields the associativity of the operators Rs (s E Q). Hence, Rs is a family of in

finitely differentiable generalized translation operators. Since I1f(t, s) = (Rsf)(t), the


operation of multiplication in V( Q) coincides with the operation of generalized convolution defined for the family of generalized translation operators Rs according to (1.2) .

• Let COO(Q) = Le(COO(Q») be the principal space of functions where the reduced

generalized translation operators act. The projector V( Q) 3 F * Oe ~ F E V (Q) is

adjoint to the projector Le. Denote by V(Q) the image of this operator.

The space V(Q) is a close subalgebra of V (Q) with two-sided identity oe and,

moreover, tjj(Q) can be identified with the quotient algebra (endowed with the quotient topology) of the algebra V (Q) with respect to the ideal orthogonal to the

principal space Coo (Q) .

Indeed, the first part of this statement is evident. Let

I = {F E V(Q)i (F, J) = 0 for all J E COO(Q)}.

Since,forany FE V(Q), we have (Oe*F,f)=(F,Le!) (t,sE Q; fE Coo(Q»), we

conclude that F E I if and only if oe * F = O. Let us show that I is a two-sided ideal.

Indeed, for any F E I and G E V( Q), we can write

and

Oe * (G * F) = oe * G * oe * F = 0

because oe is the right identity in V( Q). The equality tjj(Q) = V( Q) I I is obvious .

• Hence, tjj(Q) coincides with the space dual to Coo (Q). In what follows, the alge

bra V( Q) is called the hypergroup algebra and V(Q) is called the hypergroup al

gebra of reduced generalized translation operators. It should be noted that if too (Q)

does not coincide with Coo (Q), then the hypergroup algebra V( Q) is noncommutative even in the case where the generalized translation operators Rs (s E Q) are commuting (see Example 2 in Subsection 3.3). In this case, the hypergroup algebra of reduced generalized translation operators is commutative.

Note that we are interested in the case of reduced generalized translation operators

and, therefore, focus our attention on the algebra V(Q) (although it is clear that V(Q)


contains information both about 1J(Q) and about the process of reduction, the algebra

1J(Q) is much simpler).

1.2. Topological Bialgebras. In the case under consideration, topological bialgebras play the same role as Hilbert bialgebras in the theory of quantized hypercomplex systems. In what follows, we are mainly interested in the infinitesimal object for generalized translation operators and, therefore, only outline problems related to the theorem on realization and duality. Our attention is mainly focused on the pure algebraic scheme.

Let :J be a locally convex space and let :J' be its dual space. The spaces :J and

:J' are equipped with the Mackey topologies 't (:r. :1') and 't (:1', 1), respectively. If :J and :J' are associative topological algebras with separately continuous multiplica

tion, then the pair U = ( :r. :J') is called a topological bialgebra. Multiplication in the

algebra :J is de,noted by cp n 'I' (cp, \jf E :J) and multiplication in the algebra :J' is

denoted by F U G (cp, 'I' E :J'). Let U = ( :r.:J' be a topological bialgebra and let :J be a commutative Arens

Michael algebra (i.e., :J is a complete Hausdorff topological space such that the inequality p (cp n '1') ~ p (cp ) p ('I') holds for all seminorms p from some defining system

of serninorms and any cp, 'I' E Jj.

We denote the subset of :J' of all multiplicative functionals on :J by Q = Q(1) and equip it with the induced topology. Unlike the theory of Banach algebras, Q is not

necessarily locally compact. As is known (see, e.g., [Khe]), the Gelfand transformation

cp H (t,cp)=cp(t) (cpE :r. tE Q) maps the space :J into the space C(Q) ofcontinuous functions on Q. Since :J is an Arens-Michael algebra, its radical coincides with the kernel of the Gelfand transformation.

Let :J be semisimple. Then :J is (algebraically) isomorphic to a certain subalgebra

S( Q) of the algebra C( Q). For any t E Q, we denote by R t (Lt ) the operator in

S( Q) adjoint to the operator of right (left) multiplication by t E Q in :J'.

Assume that, in a topological bialgebra U = (:r. :J'), :J is a commutative semi

simple Arens-Michael algebra and the algebra :J' possesses the right identity which

is a multiplicative functional. Then R ( (t E Q) are generalized translation operators.

Indeed, let e E Q be a multiplicative functional which is the right identity of :J'. It

is clear that Re = id. In view of the fact that

(Rscp)(t) = (cp,tU s) = (L(cp)(s) (cp E :F. t,SE Q),

we have (Rs cp )(t) E S (Q) for any fixed t. The associativity of the operators R s is

checked in exactly the same way as in Theorem 1.1. Hence, Rs (s E Q) are generalized

translation operators.

•


If all conditions of the previous assertion are satisfied, then all operators R.~ (s E

Q) are endomorphisms of the algebra :F if and only if the operation of multiplication

in T induces the structure of a semigroup in Q. In this case, Rs/(t) = f(t Us).

Indeed, assume that all Rs (s E Q) are endomorphisms of the algebra 1'. Then

(t, SEQ),

whence it follows that t U s = r is a multiplicative functional and Q is a semigroup. By introducing evident modifications in this chain of equalities, we arrive at the con

verse assertion.

• To transform the semi group Q into a group, it suffices to require that the right iden

tity e E Q of the algebra :F' be two-sided and find an involutive automorphism B : :F~ :Fsuchthat (t,B(Rt<p»=(e,<p) for any eE Q and <pE~.

The topological bialgebra U' = (:r, :F') is called dual to the topological bialgebra

U = (:r, :F'). Ifboth :F and :F' are commutative semisimple Arens-Michael algebras

with identities which are multiplicative functionals on T and :r, respectively, then, in

the space Q C :F of multiplicative functionals on T, one can also construct a family of

generalized translation operators Rs (s E Q) which is naturally regarded as dual to

R t (t E Q). The generalized translation operators Rt (t E Q) and Ri (t E Q) are

commutative. Since (U')' = U, we have Q = Q. It should be noted that, generally speaking, this duality does not coincide with the Pontryagin duality.

Thus, let :F coincide with COO(IR) with the ordinary multiplication of functions

and let T = 1J( IR) be the group algebra of the group IR. Then Q = IR and the generalized translation operators Rs (s E Q) coincide with ordinary right translations

and Q consists of all functions e AS (A E <r). Hence, Q can be identified with the

group <r. The Gelfand transformation on 1J( IR) coincides with the Fourier-Laplace

transformation and maps 1J( IR) into an algebra of analytic functions on <r. In the case where U = (1J(Q), COO(Q)), the operation of multiplication (general

ized convolution) in lJJ( Q) is automatically continuous and can be extended to a conti

nuous mapping lJJ( Q) ® lJJ( Q) ~ lJJ( Q). The adjoint mapping

is called comultiplication.


If V( Q) possesses the right identity oe (e E Q), then, by Theorem 1.1, the comul

tiplication ~ coincides with the generalized translation, i.e.,

~f(t, s) = Rs/(t).

The associativity of multiplication in V(Q) yields the coassociativity of comultiplication ~, namely,

(id®~)~f = (~®id)~f (jE COO(Q»)

or, equivalently, the commutativity of the diagram

~ COO(QxQ)

J. .1®id

id®.1) COO(Q x Q x Q)

A linear functional E: COO(Q) --7 a: is called the right counit with respect to comulti

plication ~ if (id ® E)~ = id. It is clear that oe is the right counit for comultiplication

under consideration. It is also clear that the algebra Coo (Q) with the ordinary multiplication of functions and coassociative comultiplication with right counit contains the en-

tire body of information about the topological bialgebra 11 = (V(Q), COO (Q»). Similar

ly, the operation of multiplication of functions in Coo (Q) generates the operation of co

multiplication in V(Q) and the algebra V(Q) with generalized convolution as multiplication and comultiplication generated by the ordinary multiplication of functions in

Coo (Q) = ( V( Q»)' contains all necessary information about the dual topological bialge

bra 11' = (V(Q), COO(Q»). This construction appears to be fairly general and can be successfully applied in the theory of duality of locally compact groups and hypercomplex systems, theory of quantum groups, and other problems of analysis.

1.3. Infinitesimal Object for Generalized Translation Operators. Let us introduce the notion of infinitesimal object for infinitely differentiable generalized translation operators. We give two definitions. In the first definition, the infinitesimal object is understood as a sub algebra (infinitesimal algebra) of the hypergroup algebra. The second definition is based on the notion of generators of generalized translation operators. We establish that these definitions are equivalent and show that the linear space of an infinitesimal algebra can be regarded as the tangent space to the infinite-dimensional manifold

V( Q) at the point oe. In the sequel, we always assume that coordinates <p in the

neighborhood of the point e E Q are chosen so that <p (e) = (0, ... , 0).

Let Rs (s E Q) be infinitely differentiable generalized translation operators on a

smooth manifold Q, let V( Q) be the corresponding hypergroup algebra, and let V(Q)


be the hypergroup algebra of reduced generalized translation operators. The infinitesi

mal algebra of the generalized translation operators Rs (s E Q) is defined as a subal

gebra 1)e(Q) c t])(Q) generated by generalized functions whose supports are concen-~ ~

trated at the unit element e E Q. The subalgebra t])(Q) = t])e(Q) n t])(Q) of the alge-

bra t])e (Q) is called the infinitesimal algebra for reduced generalized translation oper-

ators. It is clear that iJ(Q) = oe * t])e (Q). The algebras t])e (Q) and iJeQ) are regarded as the infinitesimal objects for infinitely differentiable and reduced generalized translation operators, respectively.

We now give another definition of the infinitesimal object.

The right generator of order 0.= (0.1' ... ,an) of generalized translation operators

Rs (s E Q) in Coo (Q) is defined by the formula

(l.3)

It is clear that, on the right-hand side of (l.3), t, s, e E Q mean, respectively, <p(t),

<pes), and <p(e)=(O, ... ,O)E lR n, while fE Coo(Q) means fo<p-1, where <p are

some fixed coordinates in a certain neighborhood of the point e. However, we prefer the simpler notation and hope that this would riot lead to misunderstanding.

The left generator is defined similarly:

(1.4)

The algebra X; generated by all left generators is regarded as the infinitesimal object

for infinitely differentiable generalized translation operators Rs (s E Q) and the algebra

1{ generated by all right generators is assumed to be the infinitesimal object for reduced generalized translation operators.

Theorem 1.2 (Litvinov [Lit3]).

(i) The algebras t])e(Q) and X; are antiisomorphic.

(ii) The algebras iJe(Q) and 1{ are isomorphic.

Proof. Consider the collection of operators R(F) (F E t])e(Q)). Since 1)e (Q) 3

F H R (F) is a representation of the algebra t])e (Q), we conclude that

is an algebra of operators called the algebra of infinitesimal right translations.


In a similar way, we define the algebra of infinitesimal left translations as

The algebras tJJe (Q) and tJJ;- (Q) are antiisomorphic.

Indeed, assume that an element F E tJJe (Q) is such that L (F) = id. Then, for any

f E Coo (Q), we can write

whence it follows that F = oe, and the required assertion follows from the fact the map

ping 'De(Q):3 F H L(F) E 'D;-(Q) is an antirepresentation.

The algebra 'D:(Q) is isomorphic to the algebra 1Je(Q) = oe * 'De(Q). Indeed, in

view of the fact that oe * F = F for any FE tJJe(Q) , by assuming that R(F) = id, we obtain

i.e., F= oe.

Clearly,

(1.5)

Since o~a) (ex = C ex l' ... , ex n)' ex i E N U {O}) form a system of generating elements

in 'DeCQ), the generators La and Le (Ra and Re= id, respectively) generate the al

gebra of infinitesimal left Cright) translations 'D;-CQ) ('D:CQ)). Consequently, .t; = 'D;-CQ) and !i{= 'DRCQ).

• We consider several simple examples.

Example 1. If Q = G is a Lie group, the Schwartz theorem implies that the algebra

'DeCQ) = 1Je CQ) is isomorphic to the universal enveloping algebra U(g) of the Lie al

gebra g of the group G. By the Poincare-Birkhoff-Witt theorem (PBW), elements of the form

( 1 <· < < . < X X' b" ).. b'· U( ) _ II - .•• - 1 P _ n; 1 ... n IS a aSls 10 g lorm a aSls 10 g .

Section 1 Basic Concepts

Example 2. Let Q = 1R 1 and let

Then

1 (Rsf)(t) = -[J(t+s)+f(t-s)]

2

327

and the algebra 1J:(Q) is commutative and generated by a single generator R2. The

algebra of infinitesimal left translations 1J~(Q) is characterized by a more complicated structure. The generators

where

1 (Lof)(t) = -(f(t) + f(-t))

2

formabasisin 1J~(Q). Since L2k+lLn=O and L2kLn=L2k+n' we have L 2k = (L2 )k

and L2k+ I = (~)k L1, whence it follows that L1, L 2, and Lo are generating elements

in 1J~ (Q). They satisfy the relations

and, therefore, 1J~(Q) is noncommutative although the generalized translation opera

tors Rs are commuting.

Thus, to construct the direct Lie theory of generalized translation operators, it is ne-

cessary to describe the infinitesimal algebra 1Je(Q) (or 1Je(Q) if we are interested only

in the reduced case), i.e., to find the minimal collection of generating elements and establish relationships between them. The subject of the converse Lie theory can be described as the construction of the "global" object-a family of generalized translation operators-in terms of the relevant infinitesimal object, i.e., in terms of the algebra

1Je(Q) (or iJe(Q)).

As follows from Example 2, to reconstruct the original generalized translation operators in terms of a given infinitesimal object, it is not sufficient to know the algebra

iJe(Q). Indeed, the algebra iJe(Q) in Example 2 is the infinitesimal algebra of right

translations both for ordinary group translations on 1R 1 and for many other generalized

translation operators. However, if we a priori know something about the required gen-


eralized translation operators (e.g., the method of reduction or the explicit form of generators), then we can reconstruct the family of generalized translation operators for a given

algebra 'lJe(Q) in a unique way. Thus, local Lie groups can be reconstructed in a unique way for given Lie algebras Calso see Sections 2 and 3).

In numerous examples, '1JeCQ) coincides with the set of all functionals whose sup

ports are concentrated at the point e and, therefore, up to the end of this section, we as

sume that '1Je (Q) possesses the indicated property, Le.,

'1Je(Q) = { L cab~O:)1 CX=(cx1'···,CXn ), cx], ... ,cxn , kENU{O}, CaEa::}. 10:1:-:; k

We also present another definition of the infinitesimal object of a family of infinitely

differentiable generalized translation operators. It is known that the tangent space TtoM

tangent to a manifold M at a point to can be obtained as a set of all tangent vectors to

smooth curves from M passing through the point to at this point. The linear space of

the Lie algebra of a Lie group G can be regarded as the space tangent to the Lie group

G at the identity element of the group. Let us show that the linear space of the algebra

'lJe (Q) admits the same interpretation. Since some generators are degenerate, it is a

priori impossible to consider only the tangent space TeQ or its finite tensor power.

Thus, one must take the infinite-dimensional space '1J( Q) as the relevant manifold.

A path with the initial point at be E '1J( Q) is defined as a mapping 1R 1 3 t ~ Ft E

'lJ( Q) satisfying the conditions:

2) The vector-valued function 1R 1 :3 t ~ Ff is infinitely differentiable.

3) The mapping 1R]:3 t ~ supp F f is continuous in the Michael topology [Mic]

(see Subsection 2.3 in Chapter 1).

Two paths Ft and Gt are called equivalent whenever

I (Ft,f)-(Gt,f)1 = o(t) as t~O

for all fE C"'(Q).

The collection of classes of equivalent paths with the initial point at be is called the

tangent space Te '1J( Q) to the manifold '1J( Q) at the point be. It is easy to see that

the tangent space Te'1J(Q) coincides with the set of all functionals (tangent vectors)

fr E '1J(Q) of the form


(1.6)

Indeed, let us show that the mapping which associates a tangent vector F with a

path F t according to (1.6) is defined on the set of classes of equivalent paths and injec-

tive. If F t - G t, then, for all f E Coo (Q), we have

ddt [(Ft,f) - (Gt,f)] It=o ~ lim ! 1 (Ft - Gr,f)1 = 0, t~O t

whence it follows that F = {;. Conversely, if F = {;, then

( A A) . 1 o = F - G,f = hm - [(F[,f)- (G/,f)] t~ 0 t

for all f E COO(Q) and, hence, Ft - Gt .

• The tangent space Te 1J(Q) is linear. If F[ and Gt are two paths with the initial

point ()e and F and {; are the relevant tangent vectors, then the vector A F + 11 G (A, 11 E a:) is a tangent vector to the path

The tangent space Te 1J( Q) coincides with the set 1J e (Q) of functionals whose

supports are concentrated at the point e.

Indeed, if F E 1Je (Q), then F is a vector tangent to the path F, = ()e + t F, Le.,

1Je (Q) C Te 1J(Q). Let us prove the inverse inclusion. If p E Q, p *' e, then, in view of condition 3) of

the definition of the path F[ and the fact that the Michael topology is a Hausdorff topo

logy, one can indicate a precompact neighborhood Up of the point p whose intersec

tion with e is empty and such that supp Ft n Up = 0 for all t sufficiently close to zero.

Let F be a vector tangent to the path Ft. Then

(F,f)

whenever suppfCUp which means that p~ supp F. Since the point pE Q, p*,e, is


arbitrary, we arrive at the required assertion.

• 1.4. General Properties of Generators of Generalized Translation Operators. As

shown in Example 2 in Subsection 1.3, even the structure of infinitesimal algebras of very simple generalized translation operators may be quite complicated. This explains the fact that the number of established general properties of generators of arbitrary generalized translation operators is relatively small. In what follows, we establish analogs of the first direct and converse Lie theorems on the construction of a system of differential equations for a given group translation and on the reconstruction of local Lie groups according to the indicated systems of differential equations.

By differentiating the relation of associativity

(1.7)

(Xl times with respect to rl' (X2 times with respect to r2"" , and CXn times with respect

to r n and setting r = e, we get

where the superscript of the generator denotes the variable upon which it acts. Hence,

thefunction u(t, s) = (Rsf)(t) satisfies the system of equations

This statement is an analog of the first direct Lie theorem for generalized translation

operators. Since, by assumption, 1Je (Q) coincides with the set of generalized functions

whose supports are concentrated at the point e, it follows from (1.5) that

(R(F)tu)(t,s) = (L(F)su)(t,s) (FE 1Je (Q), u(t,O)=f(t)·

Let us now establish some properties of right and left generators.

The right and left generators satisfy the following commutation relations:

(1.9)

(1.10)


Thus, by applying the operator D~a) to both sides of relation (1.7) and setting s = e, we obtain (1.9). Indeed,

By applying the operator D}a) to both sides of relation (1.7) and setting t = e, we arrive at relation (1.10), namely,

• Right and left generators (not necessarily o/the same order) commute, i.e.,

Indeed, to prove this assertion, it suffices to apply the operator D;~) to both sides of relation (1.9) and set t = e.

• It is often necessary to know whether the generators of generalized translation opera

tors are linearly independent. Since e is an interior point of the manifold Q, we have

This means that left generators are linearly independent. At the same time, we are much more interested in the algebra of infinitesimal right translations. In this case, the problem of linear independence of right generators is nontrivial.


Assume that infinitely differentiable generalized translation operators Rs (s E Q)

are reduced. Then all right generators are nondegenerate and linear independent.

Indeed,

(RJ)(e) = D;a)(Rsf)(e)!s=e = D;a)(Lef)(s)!s=e = D;a)f(s)!s=e = (D(a)f))(e) .

• It worth noting that the case considered in this assertion is rather rare. Let us now outline the procedure of reconstruction of generalized translation opera

tors for a given infinitesimal object. Suppose that right and left generators acting in the space of formal power series in variables t 1, .•• , tn are known. We solve system (1.8)

and find the function (Rsf)(t) = u(t, s). The operators Rs are regarded as natural can

didates for the role of the required generalized translation operators. As a rule, it is not

necessary to use all equations of system (1.8) for finding u (t, s). Thus, in the case of Lie groups, generators of the first order uniquely determine group translations.

This heuristic construction is made rigorous by the following theorem:

Theorem 1.3. Assume that there exists a finite family of operators Ra , La ((X = (al, ... ,an)E r, aiE NU{O}, i= l,n) acting in the space offormalpowerseries

in variables t l' ... , t n and satisfying the following conditions:

(ii) (Laf)(O) = (Raf)(O) = (da)f)(O) (aE r)foranyformalpowerseriesf(t);

(iii) the system of equations

with the initial condition

u(t,O) = f(t) (1.12)

is uniquely solvable in the class of formal power series.

Then the operators Rs (s = (SI' ... ,sn)) defined in the space offormal power se

ries by the formula (Rsf)( t) = u (t' s) are generalized translation operators.

Proof. By definition, R 0 = id. Moreover, if u (t, s) is a solution of system (1.11)(1.12), then by virtue of (ii),


It remains to prove that Rs satisfy the condition of associativity. Let us first show that,

for any formal power series f (t), we have

(1.13)

Denote by Fa(t, s) and Ga(t, s) the right-hand and left-hand sides of (1.13), respectively. It is clear that Fa (t, s) satisfies the system of equations

and

Let us show that Ga (t, s) is a solution of the same system of equations with the same initial condition. (By virtue of condition (iii), this would mean that relation (1.13) is satisfied.) Indeed,

Further,

Let us now prove that the operators Rs are associative. Denote

The function <P satisfies the system of equations

(1.14)

with the initial condition <p(t, s, 0) = (RsIXt). The function 'JI(t, s, r) also satisfies

this initial condition. By using (1.13), we obtain


i.e., the function \f(t, s, r) also satisfies system (1.14) and, hence, by virtue of condition (iii) ofthe theorem, is equal to «P (t, s, r).

• Remark. The assertion of Theorem 1.3 remains true if, in its formulation, the space

of formal power series is replaced by the space Coo (U), where U is a domain in 1R n

which contains the point 0, and we require that the solution u(t, s) of system (1.11)

satisfying the initial condition (1.12) is an element of the space COO(U xU).

The statement of Theorem 1.3 belongs to Levitan and is an analog of the first converse Lie theorem. In a series of works (see the bibliography in [LevlO]), he proved analogs of the second and third Lie theorems (direct and converse) for generalized translation operators whose first-order or second-order generators are degenerate and form a

basis of a certain Lie algebra (for generators of the first order, the manifold Q is a local Lie group and the relevant generalized translation operators coincide with translations on this group). These results are contained in well-known monographs by Levitan ([Lev9] and [LevlO]) and, therefore, not presented here. We only note that Grabovskaya and S. Krein ([GrK1]-[GrK3]) gave an explicit description of the generators of generalized translation operators studied by Levitan as second-order integro-differential operators. Grabovskaya, Kononenko, and Osipov [GKO] constructed a representation of the Lie algebra in integro-differential operators of any order. These operators are then used to reconstruct generalized translation operators. As becomes clear in the sequel, in many well-known examples of generalized translation operators, generators satisfy much more general commutation relations than in the case of Lie algebras.

1.5. Algebraic Approach to the Infinitesimal Theory of Formal Generalized Translation Operators. In the present subsection, we describe another approach to the construction of (formal) generalized translation operators in terms of a given infinitesi-

mal object 1Je (Q). As earlier, we assume that 1Je CQ) is formed by functionals whose

supports are concentrated at the point e. As an infinitesimal object, we consider the

space a:: [XI' ... ,x,J of polynomials in variables XI, ... ,xn with associative multipli

cation (which does not necessarily coincide with the ordinary multiplication of polynom

ials) and the right identity £ == 1. As a global object, we take the space a:: [[ X I' ... , x,J] of formal power series in the variables X I' ... , X n with comultiplication ~ defined by

duality in terms of multiplication in a:: [XI' ... , xnl (in Subsection l.2, we demonstrated

that comultiplication generates generalized translations). Note that the Laplace transformation

maps the space 1J e (Q) into the space of polynomials, a:: [x I' ... , x,J and, hence, the


definition of infinitesimal objects introduced in Subsection 1.3 does not contradict the outlined procedure.

Thus, let CC [[x]] = CC [[xl' ... , x,J] be the space of formal power series and CC [x] = CC [Xl' •.. ,x,J be the dual space of polynomials in the variables Xl, ... ,xn. We have the

following analog of Theorem 1.1:

Theorem 1.4. The space CC [x] is an algebra with respect to a certain associative multiplication (which does not necessarily coincide with the ordinary multiplication of polynomials) and the polynomial Po(x) = 1 is the right identity of this multiplication

if and only if, in CC [[x]], there exists afamity of generalized translation operators

Rs: CC[x]] --7 CC[[x,y]] = CC [[x]] ® CC[[y]]

such that the point 0 is the identity element of these operators. The generalized

translation operators Rs and the corresponding operation of multiplication in CC [x]

are connected by the formula

( 1.15)

Proof. Necessity. Associative multiplication in CC [x] induces coassociative co

multiplication

~: CC [[x]] --7 CC [[x]] ® CC [[y]] = CC [[x, y]].

Consider a formal series (~f)(x, y) (fE CC [[x]]) and assume that the variables y = (y l' ... , y J take numerical values y = (y?, ... , y~). As a result, we arrive at a formal

series (~f)(x,yo) E CC[[x]]). We set

It is clear that the operator Ryo is linear and acts in CC [ [x]]. Let us show that the opera

tors Ry (y E CC n) form a family of generalized translation operators. It is not difficult to

see that the axiom of associativity is equivalent to the coassociativity of comultiplication, namely,

(id ® ~)~f = (~® id)~f (fE CC [[x]])

Let us show that R 0 = id. We have


(F, Rof) = (F, (i1f)(x, 0) = (F, (F, (i1f)(x, y)) = (F ® 1, i1f) = (F * 1,f) = (F, f)

(fE [[[x]]; FE [[X]).

The proof of sufficiency is evident and we leave it to the reader.

• Let [[x] 3 F H 1 * F be a projection which maps [[x] onto a sub algebra [[x].

As in Subsection 1.3, the subalgebra [[x] is regarded as an infinitesimal algebra of reduced generalized translation operators.

We fix a basis

in [[[x]], where u=(Uj, ... ,un ) isamultiindex,

In this basis, the operation of comultiplication can be written as

The condition R 0 = id means that cgo = o~. According to Theorem 1.4, the corresponding operation of multiplication in [[x] in

the dual basis ea has the form e~ * eY = cgYea. It is clear that the ordinary multiplica

tion of polynomials in [[x] corresponds to additive translations in [[ [x ]], namely,

We denote

and define right generators R Y by the formula

The correspondence [[x] 3 eY H Ry is a representation of the algebra [[xl


Indeed, the co associativity of comultiplication in (C [[x]] is equivalent to the equality

IIR aa call caR C,-I-' C = .. I-' Y ~ Y ~.

On the other hand,

and

Thus, generators satisfy the same relations as elements of the basis e a but some right generators may be degenerate.

• In Example 1, the indicated procedure leads to universal enveloping algebras U ( g)

of Lie algebras. By the PBW theorem, U(g) is isomorphic to the algebra (C [XI' ... ,

xn] with the following relations:

k X·X· = X·X· + c"Xk

I J J I IJ '

where cj are structural constants of the Lie algebra g. This enables us to introduce the

operation of group multiplication in (C [ [X I' ... , X n] ]. In the general case,.in order to construct generalized translation operators for a fixed

set of relations

where Y E r is a set of indices, ii E T m , and rn is a free tensor algebra with m generators, it suffices to establish an isomorphism of linear spaces

under which Xy is mapped into e Y (i.e., to prove an analog of the PBW theorem). Here,

I is an ideal in Tm generated by the system of relations J). If the required isomorphism

't is established, then the space (C [x I' ... , x n] inherits the structure of an associative al

gebra with right identity 1 and, by virtue of Theorem 1.4, a family of generalized trans

lation operators Rs whose right generators Ry (y E r) satisfy the relations J is defined

in the space (C [[X I' ... , X n]]. In the general case, if the indicated isomorphism exists,


the procedure of its construction is not exactly specified and remains somewhat arbitrary. To eliminate this situation, we introduce left generators La. of generalized translation operators by setting

where

It is easy to see that the correspondence ([ [x] 3 e a. 1--7 La. is an antirepresentation of

the algebra ([ [x] and the algebra generated by left generators is antiisomorphic to

([ [x].

If we now establish an isomorphism 't of the linear spaces ym I I and ([ [x I' ... , x n]'

where I is an ideal in the free tensor algebra T m generated by the relations

then, by setting

't(X)'t(Y) = 't(XY) (X, Y E Tm I I),

we equip ([ [XI'"'' x n] with the structure of an algebra with identity 1. Left genera

tors of the relevant generalized translation operators R s in ([ [[ x I' .. , , x,J] satisfy the

relations J and the indicated generalized translation operators are determined to within an isomorphism.

Possible generalizations of the PBW theorem for algebras with quadratic relations are discussed in [Ver2]. For some classes of relations, PBW-type theorems are proved in the next section.

A useful method for proving PBW-type theorems was proposed by Bergman [Berg].

1.6. Some Facts from the Theory of Topological Vector Spaces. Let '£ be a local

ly convex space and let '£' be its dual space.

The weak topology 0'( '£', '£) in '£' is defined by a family of seminorms

Similarly, the weak topology in '£ is defined by seminorms

'£3 <P 1--7 I(F, <p)1 (F E '£').


The Mackey topology 't ('E, 'E') in 'E is the topology of uniform convergence on cr ('E', 'E) - compact convex sets in 'E'. Similarly, we define the Mackey topology 't ('E', 'E) in 'E'.

The Mackey-Arens theorem states that any locally convex topology 't in 'E admits

the space 'E' as a space of 't-continuous linear functionals if and only if it is not weaker

than cr ('E, 'E') and not stronger than 't ('E, 'E').

The strong topology ~ ('E', 'E) in the space 'E' is defined as the topology of uni

form convergence on bounded subsets of 'E. The strong topology ~ ('E', 'E) is, generally speaking, stronger than the Mackey topology.

A space 'E is called reflexive if the space 'E", ~ ('E', 'E)-dual to the space 'E'

and equipped with the topology ~ ('E", 'E'), is topologically isomorphic to the original

space 'E.

A space 'E is called a Mackey space if its topology coincides with 't ('E, 'E'). A complete metrizable locally convex space is called a Frechet space. Any Frechet space is a Mackey space.

We are interested in the space Coo (Q) of infinitely differentiable functions defined

on a smooth manifold Q and equipped with the topology of uniform convergence with

all their derivatives on compact subsets of Q. The space r.D(Q) dual to COO(Q) con

sists of generalized functions with compact supports. The space Coo (Q) is a reflexive

Frechet space. Hence, the Mackey topology on r.D( Q) coincides with the strong topo-

logy ~ ( r.D( Q), COO (Q»). In what follows, we always assume that r.D( Q) is equipped with the Mackey topology, unless otherwise stated.

Let 'E and :r be locally convex spaces. In the algebraic tensor product 'E0 :r, we consider the strongest locally convex topology in which the canonical bilinear mapping

is continuous. This topology is called projective.

The completion of 'E0 :r in the projective topology is denoted by 'E ® :r. There

exists a one-to-one correspondence between linear continuous mappings from 'E ® :r into an arbitrary locally convex space (j and continuous bilinear mappings from 'E x :r into q (see [Gro)): every linear mapping 'E ® :r ---7 q is associated with its composition with the canonical mapping

Moreover,


and the dual space (COO (Q) ® Coo (Q))' equipped with the projective topology coin

cides with 'iJ(Q) ® 'iJ(Q) (see [GroD.

If f£, :J, and (j are Frechet spaces and a is a separately continuous (in the weak

topology) bilinear mapping from the space '£,' x if' into the space (j' equipped with

the weak topology, then a is a strongly continuous mapping from '£,' x :r into (j' equipped with the topology ~ ( (j', (j).

Section 2 Analog of Lie Theory for Classes of Generalized Translation Operators 341

2. Analog of Lie Theory for Some Classes of Generalized Translation Operators

As shown in Section 1, the space of polynomials (C [Xl' .,. ,XJ endowed with the structure of an arbitrary associative algebra with right identity 1 can be regarded as an

infinitesimal object of a family of formal generalized translation operators in (C [[X I' ... ,

X nJ]. The same picture is observed in the case of duality <Coo (Q), 'iJ(Q ). This means that the problem of construction of analogs of Lie theory for arbitrary families of generalized translation operators is too general. It is thus interesting to select some classes of generalized translation operators for which it is possible to construct meaningful analogs of Lie theory. In the present section, we consider three classes of generalized translation operators of this sort, namely, the Delsarte generalized translation operators, the Delsarte-type generalized translation operators, and generalized translation operators associ

ated with the hypergroup G / / H.

2.1. Infinitesimal Object for the Delsarte Generalized Translation Operators. Let G be a real or complex Lie group, let ~ be its Lie algebra, and let U( g) be the univer

sal enveloping algebra of the algebra g. We consider a compact subgroup r C Aut G of the group of automorphisms of the group G and denote by d Y the normalized Haar

measure of the group r. Recall that, for all f E Coo (Q), the Delsarte generalized translation operators are defined by the equality

(Rg2 )(gI) = J f(gI y(g2)) dy (gI' g2 E G). r

Since any continuous automorphism y E r is analytic, its differential dYe in the iden

tity element of the group G is an automorphism of the algebra g. It is easy to see that the mapping

is a representation of the group r.

This assertion immediately follows from the definition of the differential

• In what follows, for simplicity, we denote the action dYe (YE r) of the group r

342 Elements of Lie Theory for Generalized Translation Operators Chapter 3

on g by the same letter, i.e.,

y(X) = dYe (X) (X E g, Y E [').

Let XI,"" Xn E g be a basis in g. By the Poincare-Birkhoff-Witt theorem, the monomials

Xu = Xu, XU n I ... n

form a basis in U (g). We extend the action of the group [' to U (g) by setting

and extending this operation to the entire U ( g) by linearity.

The mapping

n(y): U(g) -) U(g)

introduced above preserves the canonical filtration in U (g) and is an automorphism

of U(g).

This statement is proved by induction on the degree of filtration of elements from

U (g). For n = 0, 1, the statement is evident. Suppose that the statement is true for all

elements of U (g) whose degree of filtration does not exceed m. It suffices to prove that

for all Xi E g (i = 1, ... , n). For the sake of definiteness, we assume that i = n. Then,

by the induction hypothesis, we obtain


• By using this assertion, one can easily show that the definition of the action of the

group I on U ( g) is independent of the choice of a basis in g. It is also clear that the mapping

I3Y H n(Y)E AutU(g)

is a representation of the group 1. Consider a linear operator S: U(g) ---j U(g) defined by the formula

S(Y) = f n(y) Y dy (Y E U(g)). (2.1)

r

Since the representation n (y) preserves the canonical filtration (U m (g))m <:0 (recall

that the space Um(g) is a linear span of all products Xl '" X p' where Xl''''' XpE 9

and p ~ m, and each space U m (g) is finite-dimensional), the integral in (2.1) is the ordinary integral of the vector function

which takes values in the finite-dimensional vector space U m (g), where m is the de

gree of filtration of elements Y E U m (g). A vector Y E U m (g) is called I-invariant

if n(y)Y = Y for all yE 1.

Lemma 2.1. (i) S2 = S, i.e., S is a projector in U(g).

Oi) the subspace U(g) r = S ( U (g)) is a subalgebra of the algebra U (g) and

coincides with the set of all I-invariant vectors.


Proof. We prove (i). For any Y E U(g), in of view of the invariance of the Haar measure, we conclude that

r r r

To prove (ii), we first show that

S(Y)S(Z) = S(S(Y)Z) = S(YS(Z») (Y,ZE U(g»). (2.2)

Indeed,

S(Y)S(Z) = f f n(y!) Y dy!n(Y2)ZdY2

r r

= f f n(Y2y!)Ydy!n(Y2)ZdY2 r r

(Y, Z E U(g»).

The second equality in (2.2) can be proved similarly. It follows from (2.2) that U(g) r is a sub algebra of U(g).

Let us show that U(g)r consists of all r-invariant vectors. It is clear that all r-in

variant vectors are contained in U(g)r'

Conversely, let Y E U(g)r' Then SY = Y and

n(y)Y = n(y)SY = f n(y)n(y!)Ydy! = f n(y!)Ydy! = Y.

r r

• Let V:(G/r) be an infinitesimal algebra of reduced generalized translation opera

tors.

Theorem 2.1. The algebras U(g) rand V:(G/r) are isomorphic.

Proof. In Section 1, it was shown that the algebra V: (G) generated by all group generators


is isomorphic to the algebra 'De (G) of distributions whose supports are concentrated in

the identity of the group. By the Schwartz theorem, this algebra is isomorphic to U(g)

and, consequently, 'D:(G) is isomorphic to U(g).

Let 1<:: U(g) ~ 'D:(G) be the corresponding isomorphism. We define the action

of r on 'D: (G) by setting

Let us show that, for all f E Coo (G),

(2.3)

Indeed, let HI'"'' Hn be first order generators of the group G with respect to a

fixed coordinate system in a neighborhood of the identity of the group G. Then

X - -I H X - K- I H I- K I"'" n- n

is a basis in g. It is well-known that every generator Hu can be represented in the form of a polynomial P u in generators of the first order, i.e., Hu = P u (HI' '" , Hn). Therefore,

By using (2.3), for all a = (ai' ... ,an), we can write

f (rc(y)Hu)f(g)dy = f (Hu(f 0 y))( y-I (g)) dy r r

= J DhU ) f 0 y( y-I (g)h )Ih=e dy r


= J DhU)f(gy(h»)lh=e dy = (Rr:J)(g) (g, h E G, fE Coo(G), ['

where Ra is a right generator of a Delsarte generalized translation operator, whence it

follows that KSPa(X I , ... ,X,J=Ra for all a=(al,'" ,an), Moreover, since the

polynomials P a(X I, ... , Xn) form a basis in U(g), the homomorphism K maps the al

gebra U(g)[' onto the entire algebra rJ):(G/r). Since K is an isomorphism between

U(g) and rJ):(G), we have

KerK ~ U(g)[' = 0

and K is an isomorphism between U(g)[' and rJ)eR(G).

• Denote by «XI'" Xp» the symmetrization of the monomial XI ... Xp, namely,

Let S(g) be a symmetric algebra of the vector space g, let (Sm(q»)m~O be its

canonical filtration (Sm(g) is a linear span of all products of the form XI ... Xl" where

p'5,m and XI ... XmE g), andlet

be the canonical bijection of S(g) onto U(g). We define the action of r on S(g) in a natural way, i.e.,

Lemma 2.2. The canonical bijection co: S(g) -7 U(g) is an isomorphism of rmodules, i.e., CO 0 n(y) = n(y) 0 CO for all yE r.

Proof. It is necessary to prove the equality


We proceed by induction on the canonical filtration in V(g). For elements of VI (g), the assertion of the lemma is obvious. To perform a step of induction, we apply the equality (see, e.g., [Dix3], p. 120),

(2.4)

where hi E a:: , XI"" ,Xn E g, Xh means that the relevant element Xh Egis omitted.

By applying 1t (y) to relation (2.4) and using the induction hypothesis, we obtain

- q L (( [1t(Y)X, 1t(y)Xh]

1! h

1\

X 1t(y)XI ... 1t(y)Xh •.. 1t(y)Xn))

1\ 1\

X 1t(y)Xl ... 1t(y)Xh ... 1t(y)X/ ... 1t(y)Xn» -.... On the other hand by virtue of (2.4), we have

1t(y)X((1t(y)X1 ···1t(y)Xn»

= ((1t(y)X1t(y)X1 ···1t(y)Xn))

hI 1\ - - L (([1t (y)X, 1t(y)Xh] 1t(y) XI ... 1t(y)Xh ... 1t(y)Xn» + ... , I! h

and, hence, we arrive at the required equality

•


Theorem 2.2. Assume that r is a reductive algebraic group (i.e., a subgroup of

GL(n, ([) every linear representation of which is completely reducible; in particular, finite groups and compact Lie groups are reductive}. Then the algebra U(g) r is fi-

nitely generated, i.e., has finitely many generators. If r is finite, then the algebra U(g)r is generated by the elements

fa = I~I L 1t(y)((Xa» r er

where 1 a 1 ~ 1 r I·

Proof. Let Sr be a ring of r-invariants of the algebra S(g), i.e.,

Sr = {ZE S(g) 11t(y)Z=Z}.

By Lemma 2.2, U(g)r = ro(S r)' Since r is reductive, the ring of invariants S r of the

commutative algebra S(g) is finitely generated (see, e.g., [Kra]).

We fix a set of generating elements 11"", 1m of S r and show that the elements

generate U(g)r'

Since (J) preserves filtration, i.e., ro(Sp(g» = UP(g), for any element

we have

where

and the degree of filtration of the element WE U(g) is less than p.

Let Y E U(g)r' Since Sr is finitely generated, we obtain

where P is a polynomial in the variables 11"", 1m' In view of the property of ro


proved above, we have

and the degree of filtration of the element V E U(g)r is lower than the degree of filtra

tion of Y. The fact that U(g)r is finitely generated can now be proved by induction on

the degree of filtration of U(g) and by using the fact that S r is finitely generated.

If r is a finite group, then the Noether theorem (see, e.g., [KraD implies that the

invariants generating the ring S r have the form

where \ a \ ::; \ r \. This yields the last assertion of the theorem.

• Let .9l be an associative algebra over the field a:: with identity and finitely many

generators x I' ... , X m' For any n, we denote

Let An be the linear space spanned by all elements of the form XI'" xp (xl' ... ,XpE

.9l1; p::; n). Then Ao C A Ie... is a filtration in .9l. Denote .9l (k) = .9lk / .9lk_ 1 •

The algebra .9l is called a Poincare-Birkhoff-Witt algebra (PBW-algebra) if

dim .9l(n) = en n+m-l (n = 0, 1, ... )

for any set of generators xl"'" xm. The algebra .9l is called a PBW-type algebra if,

for any set of generators xl" .. ,Xm, we have

d· liI(n) en Im.l1. < n+m-l (n = 0, 1, ... )

(see [Ver2]).

Theorem 2.3. Let .9l be a finitely generated subalgebra of a universal enveloping

algebra U(g) of a Lie algebra g. Then .9l is a PBW-type algebra.

Proof. Let Zl"" ,Zm be generators of the algebra .9l. Let us show that symmetric .


polynomials in Z" ... , Zm are linear generators of the algebra .9L (it is clear that the assertion of Theorem 2.3 immediately follows from this fact). Each monomial

inherits the filtration from U(g). The PBW theorem for U(g) implies that the degree of filtration of the element

1 '" Zi ... Zi - p' L. w(l) w(p)

• CJ) E S(p)

in U(g) is lower than the degree of filtration of the element <1>. The proof is completed by induction on the canonical filtration of U(g).

• Corollary 2.1. The algebra U(g) r is an algebra of PBW type.

In what follows, we study the simplest case of algebras U(g) r with I r 1= 2. Let

r = {id, y}, where y is an involutive automorphism of the group G. The Delsarte

generalized translation operators associated with this group are called ~rDelsarte generalized translation operators.

We reduce the representation

r 3 Y H dYe E Aut g

of the group r in the Lie algebra g, i.e., choose a basis Xl"" ,Xm, Xm + J, ••• ,Xn of

the algebra g such that y(Xk) = -Xk for 15,k5,m and y(Xj)=Xj for m<j5,n.

The existence of a basis of this sort follows from the fact that any representation of a fi

nite group is completely reducible and the commutative group r has only two one-dimensional irreducible representations.

Let

and

Since Ye E Aut g, g, is a sub algebra of the Lie algebra g and, moreover, [ go'

g 1 ] c go and [go' go] c g,. In terms of the structural constants ct of the Lie algebra


g, this means that cij = 0 for i, j > m and s ~ m, Cfk = 0 for t, i > m and 1 ~ k ~ m,

and Ckl = 0 for 1 ~ k, I, t ~ m. It is clear that

s ~ gl = id

and

By virtue of Theorem 2.2, the Lie algebra U(g) r is generated by the elements Zi = Xi

with i > m and

where l~k, l~m. The generators Z/ (m<i~n) and Zkl (l~k,l~m) are called

canonical. By direct calculations, we show that the canonical generators satisfy the following re

lations:

(2.5)

1 . 1 . 1 . + -CkIIZZ + _c l Zkl7. - -Cll ZZk 2 Ipq 2 Pq LJj 2 PI q

where summation is carried out over the repeating indices and [Zj." ZIl] denotes the

commutator [Zj.,' ZIl] = Zj., ZIl - ZIl Zj.,' By applying the last relation in (2.5) several

times and using the fact that Zkl = Zlk' we can obtain an expression for the commutator

[Zkb~]'


Thus, for I r 1= 2, U(g)r is an algebra with quadratic relations (for I r I> 2, one

can construct examples of relations whose order is higher than two). The total number of different relations in (2.5) is given by the formula

Pnm 1

= 5 C4 + 9 Cf + 4 CT + - (n - m)( m 2 + n - 1 ) 2

1 = Pmm + -(n-m)(m 2 +n-l).

2

Generally speaking, P nm is greater than the total number of relations between the basis

vectors in a Lie algebra of dimensionality

m(m -1) n+

2

(the last number is, in fact, the number of linearly independent canonical generators).

For m = 1, relations (2.5) between the canonical generators are relations of the Lie type and have the form

Thus, the elements Zl1, Z2' ... ,Zn form a basis of a new Lie algebra g. For m;:::: 2, the collection of relations (2.5) always contains non-Lie relations.

One can easily see that relations (2.5) are invariant under nondegenerate linear transformations of the form

where

det (a/ )~. :;t: 0 I]=m+!

and det (hkt)m :;t: O. kt =!

Each transformation of canonical generators of this type is associated with a nondegenerate linear transformation of the Lie algebra g. This transformation can be written in the matrix form as

where


B - (bt)m - k kt=1

and A = (ai)n . I j,j=m+1

Namely, if

is a new basis in g, then

and

Zj = Xj (l5::k, l5::m; m< i5::n).

Since the matrix of the automorphism 'Y is diagonal in the basis Xr (1 5:: r 5:: n), the generators Zj (m m) and Zkl (k, l5:: m) of the algebra U(g)r

satisfying relations (2.5) is called the infinitesimal object of ~ 2-Delsarte generalized translation operators.

Let g and 9 be Lie algebras of the Lie groups G and G, respectively, and let

r = (id, 'Y) and r = (id, y) be the subgroups of Aut G and Aut G determining the

~2-Delsarte generalized translation operators Rg (g E G) and Rg (8 E G). Two

families of ~TDelsarte generalized translation operators Rg (g E G) and Rg (8 E

G) are called isomorphic if there exists a diffeomorphism <p: G ~ G such that

<pee) = e, <p(g-I) = <p(g)-I,

and

for any g, gl' g2 E G and fE COO(G) (cf. the definition in Section 4 of Chapter 1). Denote by the differential of the mapping <p at the identity element e of the

group G. Since <p is a diffeomorphism, we have dim g = dim g. For any function f E

C"'(G), we can write


By setting gl = e in (2.7), we get <P(y(g2») = Y(<P(g2»)' This yields the relation

<P 0 Y = yo <P, where yand yare the actions of rand r in g and g, respectively.

Relation (2.7) also implies that, for any g l' g 2 E G, one of the following relations is true:

Let Ge be the connected component of G which contains the identity.

Denote

H = {hE Gel'Y(h)=h}.

Clearly, H is a closed subgroup of the group G. Hence, H is a topological Lie sub

group of the group G. Let gl be its Lie algebra. Relations (a) and (b) coincide on H.

For any h e H, we denote

and

Bh = {gE Gel <p(gh)=<p(g)<p(y(h»)}.

It is clear that Ah and Bh are closed subsets ofthe connected group Ge and, more

over, AhU B h= Ge and Ahn B h= 0. Since e E Ah and A h* 0, we have B h = 0.

Consequently, relation (a) holds for all hE Ge, which means that m) and

y Xk = - Xk (k::; m). Since 0 Y = Y 0 <1>, we conclude that the basis

in g possesses the same properties relative to the action of the group f. Note that the Lie algebras g and g are isomorphic and, therefore, their structural

- . . k -k ( .. k 1 constants in the bases Xi and Xi (i = 1, ... , n), coincIde, 1.e., cij = Cij l,}, = , ... , n). Hence, the relations between the canonical generators

and


have the form (2.5), i.e., the infinitesimal objects of isomorphic ~2-Delsarte generalized translation operators coincide.

The ~2-generalized translation operators Rg (g E G) and Rg (g E G) are called

locally isomorphic if there exists an isomorphism <1>: g --t g between Lie algebras of

the groups G and G such that 0 Y = Y 0 <1>, where y and y are the actions of the

group ~2 in g and g, respectively. The following theorem demonstrates that relations (2.5) are sufficient for the recon

struction of ~2-Delsarte generalized translation operators:

Theorem 2.4. Let Jl. be a finitely generated associative algebra whose generators

J1, ... ,Jp are canonical generators for some ~ 2-Delsarte generalized translation

operators and satisfy the canonical relations. If the quadratic part of at least one of

these relations differs from the commutator, then the ~TDelsarte generalized translation operators can be uniquely (to within a local isomorphism) reconstructed for given Jl. in a certain neighborhood of the identity of the group G. Otherwise, unique reconstruction is, generally speaking, impossible.

The proof of this theorem is quite cumbersome and we do not present it here.

Denote by V the linear span of the elements Zj (i> m) and 41 (k, I ~ m; we as

sume that Zkl = Zlk)' Consider the tensor algebra T of the vector space V. Recall that

o I T=T$T$ ... ,

where TO = ([: (or 1R. if g is real) and

Tn = V ® ... ® V (n factors).

Multiplication in T is defined as ordinary tensor multiplication. Let J be a two-sided ideal of the algebra T generated by tensors of the form


(2.8)

The associative algebra U (V) = T / J is called an algebra enveloping relations (2.5).

The composition (j of the canonical mappings V -7 T -7 U(V) is called the ca

nonical mapping of V into U ( V).

Lemma 2.3. Let (j be the canonical mapping of V into U (V) and let 't: V -7 A

be a linear mapping from V into an associative algebra A with identity such that relations (2.5) hold for the elements

Then there exists a unique homomorphism 't': U (V) -7 A such that 't' ( 1) = 1 and 't' 0 (j = 'to

Proof. The mapping 't' is unique because U (V) is generated by the identity I, (j(Z). and (j(Zkl) (i>m; k, l'5,m).

Denote by <P the homomorphism from the algebra T into A which extends 't and

is such that <P ( I ) = I. Since the elements

satisfy relations (2.5), we have <p(J) = 0 and, hence, under factorization with respect to

the ideal J, the homomorphism <p defines a homomorphism 't' from the algebra U ( V)

into A such that 't' (1) = 1 and 't 0 (j = 'to

• Let

We introduce partial ordering in the set of indices m) by setting (k, l) < (p, q) if max (k, l) '5, min (p, q), (k, /) m) and con

sider the set {m + 1, ... , n} with natural ordering. With every finite sequence of

indices 1= (<PI' ... , <pp) (each index <P A. is either (k A' I,.) or i A)' we associate an el

ement


of the algebra U (V). A sequence 1= (<1'1' .•• ,<I'~ is called increasing whenever <1'1 <

<1'2 < ... < <l'p'

Theorem 2.5. Elements of the form Y[' where I is a finite increasing sequence,

form a basis in the vector space U ( V).

Theorem 2.5 can be proved by using the Bergman diamond lemma (see [Berg]). According to Theorem 2.5, one can construct a homomorphism (0: U(V) ~ U(g)r

by setting

for any sequence 1= (<1'1' ..• , <I'~.

Since elements of the form Y/, where I is an increasing sequence of indices, form a

basis of the algebra U (V), elements of the form Z[, where I is an increasing sequence

of indices, are linear generators of U(g) r' Each element of this sort can be written as

(2.9)

where

1 <k k k< 1<'< <.< - 1 < 2 < ... < p - m, m + - 11 - ... - lq - n,

U j = 0,1, ... , OJ,j+1 E {O, 1}.

By expressing the generators Zkl and .t; ofthe algebra U(g)r in terms of the elements

Xj of a basis of the Lie algebra g, we conclude that each element Z[ of the form (2.9) admits a representation

(2.10)

where the degree of filtration of the element W is lower than that of Z/.

By the PBW theorem, Z/ are linearly independent. Hence, the elements Z/ E U(g)r

given by (2.9)form a basis in U(g)r and the algebra U(g)r is isomorphic to the al

gebra U(V) which is an algebra enveloping relations (2.5). This means that all rela

tionsin U(g)r follow from relations (2.5).

Example. Let 9 = sl2 (IR). In a basis


x = (~ y = (~ K = (0 -1

(2.11)

we define the action of an involutive automorphism cr which determines the Cartan de

composition of s12(lR), Le., cr(X) = -X, cr(Y) = -Y, and cr(K) = K, and consider the

algebra U(g) r with r = {id, cr}.

The generators of this algebra

2 Zl1 = X ,

1 Z12 = -(XY+ YX),

2

satisfy the relations which readily follow from (2.5) after evident transformations. Indeed,

Remark. The infinitesimal object of Delsarte generalized translation operators can

easily be described in the case where r consists of inner automorphisms of the group

G. If r is isomorphic to a Lie subgroup H of G, then, clearly, U(g)r coincides with

the centralizer of the Lie algebra f) of the group H in U(g).

In the case where r is the group of all inner automorphisms of a compact Lie group

G, the algebra U(g)r (in this case, it coincides with the center Z(g) of the algebra

U(g» was described by Gelfand [GeI2]. Moreover, Duflo proved that the algebra Z(g)

is isomorphic to the subalgebra S(g)G which consists of Ginvariant elements of the

symmetric algebra S(g) (note that, generally speaking, this isomorphism does not coincide with symmetrization because the latter does not preserve the operation of

multiplication). If G is a complex semisimple Lie group, then, by virtue of the

Chevalley theorem, there exists a one-to-one correspondence between Z(g) and the algebra of all polynomials over the Cartan subalgebra invariant with respect to the Weyl

group. In this case, Z(g) has 1 independent generators, where 1 = rank g.


2.2. Delsarte-Type Generalized Translation Operators and Generalized Lie Algebras. In the present subsection, we describe the infinitesimal object of Delsarte-type generalized translation operators introduced by Gurevich [Gur2].

Let G be a connected simply connected Lie group with Lie algebra g. In this case, the semigroup End G of endomorphisms of the group G is isomorphic to the semigroup End g of endomorphisms of the algebra g and is an algebraic submanifold of the

semigroup End 1R n (End cr: n).

Let r C End G be the semigroup (with identity) of endomorphisms of the group G

such that r is a closed smooth (analytic) submanifold of the manifold End G. General-

ized translation operators Rh (h E G) acting in the space CO(G) are called Delsarte

type generalized translation operators if Re = I and

(RJ)(g) = (f(A(g)B(h»), IlA B) (g, h E G; fE COO(G»), (2.12)

where IlA,B E 1J(r x r). If r C Aut G is a compact group and IlA,B =0 E ® mp then,

clearly, Rh are Delsarte generalized translation operators; here, E is the identity of the

group rand m r is its Haar measure.

The operators Rs defined by relation (2.12) are not necessarily generalized transla

tion operators. In what follows, we present a sufficient condition for Rs to be generalized translations.

We say that the space cP = COO(r) is endowed with algebraic structure if is an

associative algebra with identity 1 (A) == 1 (A E r) with respect to an operation of associative continuous multiplication

 x 3 (<p, 'II) H 

invariant under left translations in the semigroup r, i.e.,

Theorem 2.6. Assume that the space = Coo (r) is endowed with algebraic struc

ture. Consider a functional IlA,B defined in the space COO (r) ® COO (r) by the for

mula

(2.13)

Then the operators Rs (s E G) defined by relations (2.12), where II is the I""A,B

functional introduced in (2.13), form a family of generalized translation operators.


Proof. Since 1 (A) is the right identity of the operation 0, we have

(Ref) = (j(A(g) 1 (B»), j.lA,B> = U(-(g)ol))(E)=J(E(g)) =f(g) (gE G).

In view ofthe fact that the operation of multiplication 0 is left invariant in <1>, the law

of multiplication can be written in the form

(2.14)

By virtue of (2.14) and the fact that the operation of multiplication in is associative, we obtain

«<p(MA)",(MCB)8(MBD), j.lc'D>' j.lA,B>

For M = E, we get

On the other hand, it follows from (2.12) that the axiom of associativity for the operators

Rs takes the form

This relation is indeed true by virtue of (2.15). Thus, Rs are generalized translation

operators.

• It is easy to see that, for any r, the ordinary operation of multiplication of functions

on r is associated with operators of group translation. Delsarte generalized translation operators correspond to the operation of multiplication

(fog)(C) =f(C)J g(BC)dB,

r


where dB is the Haar measure on the compact group of automorphisms of r normalized by the condition

J dB = 1. r

Let us now construct an example of Delsarte-type generalized translation operators which are not Delsarte generalized translation operators. Let G = S U (2) and let X, Y,

Z be a basis in its Lie algebra su (2) such that

[X, Y] = Z, [Y, Z] = X, and [Z, X] = Y.

Consider two automorphisms A, B E Aut(su(2» given in the basis X, Y, Z by the

matrices A = diag ( - 1, 1, - 1) and B = diag ( 1, - 1, - 1). The automorphisms A and

Bare involutive and generate a commutative group

r = {E, A, B, AB =BA}.

The characters Xo == 1, Xl' X2' and X3 are determined by the relations

and form a basis in the space of functions defined on r. In <1>, we introduce an algebraic structure by the relations

X 0 0 Xi = Xi 0 X 0 = Xi (i = 0, 1, 2, 3).

The generalized translation operators associated with this algebra have the form

1 (Rsf)(t) = - [J(ts) + f(B(t)s) + f(tA(s») - f(B(t)A(s))]

2

(t, s E SU(2); fE CO(SU(2»)

By direct calculations, one can show that the first-order generators of these generalized

translation operators XI' X2 , and X3 satisfy the relations


It is clear that, by using this procedure, one can construct many examples of nontrivial Delsarte-type generalized translation operators.

Let us establish the following properties of right generators Ra of arbitrary

generalized translation operators Rs (s E G):

Lemma 2.4. (i) If (Raf)(e) = a~(D(~)f)(e) (summation is carried out over the

repeating indices; all sums are finite), then either (Raf)(e) = (D(a)f)(e) or genera

tors are linearly dependent;

(ii) if the generators Ra from (i) are linearly independent, then the relation

is equivalent to its restriction to the point e.

Proof We prove (i). Since

we have

= (~(Rsf»)(t)ls=e = a~D;~)(Rsf)(t)ls=e = L a~(R~f)(t) ~

for all t E Q. This proves (i). Let us prove (ii). It is necessary to show that the relation

implies (2.16). Indeed, by using (i), we easily obtain

(2.16)

= cY(R~Rsf)(t)ls=e = L cY D;Y)(Rsf)(t)ls=e = cY(Ryf)(t)·

Y

•


We now describe the infinitesimal object for Delsarte-type generalized translation operators. As in Subsection 2.1, let nCB) E End U(g) denote the action of an endomorphism BE r on the universal enveloping algebra U(g) of the Lie algebra 9 of the group G. The mapping

r :3 B H nCB) E End U(g)

is clearly a representation of the semigroup r. Let Xa, a = (ai' ... ,an)' be generators of right translations in the group G. By vir

tue of the Schwartz theorem, they form a basis in U(g).

Let 1t~ (B) be matrix elements of a representation 1t of the semigroup r in the ba

sis Xa. By differentiating relation (2.12) a = (aI' ... ,an) times with respect to sand

setting s = e, we obtain

(Raf)(t) = (n~(B) Xpf(A(t»), !lA,B)'

whence, in particular, it follows that the generators Ra of the Delsarte-type generalized

translation operators satisfy condition (i) of Lemma 2.4. Suppose that the generators Ra

( a E Y) are linearly independent for a collection of multiindices Y. According to Lemma 2.4 (i),

Thus,

(2.17)

These relations enable us to formulate the following procedure of finding relations for generators of Delsarte-type generalized translation operators. If it is possible to solve

364 Elements oj Lie Theory Jor Generalized Translation Operators Chapter 3

relations (2.17), i.e., to express (XyXoJ)( e) in terms of (RaR13J) (e), then, by using

the known relations for right generators of the group G, one can find relations for generators of Delsarte-type generalized translation operators at the point t = e. Lemma 2.4 (ii) enables us to omit the requirement t = e. As a result, we arrive at relations for the generators Ra.

In what follows, we restrict ourselves to the case where Y = {a II a I = I}, i.e., we assume that the first-order generators of Delsarte-type generalized translation operators

are non degenerate and linearly independent. Instead of the multi index a(k) = (0, ... , 1,

... ,0) E Y, we write the number k of the nonzero component of a(k).

Denote

(i,j, k, 1= 1, ... ,n; n=dimg).

Relation (2.17) now turns into the equality

Theorem 2.7. Assume that the first-order generators oj Delsarte-type generalized

translation operators Rk (k = 1, ... , n) are nondegenerate and linearly independent

and all matrices (di,l) are invertible.

Then the generators R1, ... , Rn are generators oj the algebra V: (G)r generated

by all Ra and, moreover,

(2.18)

where

bm dpq m ij = ij CPq ,

and C;q are structural constants oJthe Lie algebra g in the basis Xl' ... ,Xw

Proof. It is clear that the support of the convolution F * G of any functionals F,

G E VR (G)r whose supports coincide with the identity element e of the group G also coincides with the identity element (we use the same notation as in Subsection 2.1).

Thus, the generators Ra are linear generators of V: (Gk .

Let {Ra I a E 3} be a basis in V: (G)r' For any a E 3, one can indicate poly

nomials P y (Xl' ... ,Xn) such that


(Raf)(e) = «(1t&(B) Xyf)(A(t» , !lA,B)

(2.19)

Since the matrices (di,l) are nondegenerate, for any y, one can find polynomials Q; and Q y (which may be equal to zero) such that

By substituting the right-hand side of this equality for Py in (2.19) and setting t = e, by

virtue of Lemma 2.4 (i), we obtain

(2.20)

By repeating the reasoning used in Lemma 2.4 (ii), we conclude that (2.20) holds for all

t E G, i.e., Rk generates '1): (Gk . To prove (2.18), we pass in the equality

from the generators Xk of the Lie algebra g to the generators Rk • As a result, we ob

tain

(2.21)

because

( (m») (Rmf)(e) = (Xmf)(e) = Da f (e)

by virtue of Lemma 2.4 (i). By multiplying relation (2.21) by (d"ktlij ) and using Lem

ma 2.4 (ii), we arrive at relation (2.18).

• A finite-dimensional linear space V is called a generalized Lie algebra or an S

algebra if it is an algebra with multiplication C: V ® V -7 V and there exists a tensor


H: V ® V ~ V ® V which is called symmetry such that the following conditions are satisfied:

(1) H2 = id.

(2) (H 0 1)(1 ® H)(H ® 1) = (1 ® H)(H (1)( 1 ® H), i.e., H is a solution of the Yang-Baxter equation.

(3) CH=-C.

(4) C(C® 1)+ C(C® 1)(10H)(H® 1)+ C(C® 1)(H01)(10H) = O.

(5) H(C® 1) = (1 ® C)(H® 1)(1 ®H).

Theorem 2.8. The space V = I.s. {R l' ... , Rn} is a generalized Lie algebra with

multiplication C(Rj ® Rj )= btl Rm and symmetry H = (~y).

D if. I' I h H2 -'d' hPq hkl - dl L h h H' rroo. tIS c ear t at - 1 , I.e., ij pq - Uij' et us now s ow t at IS a

solution of the Yang-Baxter equation, i.e.,

To do this, we note that, by virtue of (2.14), one can write

Consider the conjugate law of multiplication

(2.22)

(this type of multiplication can be expressed in terms of the original operation of multi

plication by passing to conjugate quantities, namely, f * g = (f* 0 g*)*). Multiplication

* is associative. By using the fact that ordinary multiplication is commutative and relation (2.22), we obtain

= i j d-:-,ltsnP nk lq)1 t * S

(2.23)

Section 2 Analog oj Lie Theory Jor Classes oJGeneralized Translation Operators 367

Condition (2) from the definition of generalized Lie algebras now follows from relation (2.23) and the associativity of multiplication *. All other conditions from the definition of generalized Lie algebras admit direct verification.

• Note that the tensor H associated with a generalized Lie algebra corresponding to

Delsarte-type generalized translation operators is equivalent to the tensor

H' = (h~kl == ()!~) lJ IJ'

--"'-..

associated with an ordinary Lie algebra. Actually, in this case, the tensor H can be represented in the form

Note that finite-dimensional graded Lie algebras, including Lie superalgebras, are .. generalized Lie algebras (see [Moso]). For any generalized Lie algebra, one can con

struct a "generalized formal Lie group" ([Gur4]). It is interesting to describe generalized Lie algebras corresponding to Delsarte-type generalized translation operators.

2.3. Infinitesimal Algebra of the Hypercomplex System Ll (G, H). Let G be a Lie group and let H be its compact subgroup. In Section 2 of Chapter 2, we studied the hypercomplex system L1 (G, H). The generalized translation operators associated with this hypercomplex system have the form (see relation (2.1) in Chapter 2)

(RgJ)(g1) = J J(g1 hg2 ) dh, H

where g I, g2 E G and J E Co (G) is a biinvariant function. By differentiating this rela

tion, we arrive at the following expression for the right generator Ra of the generalized

translation operator Rg:

(RaJ)(g) = f (XaJ)(gh)dh (a == (ai' ... , an); g E G), H

where Xa is the generator of the right translations in the group G.

Since J is biinvariant, this yields

(2.24)


(Raf)(g) = J D~~)f(ghgl)lg]=edh H

= f f D~~) f(ghh1g1h1]) I g] =e dhdh] HH

= f f (Adh]X<a))f)(gh)dhdh l (g E G). (2.25)

HH

Let 9 and ~ be the Lie algebras of the groups G and H, respectively, and let

U(g) be the universal enveloping algebra of the Lie algebra 9 realized by differential

operators in COO(G). The generators Xa (u=(u1, ... ,un )) of the Lie group G form

a basis in U(g). Let

U H = {X E U(g) I AdhX=X, hE H}, [= 1.s. {~. U(g)}, and J = 1.s. {U(g) ~}.

Clearly, UH is a sub algebra of U(g) and [ and J are, respectively, the left and right

ideals of U(g). As in Subsection 2.1, one can easily show that the operator

PX = f Adh X dh (X E U(g)) H

is a projector and P U(g) = U H. Denote [H = P [ and JH =P 1.

Lemma 2.5. [H = [n UH = J n U H = JH is a two-sided ideal in UH.

Proof. Let us show that P [ c I. Consider an element X E [ of the form X = ZX1,

where Z E ~ and Xl E U(g). It is clear that AdhX = (AdhXl) ( h E H). Since

AdhZ E ~, we have AdhX E [. Consequently, Adh[ C [, whence it follows that P [ C [.

Since P[cUH, we have p[=[nuH. Similarly, we prove that PJ=JnuH. Letus

show that JH =[H. Clearly, for this purpose, it suffices to establish the equality P(ZX) =

P(XZ) (Z E ~, X E U(g)). For any Z E ~ and X E U(g), we have ZX = XZ+ adzX

and, hence, it is necessary to show that P (adzX) = 0 or, equivalently, that 1m (adz) c

Ker P for all Z E ~. Since P and adz (Z E ~) preserve the filtration in U(g), it suf

fices to prove this inclusion for their restrictions Im(adz~ Um(g» c Ker(P ~ Urn (g))

to an arbitrary layer of the filtration Um(g). Denote Pm=P ~Um(g) and ad~=adz~

Urn (g). Since Urn (q) is finite-dimensional, it can be equipped with a scalar product

( ., ·)0 and, hence, with H - invariant scalar product

(Xi'X2)H = f (AdhXI, AdhX2)odh (Xi' X2 E Um(g)). H


One can easily see that, in this scalar product, P = p* and (ad~)* = -ad~. By the

definition of the projection P, we have ImP c Ker adz for all 2 E ~. Indeed, 1m P =

UH but A~ ~ UH = id and, therefore, U H c Ker adz (2 E H). Thus, for any X E

Um(g) and E 1m ad~, we have

because 1m ad~ ...L Ker ad~.

• Theorem 2.9. The algebra V: (G, H) generated by the generators R 0; is lSO

morphic to the algebra UH / JH.

Proof. By using (2.24), we construct a linear mapping p: U(g) -t VeR(G,H), i.e.,

we associate each generator Xo; with a generator Ro; of generalized translation opera

tors Rs. By virtue of (2.25), the restriction of p to U H is a surjection. Let us show

that cr = p ~ UH is a homomorphism ofthe algebra U H onto V:(G,H). Indeed, for

any P Xo;, P X P E UH , by using the fact that the Haar measure is invariant and the func

tion f is biinvariant, we obtain

p(PXo; PXp) = f f f {(AdhlXo;)(Adh2Xp)f)(gh)dhdhl dh2 HHH

= f f f Di~) Adh2 X~f{ghhllglhl)lgl=edhldh2dh HHH

= f f D~~)D~~)f{ghglhlg2)lgl=e dhldh = Ro;Rpf(g) (g E G). H H g2=e

To prove the theorem, it is necessary to show that the kernel of the homomorphism cr

coincides with ]H. Let f(g) E COO(G) be a right-invariant function. Then the function

II (g) = (LgJ)(g) = J I(gl hg)dh H

is biinvariant for all g lEG. For = CO; Xo; E Ker cr, this yields

H H a

370 Elements of Lie Theory for Generalized Translation Operators

= f LcaD~a) f f(glh1gha)dh j la=e dh Ha H

= f f LcaD~a)f(glhlghah-I)la=edhdhl HH a

Chapter 3

= f f (Adlrl<l>f)(glhlg)dhdh j = f (f)(glhg)dh. HH H

By setting g = e, for any gl E G, we get

J <l>f(gl h)dh = 0 ( E Kera). H

Let =CaXaE UH and let fE C""(G) be right-invariant. Then

(f)(g) = LcaD~a)f(ga)la=e = fLcaD~a)f(ghh-lah)la=edh a H a

= f (Adh- I <l>f)(gh)dh = f (f)(gh)dh,

H H

(2.26)

(2.27)

i.e., the function ( f)( g) is right -invariant. If E Ker a, then, by virtue of (2.26) and (2.27), ( f)(g) = 0 for any right-invariant function. This means that E J. Thus, if

 E Ker a c UH, then E J, i.e., Ker a c JH. The inverse inclusion is obvious. Consequently, Ker a = JH.

• In exactly the same way as in Subsection 2.1, we prove that V:(G,H) is a finitely

generated PBW -type algebra.

Example. Let G = SOC 4) be a group of rotations of the space 1R 4 with an ortho

normal frame (e I' e 2, e 3' e 4) and let H = O( 2) be the complete group of rotations of

the plane (e3' e4)' We set a(k) = <5i, a(ki) = <5~ + <5} (i = 1, ... ,6; <5~ is the Kronecker

symbol), X k = Xa(k), and X ki = Xa(kl), where Xa are generators of the group SO (4) in coordinates of the second kind. Similar notation is introduced for generators of the gen

eralized translation operators corresponding to the pair (SO( 4 ), O( 2»). The generators

Rl' R22, R24, R44 are linearly independent, generate the algebra V: (G, H), and satisfy the commutation relations


(2.28)

(2.29)

Denote Z1 = R1, zl = R21 , z3 = R14 , Z4 = R44 , and V = 1.s. {ZI, zl, z3' Z4}. Relations (2.28) can now be rewritten as

(2.30)

where the tensors H = (hiJ I ) and C = (bt) are determined from (2.28). Relation (2.29)

means that

(2.31)

The linear space V endowed with the bracket [·'·]H is a generalized Lie algebra.

This algebra possesses the ideal VI = [ V, V]H' VI = 1.s. { ZI + 2z3, zl - Z4}. The alge

bra VI possesses the ideal V2 = [VI' VI]H = [ V1, V2 ]H' V1 = l.s. { Zl - Z4}.

Let V (G, H) = T / I be an algebra enveloping relations (2.30); here, T is a free tensor algebra over V and I is the ideal generated by relations (2.30). Let In be elements

of the nth tensor power in T and let Bn be the imbedding Bn: In-7 Vn( G, H). For

V (G, H), we have the following analog of the PBW theorem: The elements 1 and I3n (Zit ® ... ® liS> (s ~ 1, i l ~ ... ~ is, ik= 1, ... ,4) form a ba-

sis in V (G, H). If V I (G, H) = T / I I' where the ideal I I is generated by relations

(2.30) and (2.31), then the assertion formulated above remains true for ik E {I, 2, 3} or

ik E {I, 2, 4}. We also have

i.e., VI (G, H) is a PBW -type algebra. The algebra VI (G, H) is isomorphic to the al

gebra tJJ: (G, H).


3. Duality of Generators of One-Dimensional Compact and Discrete Hypercomplex Systems

Let G be the group of motions of a circle. The functions eint (n E ~, t E [-1t, 1t ])

are its characters; moreover, they are the eigenfunctions of the generator X of the group

G. On the other hand, the characters eint regarded as functions of n are generalized

eigenfunctions of the shift operator X in 12 (~) (see, e.g., [BerlO] or [Ber11]). In a

certain sense, the operator X can be regarded as the generator of the group dual to G

and, hence, the operators X and X can be regarded as dual to each other. A similar situation is observed for some families of orthogonal polynomials satisfying a differential equation with respect to the continuous variable and a difference equation with respect to the discrete variable. To within a coefficient, these polynomials are characters of two Hermitian hypercomplex systems which are dual to each other (see Section 3 in Chapter 1). Thus, it is natural to consider the relevant differential and difference operators as dual to each other. The existence of such duality of generators was conjectured for the fist time by Berezin and Gelfand [BeG].

The generator of a hypercomplex system can be regarded as the Fourier image of the operator of multiplication by a function uniquely defined by the characters of this hypercomplex system. In Subsection 3.1, we show that, under certain restrictions imposed on generalized translation operators associated with a compact hypercomplex system, the generator thus defined coincides with the classical generator of generalized translation operators.

Generalized translation operators associated with hypercomplex systems can be uniquely reconstructed from the corresponding generator by using a formula of the type

(R.dXt) = (X (s, X)f) established by Vainerman for generalized translation operators constructed in the Sturm-Liouville problem by the method of Levitan and Povzner (see Section 4 in Chapter 2). In this case, <pes, A) are eigenfunctions of the original SturmLiouville problem and X is the self-adjoint operator associated with this problem. In Subsection 3.2, we show that the formulas for the construction of generalized translation operators from self-adjoint operators remain applicable in more general situations. In Subsection 3.3, we establish and analyze relations between the generator of the hypercomplex system constructed in Example 2 (Section 1 in Chapter 1) and the Fourier image of the generator of the dual hypercomplex system.

3.1. Generators of One-Dimensional Compact and Discrete Hypercomplex Sys

tems. Let L] = L] ([a, b], m) be a normal commutative hypercomplex system (in a sense of Subsection 2.3 in Chapter 1), let a segment [a, b] (a, b < 00) of the real axis be its basis, let e E [a, b] be its basis unity, and let m be a multiplicative measure. We assume that the structure measure c(A, B. r) of this hypercomplex system is, generally

Section 3 Duality of Generators of Compact and Discrete Hypercomplex Systems 373

speaking, nonpositive and the hypercomplex system satisfies the axioms of quantized hypercomplex systems. For this purpose, it is sufficient that

b

f I c(A, B, r) Idm(r) ::; Cm(A)m(B) (C~ l;A, B E 1J([a, b]),

a

and that the linear space A(Q) = L2 * L2 be an algebra with respect to the ordinary

multiplication of functions (f g )(t) = f (t) g (t) dense in C (Q), where L2 = L2 ( [a, b ],

m). It follows from the theory of quantized hypercomplex systems that the dual hyper

complex system II (Q,m) is discrete; moreover, it is a commutative *-algebra (but, gen

erally speaking, not a Banach algebra). In this case, we also have analogs of the Plancherel theorem, inversion formula, and Pontryagin duality theory. Note that m(X) = II X 112"2, where X E Q is a character of the hypercomplex system L), Q is a basis of

the dual hypercomplex system, and II· 112 = II· II L:!. For simplicity, we assume that the

structure measure c (X, 'II, e) (X, 'II, e E Q) of the dual hypercomplex system is finite

in e for fixed X and 'II and I X (t) I ::; 1 (X E Q, t E [a, b]) (if the structure measure

of at least one of hypercomplex systems LI or I) (m) is nonnegative, then the last con

dition is automatically satisfied).

Assume that the characters X (t) are infinitely differentiable with respect to t and the

functionf(t,n)=x(n)(t) (XE Q,nE N) is not identically equal to zero. Suchhyper

complex systems are called one-dimensional Lie hypercomplex systems. In [SchwS], it

was proved that every hypergroup Q = [a, b] with smooth generalized translation oper

ators is Hermitian. Thus, assume that L) [a, b] is a Hermitian hypercomplex system

and that e E a [a, b]. The following statement independently established by A. Scwartz under somewhat stronger restrictions [Schw5] justifies this assumption:

Assume that L) is an Hermitian hypercomplex system and the generalized transla

tion operators Rs (s E [a, b]) associated with this hypercomplex system map C"'( [a,

b]) into COO([a, b] X [a, b]). Then e E a[a, b].

Indeed, let e E (a, b). Denote by Xk the kth-order generator of the hypercomplex

system L I :

It is easy to see that Xk is an Hermitian operator in L2 . Indeed, since the hypercomplex

system is Hermitian, we get


ci = -k J (RsfXt)g(t) dm(t)1 _ as s_e

Denote by Xk the closure of Xk . Since e E (a, b), the operator Xl is nondegenerate

On the other hand, the characters of Hermitian hypercomplex systems are real, X(t) = X (t). It follows from the inequality I X (t) I ~ 1 that X'( e) = 0 (X E Q), whence

Since the collection of characters form a basis in the space L 2 , by approximating an ar

bitrary function f E L2 by linear combinations of characters and using the fact that Xl is closed, we conclude that Xl f = O.

• Denote a# = band b# = a. Thus, for instance, if e = a, then e# = b. We enumer

ate the characters X E Q by using the set No = N U {O} so that X 0 (t) == 0 and de

note X (t, n) = XIl(t), i.e., the characters of the hypercomplex systems L1 and 11 (m)

are regarded as functions of two variables t E [a, b] and n E No. Let

N = min { kEN I X (k) ( e ) "* 0 for at least one X E Q}.

Assume that N < 00, i.e., there exists kEN such that X (k)( e)"* O. Consider the function

,,-(n)

By expanding X (t) in the Taylor series for at the point e and using the inequality

X (t) ::; X (e), we conclude that ,,-(n) is nonnegative for e = b and odd Nand nonpositive, otherwise.

A A

Suppose that (A) ,,-(n)"* "-(m) if n "* m and (B) there exists a number no E N such that X (t, no) homeomorphic ally maps [a, b] into its range.


Lemma 3.1. For condition (B) to be satisfied, it is sufficient (C) that there exist a

number no E N such that the problem

(R'bf)(r) = Kf(r), f(O) = 1 (r E No), (3.1)

where Rk (k E No) are generalized translation operators associated with the hyper

complexsystem 11(m), is uniquely solvable for all KE [-1,1].

Proof. By substituting r = 0 into (3.1), we get K = f(no) and relation (3.1) turns

into (Rflof)(r) = f(no)f(r). Thus, the character XK of the hypercomplex system

11(m) is a solution of problem (3.1). By the Pontryagin principle of duality for quantized hypercomplex systems, the multiplicative measure of the hypercomplex system dual to 11 (m) is equal to m and the characters of I) (m) lying in supp m are param~t

rized by points of the interval [a, b]. Thus, if X K E supp m, then X K (n) = X (t K' n)

and it is clear that t K (K E [- 1, 1]) runs through the entire interval [a, b]. Consider

the function \jf(t) = X (t, no). It is obvious that \jf(t) is a character of the hypercom

pIe x system L). Let us show that \jf(t) maps [a, b] onto [\jf(e#), 1] and is bijective.

Since \jf is continuous, its surjectivity is obvious and injectivity follows from (C). The

refore, \jf is a homeomorphism.

• It is easy to see that condition (C) is equivalent to the following condition: (C') there

exists a number no such that any solution of the problem (Rnof)(r) = Kf(r), f(O) = 0

(k E [- 1, 1]) is trivial. Consider the function

where A(no) *- 0 in view of condition (A), and denote by Mi,. and M A the operators

of multiplication by the functions A(n) and AU) in the spaces 12(m) and L2, re

spectively. We define self-adjoint operators X and X as follows:

(3.2)

where :r is the Fourier transform of L2 on 12(m). Clearly, the spectrum 0- (X) of the

operator X is equal to the closure of the set of values of the function ~(n) and 0-( X) is the set of values of the function A (t). In view of conditions (A) and (B), the spectra

of X and X are simple.


The operator X is bounded and given by the formula

b 00

(Xf)(n) = J A(t)X(t,n) I f(k)x(t, k) m(k) dm(t), a k=O

and, in view of the fact that the series in the integrand is uniformly convergent, we get

1 00 b

Xf(n) = I~ II f(k)m(k)J x (t,no)x(t,n)x(t,k)dm(t) A(no) k=O a

( 1)00 b + 1-1 ~ I I f(k) m(k) J X (t, k)X (t, n)dm(t).

A(no) k=O a

Since

b

c(p, q, r) = m(p)m(q) J X (t,p)x (t, q)x (t, r)dm,

a

(Rpf)(q) = m(p;m(q) I fer) c(p, q, r) mer), r

and the characters in L2 ([ a, b], m)) are orthogonal, we arrive at relation (3.3).

• In what follows, we use the theory of expansion in generalized eigenvectors (see,

e.g., [BerlO], [Ber11], and [BeKoD.

Lemma 3.2. The chain

(3.4)

is suitable for expansion in generalized eigenvectors of the operator X, i.e., 1R~ is

a nuclear space densely and topologically imbedded in 12(m); moreover, it is the

projective limit of Hilbert spaces and X continuously maps 1R ~ into 1R ~.


Proof. It is necessary to prove only the last assertion. In view of (3.3) and the fact

that c(p, q, r) is finite, IR~ is invariant under the action of X. To show that X is

continuousin IR~, we fix a weight 't= ('tk)k=O anddenote 11'II,tln = 11·111 2('tln). Since

IR ~ = pr lim 12 ('t), it suffices to find a weight 't' such that II X f tm ~ C' II f II 't'm (f E

IR~). For any f E IR ~, we can write

Ilxfll~m = L I (Xf)(k)1 2 'tk'n(k) k

Consider the first sum on the right-hand side ofthis inequality. We have

2

L I (Rnof)(k) 12 'tiiz(k) = A/ L Lf(r)c(no,k,r)m(r) mA't(kk) k m (no) k r

Let N(k) = 1 + max {II c(no, k, r) :;t O}. It is clear that

Since f and c are compactly supported, both sums in this expression are finite. Therefore,

and

L I (RnOf)(k) 12 'tk'n(k) k

Thus, we can take


as the required weight.

• A

Remark 1. The operator X is self-adjoint in 12(m) and, hence, it is a Carleman

operator (see, e.g., [BerlO]). Thus, almost all its generalized eigenvectors with respect to

the spectral measure lie in the space 12 eel m), where

If, in addition,

00 1 I- < 00

k=O 'tk

then, as becomes clear somewhat later, necessarily 12 ('t-1 m) contains all its general

ized eigenvectors.

Remark 2. The condition that the structure measure c(p, q, r) of the dual hypercomplex system is compactly supported is necessary, in fact, only to take the standard

space 1R ~ as a space of test functions for the expansion of X in generalized eigen

functions. This condition can be removed if we use X to construct a nuclear rigging of the space 12(m) (see, e.g., [BeKo]) and require that the series

00

I c(p, q, r)m(r) r=O

be absolutely convergent for fixed p, q E No.

Theorem 3.1.

(i) For any fixed n, the character X (t, n) regarded as a function of t is an ei

genfunction of the operator X corresponding to the eigenvalue 'A(n).

(ii) For any fixed t, the character X (t, n) regarded as a function of n is a

generalized eigenvector of the operator X with respect to chain (3.4) and if


condition (C) is satisfied, then any generalized eigenvector of X is a multiple of X (t, n) for some t E [a, b ].

Proof. We prove (i). It is clear that X (t, n) E tJJ(X) and, hence,

b

(Xx(-,n))(t) = L~(k) J X(s,k)X(s,n)dm(s)x(t,k)m(k) k a

= L~{k) 8k,nllx(-,n)ll~ X(t, k) m(k) = A,{n) X(t, n). k

We prove (ii). For any 11(p)E lR~, we have

The first series on the right-hand side of this equality takes the form

00

L X(t, k)Rno11(k) m(k) = k=O

= L ~(r) L c(r,no,k)X(t,k)m(k) r m(no) k

r

Here, we have used the fact that the dual hypercomplex system is normal and commuta

tive, and t~ fact that, for compactly supported 11 (r) and c(p, q, r), both series contain finitely many terms). As a result, we obtain


Let us now prove the last assertion in (ii). Let f(n) be a nontrivial generalized ei

genvector corresponding to the eigenvalue "A. For any 1'\ E 1R ~, we can write

(3.5)

For r = 0, we get f(no) + f(O) [(1- "A) 1 5:,(no) 1- 1] = 0, whence

By substituting the value of "A in (3.5), we obtain system (3.1). Without loss of general

ity, we can assume that f(O) = 1 (if f(O) = 0, then, by virtue of (C'), f == 0). The characters X(t, n) of the hypercomplex system it(m) are solutions of (3.1). To complete

the proof, it remains to use condition (C).

• Note that (ii) implies that all generalized eigenvectors of the operator X lie in the

space 12 ( 't-1 m), where

L _1 m(k) < 00.

k 'tk

The domain '1J (X) of the operator X is an ideal with respect to multiplication * in ~ and, moreover,

(Xf* g)(t) = (X(j* g»)(t) (fE '1J(X), g E L2 ), (3.6)

(3.7)

where X+ is the operator adjoint to the operator X ~ lR'O with respect to chain (3.4)

(see [BeKo]).


Indeed, we have

Here, we have used the properties of Fourier transforms. Relation (3.7) is proved in exactly the same way.

• Theorem 3.2. The generalized translation operators (RsfXt) (t, s E [a, b]) and

(Rpf)(q) (p, q E No) are uniquely reconstructed from the operators X and X by

the formulas

(RsfXt) = (x(s, ~-\X»f)(t) (t, s E [a, b]; fE L 2 ), (3.8)

Proof. Let us prove (3.8). We have

b

:F(Rs (' »)(p) = f f(t)(RsX)(t,p)dm(t) = X(s,p)( :Ff)(p) (fE L 2 ([a, b], m».

a

Note that

(Xf)(t) = L~(k)(:Ff)(k)x(t,k)m(k) k

is, in fact, the spectral decomposition of the operator X. Therefore, for all f E L2 ([ a, b], m), we can write

00

(X(s, ~-\X»f)(t) = LX(s,~-l(~(k»m(k) X(t, k)( :Ff)(k) k=O

= L Xes, k)X(t, k)( :Ff)(k) m(k) k

= L (:F(Rsf(·»)(k)x(t,k)m(k) k

382 Elements of Lie Theory for Generalized Translation Operators

The proof of (3.9) is similar. Indeed, it can easily be seen that

By using the spectral decomposition of the operator X

J K( :r-If)( A-I (K) X( A-I (K), k)dll(K),

{1C=A(t)}

where dll(K) = dm( A-I (K), we obtain

b

(X(A-ICX),p)f)(k) = J X(t,p)(:r-If)(t)x(t,k)dm(t)

a

b

= J :r- I (RpfC ))X(t, k)dm(t) = (Rpf)(k).

a

Chapter 3

• Note that the idea of reconstruction of generalized translation operators according to

formulas ofthe types (3.8) and (3.9) belongs to Vainerman. Theorem 3.1 actually justi-

fies the following definition: The operators X and X are called generators of the dual hypercomplex systems LI and II (m), respectively, and regarded as dual to each other.

The following statement establishes the relationship between X and X:

Theorem 3.3.

(i) Ker X = { v I v = const} and 1.,( t) is a unique solution of the problem

~

(X A)(t) = sign A(no) X(t, no), A(e) = 1; (3.10)

(ii) {v I v = const} C Ker (X+ -1) (in the case where condition (C) is satisfied,

the inclusion turns into the equality) and A(n) is a solution of the problem

(X+~)(n) = ~(n) + sign ~(no), ~

1.,(0) = 0, (3.11)

A+ A where X is the operator adjoint to X<SlR'O with respect to chain (3.4).

Furthermore, this solution is unique if condition (C) is satisfied.


Proof. We prove (i). Let v = const. Clearly, Xv =v(XX)(t, 0) = v~(O) = O. On the other hand, if f(t) E Ker X, then Mi,.:Ff == 0 or

~(n) f f(t)x(t, n)dm(t) = 0

for all n E No. Since ~(n) '# 0 for n '" 0, we have

ffU)X(t,n)dm(t) = 0

for all n '" 0, whence, since the family of characters is complete in L2 ([ a, b], m), we

conclude that f (t) = vx (t, 0) = v. This implies that problem (3.10) is uniquely solvable.

As follows from the explicit expression for A(t), this function satisfies relation (3.10). Let us prove (ii). Its first statement is obvious. Suppose that condition (C) is satis-

fied and v E Ker (g+ -1). By using the same arguments as in Theorem 3.1 (ii), for all

fE JR.;, we obtain

o = (v, Xf - f)/ A) = I ~ 1 I L fer) mer) [(Rnov )(r) - vCr)]. 2(m A(no) r

Since f is arbitrary, this enables us to conclude that vCr) satisfies system (3.1) with

K = 1. By virtue of (C), we have vCr) = yeO) = const. Let us prove (3.11). Since the structure measure is compactly supported, it follows from the definition of the number N that

1 "" ~ aN mer) ( 1) ~ = I~ l~c(no,p,r)-wX(t,r)ln=e ~ ~ + l- IA I A(p) A(no) r at m(no)m(p) A(no)

:t: (I c(no, p, r)xCt, r)m(r»). ( ) A

= ~ 1 r n=e + 1 _ A 1 A(p) IA(no)1 m(no)m(p) IA(no)1

A A

A(p) + sign A(no)'


The unique solvability of problem (3.11) follows from the form of the kernel of the

operator X+ - 1.

• It should be noted that, without loss of generality, one can assume that N = 1. We

set <p (t) = I t - e IN / N!. It is clear that <p (t) is a homeomorphism of [a, b] into [0,

(b - a)N/ N!] (at the same time, <p is not a diffeomorphism). In the segment [0, (b

a)N/N!], we construct an Hermitian hypercomplex system L1 ([ 0, (b - a)N/N!] , ml) isomorphic to L1 ([ a, b], m) by setting

(LqJ(s)i)(<p(t)) = (Ls(f°<p»(t),

e1 = ° (t,sE [a,b]; fEC[O,(b-a)N/N!]),

(b_a)N / N! b

f f(P) dm l (p) = J fO<p(t)dm(t). o a

The functions 'l'n= Xn 0 <p-1 are, clearly, the characters of the hypercomplex system

L1([0, (b-a)N/N!],ml)' whence we conclude that ",:(e1)=x(N)(e)= ~(n) (nE

No). Let XI = !!i-1M}..!Fi be a generator of the hypercomplex system L1 ([ 0, (b

q.)N/N!], m1), where :Fi is the Fourier transformation with respect to the characters

",(t, n) = '" n(t)· The operator Ff(t) = (f 0 <p )(t) is a unitary operator from L2 ([ 0,

(b-a)N/N!],mt) into L2([a,b]). It is clear that Xl = F- 1XF. Let us show that, under certain restrictions imposed on the generalized translation

operators Rs (s E [a, b]), the operator X coincides with a generator of order N in the sense of (1.3), i.e., with

Without loss of generality, we can restrict ourselves to the case N = 1. Assume that the

measure m is absolutely continuous with respect to the Lebesgue measure d t, that the

relevant density is a sufficiently smooth function, and that, in a certain neighborhood U of the point e, one can always find smooth functions a(s) and ~ (s), a ~ a(s),

~(s)~b, a(e)=e, ~(e)=e#, suchthat

(Rsf)(t) E COO([e, a(s)] x [e, s])

n COO([a(s), ~(s)] x [e, s]) n COO([~(s), e#] x [e, s]»


for any s E U and f E COO([a, b]) and we have

a sup -Rsf(t) = M(t) E Loo([a, b], m). SEU as

In relations presented above, the first component of the products such as I x [e, s] corresponds to the variable t and the second component corresponds to the variable s. For e=b andsufficientlysmaliis-el, the expressions [e,s], [e,a(s)], [a(s),p(s)], and [pes), e#], denote [s, b], [a(s), b], [pes), a(s)], and [a, P(s)], respectively.

This quite cumbersome condition appears to be necessary because the "natural" requirement of smoothness of Rsf(t) in both two variables contradicts the existing exam

ples.

Theorem 3.4. The generator X I is essentially self-adjoint and its closure coin

cides with the operator X defined by formula (3.2).

Proof. Suppose that operator XI is Hermitian. Let us show that Xl is essentially

self-adjoint. Let g(t) E Ran (Xl + i).l. Then, for all n E No, we can write

b b

o = f [(XIX)(t, n) + iX(t, n)] g(t)/l(t)dt = f (X' (e, n) + i)X(t, n)g(t) /l(t)dt.

a a

At the same time, since the characters are real, we have X' (e, n) ::/:. i, whence it follows

that (g, X (., n)h = 0 (n E No). Since the system of characters is complete in L2, we

conclude that dim Ran (Xl + i).l = O. Similarly, dim Ran (XI - i).l = 0 and, consequently, Xl is essentially self-adjoint.

LetfE V(X I ). Then

b a (M~:Jf)(x ) = ff(t)-(RsX )(t)1 _ /l(t)dt

n as n s_e a

b

= f f(t)(Xlx n) (t)/l(t)dt

a

b

= f (Xd)(t)Xn(t)/l(t)dt = (:J(Xd»(xn) a

because XI is Hermitian and X (t)E V(XI ). This implies that XI c :J-IMI)..:J =X n

and, hence, in view of the essential self-adjointness of Xl' its closure coincides with X.

386 Elements oj Lie Theory Jor Generalized Translation Operators Chapter 3

To prove that Xl is Hermitian, we need the following formula:

a ~(s) ~(s) a J(t ) as f J(t, s)dt = f as's +~' (s)J(~(s), s) - a'(s)J(a(s), s), (3.12)

a(s) a(s)

where J (t, s), a (s), and ~ (s) are sufficiently smooth functions, and dt is the Lebesgue measure.

Lemma 3.3. The generator

is an Hermitian operator.

Proof. By using the Lebesgue theorem and relation (3.12), we obtain

(XJ,g)2 b a

= f as (Rsf)(t)ls=eg(t) Jl(t)dt a

~(s) a _ = lim f as (Rsf)(t)ls=eg(t) Jl(t)dt

s--te aes)

[ a b _ a aes) _

= lim - f (Rsf)(t)g(t)Jl(t)dt- - f (Rsf)(t)g(t)Jl(t)dt s--te as as

a a

b

- ~ f (Rsf)(t)g(t)Jl(t)dt - ~' (s)(Rsf)(~(s» g(~(s» Jl(~(s» as ~(s)

+ a'(s)(RJ)( a(s»g(a(s))1'( a(s» 1

[aeS) a _ b a _ ]

= (f * g)'(e) - lim f -(Rsf)(t)g(t)Jl(t)dt + f -a (Rsf)(t)g(t)Jl(t)dt . s--te as f.I S

a pes)

For s E U, in view of the fact that a(s) is continuous, we get


a(s) a f -(Rsf)(t)g(t)~(t)dt as a

a(s) I a I ~ J sup -(Rsf)(t) I g(t) I ~(t)dt seU as

a

a(s)

a(s)

= J M(t) I get) I ~(t)dt a

~ II MIIL..,([a,bl,m) J I g(t) I ~(t)dt ~ 0 as s ~ e (f, g E COO([a, b])).

A similar estimate holds for

whence

a

b a J -(Rsf)(t)g(t)~(t)dt, as ~(s)

(Xd, g)2 = (f * g)'(e) (j, g E COO([a, b])) (3.13)

but (f*g)'(e) = (g*])'(e). By using (3.13) and the fact that the generalized transla

tion operators are real, we prove that the operator Xl is Hermitian. Indeed,

(Xlf, g)2 = <1 * g)'(e) = (X\g,fh = (X\g,f)2 = (f, X l g)2 (f, g E COO([a, b])) .

• Example 1. Let [a, b] = [0, n]. Consider an Hermitian hypercomplex system gen

erated by generalized translation operators

1 (Rsf)(t) = -(f(1 t-sl) + f(n -In - t- s DJ

2 (3.14)

with the Lebesgue measure as a multiplicative measure. The functions X (t, n) = cos n t

are characters of this hypercomplex system, ~(n) = -n 2, and A. (t)= cos t. The gener

ator X is a differential operator generated by the second derivative d 2 / d t 2 and the

boundary conditions f' (0) = f' (n) = 0 and the operator X has the form

A 1 (X f)(n) = -(fen + 1) + f(1 n -11)].

2


It is clear that the generalized translation operators from (3.14) satisfy the smoothness conditionfromTheorem3.4,where a(s)=s, ~(s)=It-S, and X coincides with the closure ofthe second-order generator restricted to 1'(0) = 1'(It) = 0. It is easy to see that the mapping [0, It] 3 t H cos t = x establishes an isomorphism between the hyper

complex system L1 ([ 0, It], d t) constructed from the generalized translation operators (3.14) and the hypercomplex system

constructed from the Chebyshev polynomials of the first kind. Under this isomorphism,

the second-order generator X2 of the hypercomplex system L 1 ( [ 0, It ], d t) transforms into the first-order generator

of the hypercomplex system

~([-1' 1], b).

Example 2. Consider the Hermitian hypercomplex system L1 ([ - 1, 1], ( 1 - t) <X (1 +

t)~ dt) constructed from the Jacobi polynomials Pn(t) = p~<x,~)(t), (t E [-1, 1], a ~ ~,

a + ~ ~ -1) orthonormal with respect to the measure dm(t) = (1 - t)<X (1 + t)~ dt (see Subsection 3.5 in Chapter 2). Generally speaking, the structure measure of this hypercomplex system may take negative values (see [Gas3]).

The functions x(t,n)=Pn(t)/Pn(l) (tE [-1, l],nE No) are characters of this

hypercomplex system. The dual hypercomplex system 11 (m) (m(n) = Pn2(1») is de

scribed in Section 3 of Chapter 2. Both hypercomplex systems L 1 ([-1, 1], m) and 11 (m) are Banach algebras (see Section 3 in Chapter 2); moreover, the structural con

stants of the hypercomplex system 11 (m) are nonnegative for a ~ P and a + ~ + 1 ~ 0. Hence, all conditions formulated at the beginning of this subsection are satisfied. It is easy to see that

A

A(n) = ~ (t, n) I = n(n + a + ~ + 1) dt X t=] 2(a + 1)

(see [Sze], p. 75) and AU) = t. Clearly, the generator X is generated by the differential equation for Jacobi polynomials


X = - (l-t2 )-2 -[a-~+(a+~+2)t]- . 1 {d 2 d } 2(a+l) dt dt

(The equality (X X)( t, n) = X' ( 1, n) X (t, n) is evident and it remains to use the fact that

the family of of characters is complete in Ll ([ -1, 1], m).) Let us show that

where

(Xf)(n) = _1-L(f(.)P.(l») (n), Pn(l)

(3.15)

is the difference operator in l2 for which Jacobi polynomials are polynomials of the first

kind. Indeed, for f E [2' we can write

(Xf)(n)

1 = --[anf(n + 1 )Pn+l (1) + ~nf(n)Pn( 1) + an_den -1 )Pn-l (1)].

Pn(l) .

• Thus, the generator X is unitary equivalent to L.

3.2. General Case of the Construction of Generalized Translation Operators from a Generator. In the present subsection, we show that relations (3.8) and (3.9) remain true in a more general case where it is, generally speaking, impossible to construct a hypercomplex system from the collection of generalized translation operators Rs (or Rp).

Consider an unbounded self-adjoint operator X with simple spectrum and complete

system of eigenvectors acting in L2 ( [a, b], m), where m is a regular positive Borel

measure. Assume that X is semibounded, Ker X = {A I A = const }, and the eigenfunc

tions of X are continuous and real. We enumerate the points ofthe discrete spectrum so

that Ao = 0, and set e = a if X is semibounded from below and e = b if X is semi

bounded from above. Let X (t, n) be the eigenfunction of the operator X corresponding

to an eigenvalue An and normalized by the condition X (e, n) = 1. Suppose that

I X (t, n) I ~ C. Denote by A-I the function inverse to the function No:3 n ~ An E IR.

Theorem3.5. The operators (RsiXt) = (X(s, A-1(X»f)(t) (fE L 2 ) form a

commutative family of generalized translation operators with the following properties:


(i) (Rs J)(t) = (Rsf)(t) m-almost everywhere in t;

(ii) (Rs/)(t) = (Rtf)(s) mxm-almosteverywhere;

(iii) 1/ Rs 1/ s; C;

(iv) the operator valued function [a, b] 3 S H Rs is weakly continuous;

b b

(v) f (Rsf)(t)dm(t) = f f(t)dm(t) (fE C[a, b]) and the measure m is strongly

a a

invariant;

(vi) (RsX)(t,n) = X(s,n)x(t,n).

Proof. It is clear that XU, 0) = 1. To prove that Rs are commutative generalized translation operators, it is necessary to check the following properties:

(a) Re = /;

(b) (Rsf)(t)E L2 ([a,b], m) m-almosteverywherein t;

(c) (R:(Rsf)«(t) = (R~(RJ»(t) m x m-almost everywhere;

(d) (R~Rsf)(t) = (R~(RJ»(t) m x m-almost everywhere.

Since X (t, k) is a complete set of eigenvectors, we conclude that

(Xf)(t) = L Ak(J, X(-,k»2X(t,k)llx(-,k)llz2 kENo

= L Ak(J, X(·, A-10lok»2X(t, A-I (A.k» Ilx(-.A-\Ak»II~ Ak

is the spectral decomposition of the operator X. Hence,

(Rsf)(t) = X(S,A-1(X»f)(t) = L X(s,k)x(t,k)(J,X(-,k»2 Ilx(-.k) 11;2. (3.16) k

This immediately implies property (iii) (because II Rs II = II X (s, A-I (X» II < C) and

property (ii). Condition (b) follows from (ii), and (d) follows from the fact that Rr and


Rs are functions of single operator. It is also clear that Re = I. Let us check the associativity of generalized translation operators, condition (c). Denote

Since the eigenfunctions of the operator X are orthogonal, we get

(x (r, A-I (X» X (s, A-I (X»f)(t)

= L X(r,p) L Ck(X(-,k),X(-,P»2X(s,k)X(t,k) p k

= L (f, xC, k»2x(r, k)x(s, k)X(t, k), k

whence we readily arrive at (c). Property (i) follows from the fact that the eigenfunctions of X are real.

We check (iv). Since X (s, A-I (X»2 ~ C IIfll211 g 11 2, by using the Lebesgue the

orem, we find that

b

(X(S,A- I (X»f,g)2 = L Ckx(s,k)J X(t,k)g(t)dm(t) k a

= L (f,X(-,k»2 (g'X(·,k»2I1xCk)II~2X(s,k) (f,gE L2) k

This yields (iv) because the last series is uniformly bounded and, hence, uniformly convergent. It follows from (3.16) that

(RsX)(t,n) = LX(t,k)x(s,k)(x(-,n),x(-,k»21IxLk)II~2 = x(t,n)X(s,n). k

Finally, by using the relation

and the Lebesgue theorem, we conclude that the measure m is invariant, i.e.,

b

J L (f, X(·, k»2 X(t, k)X(s, k) II xC k)II~2 dm(t) a k


b

= I (f,x(-,k»2 X(s,k)llx(-,k)1122 f x(t,k)dm(t) k a

b

= (f,x(·,0»2 = ff(t)dm(t). a

Since R; = Rs, the measure m is strongly invariant.

• A similar theorem is true in the discrete case.

Consideraboundedself-adjointoperator X (1IXII=I) acting in the space 12(m),

where m( k) > 0, with simple spectrum homeomorphic to the segment [a, b] c IR. As

sume that this homeomorphism is determined by a function [a, b] 3 t H ~(t) E cr(X)

such that SUPA(t)=A(e)= 1. (Clearly, eE Cl[a,b].) The operator X as a self-adjoint operator in 12(m), is a Carleman operator and, hence, for decomposition in its general

ized eigenvectors, we can take the chain

where p = (Pn );=0 is an arbitrary weight satisfying the condition

00 1 I- < 00

k=O Pk

Without loss of generality, we can assume that

00 m(k) I- < 00.

k=O Pk

(3.17)

Let us now extend rigging (3.17) by using a nuclear space V c 12 (pm) with the fol

lowing properties:

1) X continuously maps V into V;

2) for any A(t) E cr(X), the dimensionality of the space N(A(t» generated by

generalized eigenvectors correspond to an eigenvalue A (t) is equal to one.


Denote by X (t, n) E N (A (t» a generalized eigenvector normalized by the condition X (t, 0) = 1. Let X (t, n) be continuous real functions of t for fixed n. Assume that X (e, n) == 1 and there exists a number C ~ 1 such that I X (t, n) I:::; C. As in Theorem 3.5,

we denote by A-I the function inverse to the function [a, b] '3 t H A(t) E a(X).

Theorem 3.6. The operators RpX(q) = X(A-1(X), p)f(q) form a commutative

family of generalized translation operators with the following properties:

(ii) (Rpf)(q) = (Rq.D(p);

(iv) the eigenvectors X (t, n) regarded as functions of t are orthogonal with re

spect to the measure dll(t)= iIx([2dp (A(t)), where 11·11- is the norm in

12 (p -I m) and p is the trace measure of the operator X.

If, in addition, the functions X (t, n) are complete in L2 ([ a, b], 11), then

(v) the measure m is invariant, i.e.,

q q

The last condition is, in particular, satisfied if the linear span of the functions

X (t, n) is uniformly dense in C ([ a, b ]).

Proof. As in the proof of Theorem 3.5, it is necessary to check not only conditions (i)-(v) but also (a)-(d). It is clear that Ro = I and the operators Rp are commuting. By using formula (3.5) from Chapter 3 in [BeKo], we obtain

b

(X(A- 1 (X»),p )f)(q) = J X(t,p)X(t, q)(X(t" ),f)o Ilxt [2dp (A(t», a

XU, k) <p(A(t» = Ilx(t, ')11- and

1 Jf(k) = ,-::-fk'

\/Pk


This immediately implies (i)-(iii) and (b). The relation

(XO,,-I (X», p)q (X(A -I (X», q )f)(r)

= (X(A- 1 (X»,p )(X(A- 1 (X», r)f)(q)

b

= f X (t,p)y..(t, r)x(t, q)(xU,' ),f)o II X(t,) [2dp (A(t» a

implies associativity of the generalized translation operators Rp.

Let us prove (iv). By using formula (3.7) from [BeKo, Chapter 3], for all f, g E

12(m), we obtain

b

(J, g)o = (E([ a, bnJ, g)o = f (J, X(t,· »o(g, X(t,' »0 Ilx(t,)[2 dp(A(t»

a

We set f(k) = o~ and g(k) = o~, where o~ is the Kronecker symbol. Then

b

J X(t,p)X(t, q)dp(A(t» = o~ m:p) a

which immediately implies (iv) and the equality II X(', p) II~(~) = m(p).

To prove the last assertion of theorem, it suffices to consider the case f(k) = o~. It is clear that

b

I Rq o~ m(k) = I f X (t, p )x(t, q)x (t, k)d~(t) m(p)m(k). k k a

On the other hand, since the functions X (t, k) are complete in L2 (Jl), we have

b

X(t,p)X(t, q) = I f x (t,p)x(t, q)x(t, k)d~(t)X(t, k) IlxC k)II~. (3.18) k a

almost everywhere with respect to the mt{asure m. Since the functions X (t, n) are continuous in t and uniformly bounded, both sides of relation (3.18) are continuous in t.

By setting t = e in (3.18) and using the fact that m(k) = IlxC k)II~, we obtain


b

L f X(t,p)X(t,q)X(t,k)d~(t)m(k) = 1, k a

whence it follows that

L R:(~;)m(k) = m(p) = L ~;m(k). k k

• The conditions of Theorems 3.5 and 3.6 are satisfied by the generators X and X of

one-parameter Lie hypercomplex systems, by the operators appearing in the regular Sturm-Liouville problem, and by singular Sturm-Liouville operators defined on an interval studied in [AcTl].

We now analyze the case of regular Sturm-Liouville problem in more details. Consider a differential expression

d2 L = - dt2 + q(t), q E C([ 0, nD.

Let ~ (t) be a solution of the Cauchy problem

Denote

-~"+qm = 0, ~(O) = 1, ~'(O) = h~(O) (h '1= 00).

~'(n) = H. ~(n)

Under the assumption that H'I= 00, we consider the Sturm-Liouville problem

Ly = -y" + qy = -Ay,

y'(O) = hyCO), y'(n) = Hy(n).

(3.19)

(3.20)

Let <p(t, An) be eigenfunctions of problem (3.19)-(3.20), i.e., solutions of problem

(3.19)-(3.20) satisfying the condition <p(0, An) = 1. It is known that the eigenvalues An

are distinct and have the following asymptotic behavior in n:

and


uniformly in tE [O,n] (see, e.g., [LeS)). Clearly, A.o=O and cp(t,A.o)=J.l(t). According to the Sturm theorem on zeros of solutions, we conclude that J.l (t) > 0 (t E [0, n]). Consider a unitary operator

and denote X = VLV- 1, where L is a self-adjoint operator in L2 ([ 0, n], dt) associated with problem (3.19)-(3.20). It is easy to see that X satisfies all conditions of Theorem 3.5. Its eigenfunctions have the form

( ) cp(t, A.n ) X t, n =

J.l(t)

Thus, it is possible to construct generalized translation operators satisfying conditions (i)-(vi) for the regular Sturm-Liouville problem. Note that these generalized translation operators can also be constructed by the method of Povzner (see Section 4 in Chapter 2). It is thus interesting to determine the classes of potentials q (t) for which it is possible to construct commutative quantized hypercomplex systems from the generalized translation operators associated with the Sturm-Liouville problem.

3.3. Analog of the Canonical Commutation Relations for the Delsarte Generalized Translation Operators. First, we recall necessary facts about canonical commutation relations. A pair of self-adjoint operators A, B acting in a Hilbert space J{ is called a representation of canonical commutation relations (CCR) if

AB - BA = -iI. (3.21)

The situation where both A and B are bounded is impossible. Moreover, relation (3.21)

should be understood on a dense subset of analytic vectors of the operators A and B

(recall that a vector cp E J{ is called a joint analytic vector of a finite family of oper-

ators {AJ:l if there exists s > 0 such that

or, which is the same, in the Weyl form eitA eisB = eits eisB eitA . If there are no nontrivial

subspaces of J{ invariant under the action of all operators eitA and eisB (t, s E IR 1) ,


then the considered representation of CCR is called irreducible. According to the von Neumann uniqueness theorem, there exists a unique (to within unitary equivalence) irreducible representation of CCR, namely,

A = 1 d

i dt' B = M"

where M t is the operator of multiplication by t. This representation is called the Schro

dinger representation. Note that the operator A = 7 :t is a generator of the group IR

and B = M t is the Fourier image of a generator of the dual group which is also equal to

IR. In this case, Rt = e j tA is an operator of translation in IR and Ts = e j sB is the Fourier

image of an operator of translation in the dual group. Our aim is to replace the group IR

by a one-dimensional hypercomplex system, the operator A by a generator of this hypercomplex system, and the operator B by the Fourier image of a generator of the dual hypercomplex system (Fourier transformation is carried out over the characters of this hypercomplex system) and to establish relationships between these operators. These relations can be regarded as an analog of CCR.

First, we note that the constructions realized in Subsection 3.1 can easily be generalized to the case of one-dimensional hypercomplex systems with locally compact bases. Actually, as shown in [Schw5], to within an isomorphism of hypercomplex systems, one can assume that the semiaxis [0,00) is a basis of a one-dimensional Hermitian Lie hy

percomplex system and e = ° is its basis unit. Assume that a basis of the dual hyper

complex ·system also coincides with the semiaxis Q = [ 0, 00) and, hence, its characters

are given by a smooth function X(t,'t) (tE Q,'tE Q). Thefunctions A(t) and 5:.('t)

(t E Q, 't E Q) determining the generators X and X, respectively, can be found by differentiating the function X (t, 't), namely,

where

and

and

k = min{n: a:xl ,*0 forsome tEQ} a 't t=O

l = min { n: an~ I "# ° for some 't Eli} . at t=O

All results of Subsections 3.1 and 3.2 remain true in this case. We omit the proofs of relevant assertions and consider the following simple example:


Let L] ([ 0,(0), dt) be the hypercomplex system constructed in Example 2 in Section 2 of Chapter 1. The corresponding Delsarte generalized translation operators have the form

1 Rsf(t) = "2 U(t+s) +f(lt-sl)], fE C([O,oo)).

Recall that the characters of this hypercomplex system are X (t, 't) = cos 't t (t, 't ~ 0). It

is clear that the dual hypercomplex system Lt (Q, m) has the basis Q = [ 0, (0) and the

multiplicative measure dm('t) = d't. Moreover, the generalized translation operators associated with this hypercomplex system are also Delsarte generalized translation operators

1 (Rd)(Il) = "2[J('t+cr)+f(I't-cr D] (IE C([O,oo»)).

This means that the hypercomplex system L] ([ 0, 00 ), d t) is self-dual. The genera

tor X = !fM_t2 !f of this hypercomplex system is generated by the differential expres-2

sion ~ and the initial condition 1'(0) = 0 (JE C OO([ 0,00 )), where !f is the Fourier

dx cosine transformation, M_ t2 is the operator of multiplication by -'t2 in L2 ([ 0, 00 ),

d't), and the generalized translation operators have the form Rs = cos s.JX (s E ([ 0,

00)). The generator of the dual hypercomplex system X is unitarily equivalent to the

operator of multiplication by - t 2 in the space L2 ([ 0, 00 ), d t). Denote by Tt the

Fourier images ofthe generalized translation operators Rt ('t E [0,(0) = Q), i.e., T t = r] Rt!J It is easy to see that (Ttf)(t) = cos 'ttf(t) (J E L 2).

Let A = - X and B = Mt2 = - r 1 X !J It is clear that there are no quadratic rela

tions between A and B. At the same time,there are (exactly) two linearly independent

relations of the third order for A and B:

[A,[B,A]] = 8A, (3.22)

[B, [A, B]] = 8B. (3.23)

This statement can be checked by direct calculation. For the present, relations (3.22)(3.23) are regarded as formal relations on a set of smooth functions.

Let us show that the operators A = -X and B = M t2 have a dense set of joint

analytic vectors. Indeed, it is clear that the functions V2k(t)=H2k(t)e-t2/2 (kE No),

where H2k (t) are orthonormal Hermite polynomials, form a basis in the space L2 ([ 0,

00), dt). Let us show that t])= l.s. {V2k(t)1 kENo} is a space of joint analytic vectors

of the operators A and B. It is evident that t])c t])(A) n t])(B) and is invariant under


the action of A and B. By using the recurrence relations for Hermite polynomials, we obtain

= -~(n + 1)(n + 2) vn+2 - n + - vn + -~n(n -1) v n-2 1 ( 1) 1 2 2 2

(n = 2k).

By induction on k = 0, 1, ... , we prove that, for all n ~ 2,

(3.24)

where, for any i = 1, ... , k, the operator Xi is equal either to A or to B. For k = 0

and k = 1, the assertion is evident. By the induction hypothesis, we obtain

::; [1(S+1)(s+2k)k-l + s(s+2k-2)k-1

Here, we have used the fact that

1 ~(n+l)(n+2) < -(2n+3) and s+2k-4 > 0 forall n~2.

2

By using (3.24), one can easily show that, for all n ~ 2 and t < (4e(n + 3) r l , the series


00 -k k[ 1 ] ~ L _e_(~+2) 2+-(k-l) k=) .J21tk k 6

is convergent. The case n = ° is investigated analogously (recall that we consider only

even n).

• We can now give the following definition: A pair of positive self-adjoint operators A

and B acting in a separable Hilbert space Ji such that relations (3.22)-(3.23) hold on a dense set of joint analytic vectors of A and B is called a representation of relations (3.22)-(3.23).

A representation of relations (3.22)-(3.23) is called irreducible if the space Ji has no

nontrivial closed subspace Jio such that E A (~) Jio c Jio and E B (~) Jio c Jio (~ E

([ 0,00 »), where E A and E B are the resolutions of the identity for the operators A and

B, respectively. It is easy to see that this is equivalent to the condition that the space Ji has no nontrivial closed subspace Jio c Ji invariant under the action of U (t) = cos t.fA and V(s) = cos s rs or, which is the same, that any bounded operator A

commuting with all U(t), V(s) (t, s E ([0,00») is a multiple of the identity operator,

i.e., A = 'AI. Let us show that the representation of (3.22)-(3.23) by the operators Au = - u"

(u'(O)=O) and B=Mt 2 is irreducible.

Indeed, assume that a bounded operator A commutes with T", and Rs (t, s E ([0,

00»). Then, by using the relation [A, Tt ] = 0, we conclude that A commutes with the

operator of multiplication by the Fourier (cosine) image of an arbitrary function f E

L) ([0, 00), dt). Since :F(L) ([0, 00), dt») is dense in Coo ([ 0,00» and the operator A

is continuous, A commutes with the operator of multiplication by an arbitrary function

from Loo ([ 0, 00 ), d t). By Riesz-Neumann theorem, A coincides with the operator of

multiplication by a function f E Loo ([ 0, 00 ), d t). The condition [A, Rs] = ° now takes

the form

u(t+s)[f(t)-f(t+s)] + u(\t-s\)[f(t)-f(\t-s\)] = ° (t,sE ([0,00»),

whence it follows that f(t + s) = f(t) for almost all (t, s) E [0,00) x [0, 00), i.e., the

function f(t) is constant almost everywhere. Hence, the representation under consider

ation is irreducible.

•


Theorem 3.7. If A # 0, then there are no representations of relation (3.22) by bounded self-adjoint operators. The same assertion is true for relation (3.23).

Proof We consider only relation (3.22) because relation (3.23) is analyzed in exactly the same way. The proof is based on the use of the identity

(3.25)

established below. If A and B are bounded operators, then identity (3.25) implies that,

for any n = 1, 2, ... ,

whence it follows that II A IIII B II 2 2n2 (n = 1, 2, ... ) in contradiction with the assumption that both operators A and B are bounded.

Let us now prove (3.25). Denote

Then relation (3.22) can be rewritten as ~l = 4A + TJ. It is obvious that

and (k=O, ... ,n; s=O, 1, ... ).

If p + q = 2m, then we have

By using these properties of Tt, we arrive at the following recurrence formula:

n 2n-l 1 (n n) Tk = 4A + "2 Tk+1 + 1k-l ' (3.26)

Indeed,


4A2n-1 1 ('Tn 'Tn ) 2n-1 1 (n n) = + 2" .L2n-k-1 + .Lk_1 = 4A + 2" Tk+1 + Tk_1 .

The proof of the theorem is completed by the following lemma:

Lemma 3.4. The equality

Tn = 4kA2n- 1 + Tn n n-k (3.27)

holds for all k = 0, ... , n.

Proof. We proceed by induction on k. According to (3.26),

Tn = 4A2n- 1 + .!..(Tn +Tn ) = 4A2n- 1 + Tnn_l , n 2 n+1 n-l

i.e., equality (3.27) holds for k = 1. By using the induction hypothesis, we write

_ 2 2n-1 1 ( n n) - 4(k + l)A + 2" Tn-(k+l) + Tn-(k+l) .

By applying the induction hypothesis once again, we get

Tn = Tn _ 4(k_I)2A2n- 1 n-(k-I) n ,

whence it follows that

Tnn = 4(2k2+2_(k_l)2)A2n- 1 Tn 4(k 1)2A2n- 1 Tn + n-(k+l) = + + n-(k+I)'

• By setting k = n in relation (3.27), we obtain (3.25).

• To establish all irreducible representations of relations (3.22)-(3.23), one can use the

method of many-dimensional dynamical systems (see [VaSIl, [VaS2]). For the sake of

brevity, we omit the proofs and present only the final result (it was obtained by Vaisleb).

Let (vn);=o be an orthonormal basis in a Hilbert space J-f. Then any irreducible representation of relations (3.22)-(3.23) by positive operators is unitarily equivalent to one of the following representations:


(a) AVn = ..In(n+A-l)vn_l + (2n+A)vn + -J(n+l)(n+A)vn+l'

BVn = -..In(n+A-l)vn_l + (2n+A)vn - -J(n+l)(n+A)vn+l (A,~o);

BVn = ..In(n+A-l)vn_l + (2n+A)vn + -J(n+l)(n+A)vn+l (A~O);

(c) dimJ{= 1, A=B=O.

Thus, the von Neumann theorem is no longer true for relations (3.22)-(3.23). The representation

analyzed above is unitarily equivalent to representation (b) with 1..= 1/2. One can easi

ly prove this fact by choosing a basis in L2 ([0, <Xl], dt) in the form vk = H2k(t)e-t2/2,

where H2k are Hermite polynomials. Note that, in case (a) with A > 0, the operator A

is defined by the Jacobian matrix and the corresponding polynomials of the first kind are

the Laguerre polynomials Ln(x, a) (A = a + 1, a > - 1). Thus, A is unitarily equiva

lent to the operator Mt of mUltiplication by the variable t in the space L2 ([ 0, <Xl ),

to.e- t dt). Under this equivalence, the operator B turns into the operator

-4ty"+ 4(t-a-l)y'- ty +2(a+l)y (YE COO([O,<Xl»).

It is easy to see that the unitary operator

transforms B into the Bessel operator

B =

and A into the operator Ii of multiplication by x 2 in the space L2([0, 00), x 2o.+1dx).

For a ~ -1/2, the Bessel operator B can be associated with generalized translation operators which preserve positivity and, by using these operators, one can construct the

hypercomplex system L, ([ 0, 00), t 2o.+ 1 dx). The functions


where la is a Bessel function, are characters of this hypercomplex system. The opera

tor A is the Fourier image of a generator of the dual hypercomplex system. Hence, although the uniqueness theorem is not true for relations (3.22) and (3.23), the class of hypercomplex systems who"se generators satisfy relations (3.22) and (3.23) is not broad.

If the requirement that the operators A and B in representations of (3.22) and (3.23) must be positive is omitted, one can also count all possible representations. The number of representations of this sort is greater than the number of representations considered above.

Remark. By setting

x = ..£A 2 '

Y = ~B Z [X Y] 2i' = , ,

we conclude that relations (3.22) and (3.23) can be reduced to relations between ele

ments of the basis X, Y, Z of the Lie algebra Sl2(lR). By choosing, in the collection of

unitarizable irreducible representations of slilR) (it is well known that such representations admit a complete classification (to within unitary equivalence)), representations

for which the operators A and B are positive, we can obtain the collection of irreducible representations of (3.22), (3.23) without using the indicated method of dynamical systems.

Supplement. Hypercomplex Systems and Hypergroups: Connections and Distinctions

The supplement consists of two parts. The first part (Sections 1 and 2) is devoted to the presentation of the results obtained by Berezansky and S. Krein in 1950-1953. These results were published in 12 papers ([BKrl]-[BKr3] and [Berl]-[Ber9]; see also a brief survey [BKr4]). The second part (Sections 3-5) contains the discussion of some recent results and the analysis of the relationship between hypercomplex systems and the theory of hypergroups.

1. Hypercomplex Systems with Locally Compact Basis. Definition and Properties

1.1. Finite-Dimensional Case. A finite-dimensional hypercomplex system over the

field of complex numbers is defined a finite-dimensional algebra A with a fixed basis

Q = {el' ... ,ed} (d EN). Hence, J = L:=l Jpep for all J E A. To introduce the

operation of multiplication * in A, it suffices to define it for elements of the basis, i.e.,

d

ep*eq = LC;qer (p,q=I, ... ,d), (S.l) r=1

where {c;q t = 1 is a cubic matrix of structural constants C;q E (C. One can easily p,q,r

write relations which make multiplication in A associative.

We identify the vector ep with p (hence, Q = { 1, .,. ,d}) and interpret the coordi

nate Jp of a vector J E A as the value of the function at the point p = e p' With this in

terpretation, the hypercomplex system A coincides with the space of complex-valued

functions J defined on the basis Q: Q:3 P ~ J(p) E ([ with ordinary linear operations

and multiplication defined by the formula which follows from (S.l), namely, for any J and g

d

(j* g)(r) = L J(p)g(q)c;q = L J(p)g(q)c(p, q, r) (S.2) p,q=1 p,qEQ

406 Hypercomplex Systems and Hypergroups Supplement

(r E Q; c(p, q, r) = C;q = (p * q) (r)).

The general idea is to replace the basis {I, ... ,d} by a general locally compact

space Q. In this case, the elements of the hypercomplex system are regarded as com

plex-valued functions on Q, the matrix of structural constants is replaced by a function

Q x Q x Q :3 (p, q, r) H c(p, q, r) E a:: , and multiplication (convolution) is defined by analogy with (S.2) as

(j*g)(r) = f f J(P)g(q)c(p,q,r)dm(p)dm(q) (rE Q), (S.3)

Q Q

where m is a suitable measure. However, in some important cases (e.g., for the ordinary convolution of functions on

lR), multiplication in (S.3) makes sense only if c(p, q, r) is a generalized function. Therefore, it is reasonable to define not a "structure function" c(p, q, r) = (p * q )(r),

but a "structure measure" c (A, B, r) which is equal to the convolution of two indicators

KA (.) and KB (.) of sets A, B e Q. Thus, instead of "a continuous analog" of the struc-

tural constants C;q = c(p, q, r) of finite-dimensional hypercomplex systems, we consi

der their matrix analog

c(A, B, r) = L c(p,q,r) (A,BeQ; rE Q). (SA)

pEA.qEB

1.2. Definition of Hypercomplex Systems with Locally Compact Basis. Let Q be a

complete separable locally compact metric space of points p, q, r, ... , let 'B(Q) be the

a-algebra of Borel subsets, and let 'Bo( Q) be the subring of 'B(Q) which consists of sets with compact closure. We consider the Borel measures, i.e., positive regular mea

sures on 'B(Q) finite on compact sets. We denote by C(Q) the space of continuous

functions on Q and let C b( Q), Coo (Q), and Co (Q) be spaces consisting of bounded,

approaching zero at infinity, and compactly supported functions from C(Q), respectively.

A hypercomplex system with basis Q is defined by its structure measure c(A, B, r)

(A, B E 'B(Q); r E Q). For fixed Band r (A and r), the structure measure c(A, B, r)

is a Borel measure in A (in B) with the following properties:

(HI) the function c(A, B, r) lies in Co(Q) for all A, B E 'Bo(Q),;

(H2) the following associativity relation holds for all A, BE 'Bo(Q) and SEQ:

f c(A, B, r)drc(E r, C, s) = f c(B, C, r)drc(Er' C, s). (S.5)

Q Q

Supplement Hypercomplex Systems and Hypergroups

The structure measure is called commutative if

(H3) c(A, B, r) = c(B, A, r) (A, B E tJ3o(Q».

A measure m is called a multiplicative measure whenever

J c(A, B, r)dm(r) = m(A)m(B) (A, B E tJ3o(Q»·

Q

(H4) We suppose the existence of a multiplicative measure.

Theorem 1.1. For any f, g ELI (Q, m), the convolution

(j*g)(r) = f f(P)dp(fg(q)dqC(Ep,Eq,r)} Q Q

407

(S.6)

(S.7)

(S.8)

is well defined. This formula generalizes the definition of convolution (S.3). The space L] (Q, m) with this convolution is a Banach algebra (under condition (H3), the algebra

is commutative}. This Banach algebra is called a hypercomplex system with basis Q.

It is obvious that c(A, B, r) = (KA * KB)(r) (A, BE tJ3o(Q». One may construct hy

percomplex system both with and without unity. If the unity is not included in LI (Q,

m), then it is convenient to join it formally to LI (Q, m). The unital algebra obtained as

a result is denoted by .£1 (Q, m). We especially emphasize two points in the axiomatics of hypercomplex systems: the

nonnegativity of the structure measure (hence, finite-dimensional hypercomplex systems have nonnegative structural constants) and continuity and finiteness of the convolution

(KA * Ks)(-) (A, BE tJ3o(Q». These properties of hyper complex systems are essential. Let us now present an example of a hypercomplex system. A large part of the theory

of hypercomplex systems is constructed by analogy with this example. Weare speaking about the group algebra of a unimodular locally compact group G with the basis Q = G. The structure measure is defined by the equality

c(A,B,r) = p(B-IrA) (A,BE tJ3(Q); rE Q), (S.9)

where p is the Haar measure on G. The multiplicative measure m is p. It is obvious that the ordinary convolution of functions summable with respect to the Haar measure on the group can be rewritten in the form (S.8). Therefore, the group algebra LI (G, p) is an example of a hypercomplex system with the locally compact basis Q = s.

In Section 1, we consider only commutative hypercomplex systems, i.e., we always assume that (H3) is satisfied. For such hypercomplex systems, by using the nonnegativ-


ity of the structure measure and by imposing some additional restrictions, we can prove the existence of a multiplicative measure (see Subsections 1.5 and 3.1).

1.3. Characters and Maximal Ideals. A nonzero function Q:3 P f-+ X (p) E <t measurable and bounded almost everywhere is called a character of the hypercomplex

system if, for any A, BE $o(Q),

I c(A, B, r)x(r)dm(r) = X(A)X(B)x(C) = I x(r)dm(r), C E $o(Q)· (S.IO)

Q e

The set of characters is in one-to-one correspondence with the set of maximal ideals

of the Banach algebra L1 (Q, m) (or the algebra £1 (Q, m), where we consider only

proper ideals of L1 (Q, m )). This correspondence between a character X and a maximal

ideal M is described by the equality

J(M) = I J(p)x(p)dm(p) (fE L1 (Q, m)). (S.1I)

Q

We denote by X the set of characters endowed with the topology of the space of

maximal ideals. The set X is a locally compact metric space if L1 (Q, m) is an algebra

without unity and it is a compact space if L 1 (Q, m) is an algebra with unity. Relation

(S.7) now implies that the function f(P) == 1 (p E Q) is always a character of a hyper

complex system. It follows from (S.Il) that II X IlL.., (Q,m) = II M II::::; 1.

2.4. Normal Hypercomplex Systems with Basis Unity. Consider a class of hypercomplex systems for which, by analogy with group algebra, one can construct meaningful harmonic analysis. For these systems, there exists an analog of the inverse elements for points of the basis and there exists a point of the basis which plays the role of "unity located in the basis", (i.e., the identity element in the case of group algebras).

(H5) A hypercomplex system is called normal, if there exists an involutive homeo

morphism Q:3 P f-+ p* E Q such that meA) = m(A*),

c(A,B, C) = c(C,B*,A), and c(A,B,C) = c(A*,C,B) (A,BE $o(Q)),

where c(A, B, C) = Ie c(A, B, r)dm(r).

It is worth noting that, for normal hypercomplex systems, the mapping L1 (Q, m) :3

f (p ) f-+ j* (p ) = J (p*) ELI (Q, m) is an involution in the Banach algebra L1 (Q, m ).

Consider some properties of normal hypercomplex systems.

Supplement Hypercomplex Systems and Hypergroups 409

Theorem 1.2. The multiplicative measure in a normal hypercomplex system is

unique. Characters of this system are continuous. The radical (i.e., the intersection of

all maximal ideals) coincides with the set of annihilators (i.e., elements a E Ll (Q, m)

such that a * f= 0 for any fE L] (Q, m)).

A character X of a normal hypercomplex system is called Hermitian if it corres

ponds to a symmetric maximal ideal !vf = M. In other words, X is Hermitian if X (p* ) = X (p) (p E Q). Denote the collection of all Hermitian characters by Xh C X.

In normal hypercomplex systems, one can introduce generalized translation operators

Tp (p E Q) as follows:

where B 11( s) is a ball in Q of radius 1 / n centered at the point s. We can prove that

the limit in (S.12) exists for fE Loo(Q, m). Convolution (S.9) can be rewritten as rela

tion (3) in Introduction. Roughly speaking, if the function c (p, q, r) is as in (S.3), operator (S.12) takes the form

(Tpf)(q) = J c(p, q, r)f(r)dm(r) (p E Q).

Q

In the case of group algebras, we have (Tpf)(q) = f(qp) (p, q E Q = G).

In normal hypercomplex systems, c(A, B, r) :::; meA) (A, B E ~o(Q); r E Q).

Hence, there exists the Radon-Nikodym derivative dc ( " B, r) / dm ( . ) (p) = c' (p, B, r)

defined for m-almost all p E Q. It can be interpreted as the convolution of the 8-function 8p with the indicator KB(-), i.e., c'(p, B, r) = (8 p * KB)(r) E [0, 1]. This convolution is a measurable bounded compactly supported function.

(H6) A point e E Q is called the basis unity of a normal hypercomplex system if

c(A, B, e) = m(A* n B) (A, B E ~o(Q)). (S.13)

In the case of group algebras, the identity element of the group is the basis unity of

the hypercomplex system L] (G, p). If Q is discrete, then u (p ) = 1/ m (e ) 8~ is the unity of the algebra L] (Q, m).

Theorem 1.3. A normal hypercomplex system with basis unity is semisimple.

A point e E Q is a basis unity if and only if X (e) = 1 for all characters and the

system of characters is complete in Co(Q) in a sense of L ] (Q, m) (i.e., if

J g(r)x(r)dm(r) = o for gE Co(Q) andallcharacters X, then g=O).


We develop harmonic analysis for normal hypercomplex systems with basis unity, including some generalizations of harmonic analysis on commutative compact and discrete groups (the Plancherel theorem, positive definite functions, duality, and almost periodic functions). In the remaining part of Section 1, these results are mainly presented for hypercomplex system with compact or discrete bases.

1.5. Harmonic Analysis for Hypercomplex Systems with Compact Basis. Let Q be a compact basis. We have the following theorem on the existence of a multiplicative measure:

Theorem 1.4. Assume that the structure measure c (A, B, r) of the general com

mutative hypercomplex system with compact basis Q satisfies the condition c (Q, Q,

r) > 0 for any open set 0 C Q and any r E Q. Then there exists a multiplicative measure (i.e., in this case, (H4)followsfrom (HI), (H2), and (H3)).

For normal hypercomplex systems, the following theorem on completeness of char

acters is true:

Theorem 1.5. Let L1 (Q, m) be a normal hypercomplex system with basis unity

and compact basis Q. The space of characters X is discrete and countable, i.e., X =

{X (p)} = (Xj(P))j=l (if Q is a finite set, then X is finite). All characters are Her-

mitian andform a complete orthogonal system in L 2 (Q, m). Denote Q = X h = X.

This set is called the dual basis. The Parseval equality takes the following form:

J f(P)g(p)dm(p) = Iicx)g(x)m(x) forany j,gE L 2 (Q,m), Q XEQ

(S.14)

where

lex) = ~ f(P )x(p) dm(p) and m (X) = (~I x(p) I' dm(p)T,

and m is a Planche rei measure.

Corollary 1.1. We have

c(A, B, r) = I X(A)X(B) x(r) m(x) (A, B E 'B(Q); r E Q). (S.15)

XEQ

Note that it follows from (S.14) and (S.lO) that (c(A, B, ')m(x» = X(A)X(B). By

reconstructing c(A, B, r) from (c(A, B,') m(x)) and using the positivity of c(A, B, r), we obtain (S.15).


A continuous bounded function Q 3 P ~ <p (p) E <I: is called positive definite if the inequality

n

L (Tpj<p)(pZ)~j ~k ~ 0. j,k=1

Theorem 1.6. A function (X) m(x) are uniquely determined by <po

(S.16)

(S.17)

This theorem admits the following equivalent formulation: If the Fourier coefficients

<7> (X) of the function <p E Loo(Q, m) are nonnegative, then the expansion of <p in a

series in X is absolutely and uniformly convergent.

1.6. Harmonic Analysis for Hypercomplex Systems with Discrete Basis. Consider a normal hypercomplex system L1 (Q, m) with a countable discrete basis Q and basis

unity (in this case, it is the unity of the algebra L1 (Q, m». Hence,

The structure measure of this hypercomplex system is given by (SA) in the form of

an infinite structure matrix c(p, q, r) = (Op* 0q)(r) ~ ° (p, q, r E Q). It is now con

venient to weaken condition (HI): we do not require, that c (p, q, r) is compactly

supported for fixed p, q E Q. The existence of a multiplicative measure (i.e., (H4» is supposed. The definition of characters and the normality condition now take the form

L c(p, q, r)x(r)m(r) = XCp)m(p)x(q)m(q), rEQ

(S.18) c(p, q, r)m(r) = c(p, q*, r)m(r).

The theorem on completeness of characters now admits the following formulation:

Theorem 1.7. The spaces X and Xh of characters and Hermitian characters of

a normal hypercomplex system with discrete basis and basis unity are compact. There


exists a Borel measure m on X h (Plancherel measure) such that the following

Parseval equality holds for all J, g E L2 (Q, m) = 12 (m ) :

Lf(p)g(p)m(p) = J j(X)g(x)dmCx), PEQ Q

fCx) = Lf(p)x(p)m(p), (S.19) PEQ

where Q = supp m is the dual basis.

By setting g = Op in (S.19), we obtain the following inversion formula for f:

f(p) = J jCX)x(p)dm(x). (S.20)

Q

Corollary 1.2. We have

c(p,q,r) = m(p)m(q) JX(p)x(q)XCr)dm(x) (p,q,rE Q). (S.21) Q

Positive definite sequences Q :3 p~ <P (p) E <C are defined by (S.16). The Bochner-Herglotz theorem for sequences of this sort can be formulated as follows:

Theorem 1.8. A function <P (p) is positive definite if and only if

<P (p) = J X(p )dcr(X) (p E Q), (S.22)

xh

where the positive Borel measure cr on Xh is uniquely determined by <po

1.7. Construction of Hypercomplex Systems with Compact and Discrete Bases by Using Orthogonal Systems of Functions. The formulas presented in Subsections 1.5 and 1.6 enable one to construct hypercomplex systems with compact and discrete bases.

Theorem 1.9. Let Q be a compact space with finite Borel measure cr positive

on open subsets of Q, let Q be a countable set and let L = (<p /p») i=Q be an or

thogonal (with respect to cr) system of continuous functions on Q which is complete in L 2 (Q, m). Suppose that

(i) if <Pi E L, then <Pi E L;


(ii) one can indicate points e E Q and e E Q such that <P j (e) = 1 for all j E Q and <peep) = 1 for all p E Q.

Then thefoUowingfonnulas similar to (S.15) and (S.2i)

c(A, B, r) = L <p/A) <pj(B) <p/r) mU), jEQ

c(p, q, r) = m(p)m(q) f <pp(t) <pit) <Pr(t) dcr(t) (p, q, r E Q) Q

(S.23)

(S.24)

define the structure measures of normal hypercomplex systems with basis unity in the cases of a compact basis and a discrete basis, respectively. It should be additionally

assumed that the sums in (S.23) in the first case and the integrals in (S.24) in the second

case are nonnegative.

A system of functions <P j with properties outlined above is called an A 05ystem. If

we it is necessary to construct a single hypercomp1ex system with the structure measure (S.23) or (S.24), the requirements imposed on the measure cr and a system of functions can be weakened.

1.8. Duality for Hypercomplex Systems with Compact and Discrete Basis. The results presented in Subsections 1.5-1.7 enable one to construct duality theory for normal hypercomplex systems with compact and discrete bases. It can be regarded as a generalization of the classical duality theory for commutative compact and discrete groups.

Let LI (Q, m) be a hypercomplex system with compact basis, let Q be the dual countable basis (the collection of all characters X, <P, "IjI, ... ), and let m be a Plancherel

measure of the form (S.14). The space L I ( Q, m) = II (m) becomes a hypercomplex system with discrete basis if we define the dual structure measure c by the formula

c(X, <P, "IjI) = m(X) m(<p) f X(p)q>(P) "IjI(p)dm(p) (X, <p,"IjI E Q) (S.25)

Q

and assume that the integrals in (S.25) are nonnegative. This dual hypercomplex system is normal if we set X* = X and possesses the basis unity e == 1.

Every point p E Q defines a function on Q: Q 3 X ~ X (p) E <I:. It is easy to see that this function is an Hermitian character of the hypercomplex system II (m). There-


A A A A

fore, Q C Q, where Q is the collection of Hermitian characters of the dual hypercom-

plex system I I (m). We say that hypercomplex systems are in duality if Q = Q.

Theorem 1.10. Let L 1 (Q, m) be a normal hypercomplex system with compact

basis and basis unity. The discrete set Q of characters is a basis of the dual normal

hypercomplex system II ( Q, m) if and only if the pointwise product <p I (t) <P2 (t) of

any two positive definite functions <P 1 (t), <P2 (t) (t E Q) is positive definite. The set A

Q of Hermitian characters of the dual hypercomplex system coincides with the original

compact set Q if and only if the pointwise product <PI (X)<P2 (X) of any two positive

definite sequences <PI (X) and <P2 (X) (X E Q) is positive definite. In this case, the

Plancherel measure m coincides with the original measure m.

Note that, since supp m = Q, we have supp Jz = Q. Hence, Q = Q is the dual

basis for 11 (m) in a sense of Theorem 1.7.

A similar theorem holds if we start with a hypercomplex system II (m) with a dis

crete basis Q, the structure measure c (p, q, r) (p, q, r E Q), and characters X, <P, \If, ....

In this case, for the dual compact basis supp m = Q, we can construct the hypercom

plex system LI ( Q, m) with the structure measure such that

c(A, B, X) = L p(A)p(B) X(p) m(p), peA) = f \If(p) dm(\If) (S.26) peQ A

for all A, BE t.B(Q) and X E Q.

The set of characters of the hypercomplex system L I ( Q, m) coincides with Q and the Plancherel measure of the dual hypercomplex system coincides with m.

Note that the original proof of Theorem 1.10 was based on the use of another definition of normality in the discrete case (see Subsection 1.10). For the standard definition of normality, the last condition of positive definiteness of pointwise products of sequences can be omitted.

1.9. Hypercomplex Systems with Complex-Valued Measures. Hypercomplex systems of this sort can be regarded as a generalization of the notion of finite-dimensional

hypercomplex systems in a sense that the structural constants C;q may be complex. All

definitions are similar to the definitions in Subsections 1.2-1.4; c(A, B, r) is, in this

case, a complex Borel measure (or charge) of A (B) for fixed B (A) and r. The mul

tiplicative measure is a nonnegative Borel measure m such that the following inequality holds (instead of (S.7)):

f v(A,B,r)dm(r) ~ m(A)m(B) (A,BE t.B(Q); rE Q), (S.27)

Q


where v (A, B, r) is the total variation on A X B of the charge cr defined on I}3(Q X Q)

by the formula cr(A X B) = c(A, B, r). It follows from inequality (S.27) that the

convolution (S.8) exists for all J, gEL 1 (Q, m) such that

The basis Q can be, in particular, compact or discrete. After appropriate reformulation, the major part of the results presented in Subsections

1.2-1.7 remain true for hypercomplex systems with complex measures (It is clear that the existence of a multiplicative measure is supposed). However, it is quite difficult to construct the relevant duality theory because inequality (S.27) for the dual hypercomplex system is violated (for more details, see Section 5).

1.10. References. The results presented in Subsections 1.2-1.4 for the case of a compact basis were obtained in [BKr1]-[BKr5] and [Ber9] (the locally compact case was considered in [Ber6]). The results of Subsections 1.6-1.9 with another definition of normality in the discrete case (c(p, q, r) = c(r, q*,p) instead of (S.18» were established in [Ber2]-[Ber9]. The theory of almost periodic functions in hypercomplex systems is discussed in Subsection 2.6.

2. Examples of Hypercomplex Systems

2.1. Group Algebra of a Locally Compact Group. This example was mentioned in Subsection 1.2. The structure measure is defined by (S.9). The facts presented in Subsections 1.2-1.8 imply the relevant facts from harmonic analysis on such groups.

2.2. Center of the Group Algebra of a Commutative Compact Group. Let LI (G,

p) be the group algebra of a compact group G (p is the Haar measure on G). The center of LI (G, p) is formed by functions which are constant on the conjugacy classes

p = {h -I g h I h E G} (g E G). Let Q be the collection of such classes endowed with the quotient topology relative to the mapping G :3 g ~ g = P E Q. The center of the group algebra of the commutative compact group G gives an example of a normal hy

percomplex system LI (Q, m) with basis unity and compact basis. The structure mea

sure is defined by the equality c(A, B, r) = p( Ag n .8), where A = {h E G I Ii E A}

and g E r, the multiplicative measure m is the image of p, the basis unity coincides

with the group unity e, and P* = p-I. The characters of the hypercomplex system

L1(Q,m) have the form X(p)=X(g)n- l , where gEp and X is a character of an irreducible representation of G, n = dimX. The results of Subsections 1.2-1.8 yield the representation theory and the harmonic analysis of compact groups, including the Tannaka-M. Krein duality principle.


2.3. Harmonic Analysis of Spherical Functions. Let G be a compact group, let H

be a closed subgroup of G, let Q = G / H be the space of double cosets p = {hi g h21

hi, h2 E G} (g E G), and let G :3 g ~ g = P E Q be a natural mapping. The func

tions in the group algebra L 1 (G, p) constant on classes the p form a subalgebra of Ll (G, p). We suppose that this sub algebra is commutative (i.e., (G, H) is a Gelfand pair). Then we can introduce a normal hypercomplex system with compact basis Q and basis unity. Thus, the structure and multiplicative measures are defined by the same for

mulas as in Subsection 2.2 but with the new natural mapping, P* = p-l, and the basis unity e = {H}. Characters of such hypercomplex systems coincide with spherical functions. The results presented in Subsections 1.2-1.8 are, in fact, basic results of harmonic analysis on Gelfand pairs.

2.4. Hypercomplex systems Associated with the Sturm-Liouville Operator. We consider the Sturm-Liouville expression on a semiaxis

Ly = -y"+a(p)y, pE [0,00), (S.28)

where a (p) is a real continuous function of bounded variation on [0, 00). Let a(p) be a nonnegative nonincreasing function such that

I a(p') - a(p") I ~ a(p') - a(p") (0 ~p' <p" < 00)

(e.g., a(p) = Var;a). Consider the equation -y" + a(p) = 0. Denote by /l(p) the

solution of this equation such that /leO) = I and /l'(O) = 0. By using the well-known method of Levitan and Povzner, we can define the convolution of functions on [0, 00 ).

We extend the function a (p) to the entire real line as an even function and consider the following hyperbolic partial differential equation:

For a two times differentiable function f (P), we set

(Tpf)(q) = u(p, q)//l(p)/l(q) (p, q E [0,00»,

where u (p, q) is a solution of the last equation satisfying the initial conditions u (p, 0) =

f(P) /l(p) and u~ (p, 0) = ° (the function f(p) is supposed to be extended to the entire real line as an even function). By using the Riemann method for hyperbolic equa

tions, we can extend the operator Tp to a broader class of functions f (P ).

We define convolution by the formula

00

(f*g)(r) = f (Trf)(q)g(q)/l2(q)dq (rE [0,00». (S.29)

o


This operation generates the hypercomplex system Ll (Q, m) with the basis s = [0,

00) and the multiplicative measure dm (p) = J..l2 (p) dp. The real structure measure is given by seA, B, r) = (KA * KB)(r). The hypercomplex system Ll (Q, m) is normal

(p * = p) and possesses the basis unity e = O. Assume that a(p) ~ ° as p ~ 00. The characters of the hypercomplex system co

incide with <p (t, A,) I J..l (t), where <p (t, A) are eigenfunctions of the Sturm-Liouville

operator defined by (S.28) with the initial condition y'(O) = 0 and A belongs to the

spectrum of this operator. The hypercomplex system Ll (Q, m) possesses the nonnega

tive structure measure (which satisfies (S.7» if the function a( p) is nonincreasing. In

this case, a = a.

2.5. Hypercomplex Systems Associated with Orthogonal Polynomials. Let E be a compact subset of the real line, let cr be a finite Borel measure on E positive on open

sets, and let Po (t), PI (t), ... be the relevant system of orthonormal polynomials. Let

b = sup E. It is obvious that P n( b) :;:. O. According to Subsection 1.7, by using this sys

tem of polynomials, one can construct a hypercomplex system with the discrete basis

Q = {O, I, ... } by setting

= m(q) J <pp(t)<Pit)<Pr(t)dcr(t), E

(S.30)

for all p, q, r E Q. We suppose that the integrals in (S.30) are nonnegative. This hy

percomplex system is normal p* = p and possesses the basis unity e = O. The procedure of construction of hypercomplex systems associated with orthogonal

polynomials can be generalized to the case of real c(p, q, r) as in Subsection 1.9. We can also present the procedure for choosing constants m (p) (p E Q) guaranteeing the validity of inequality (S.27) in the case of discrete bases. Moreover, for such hypercomplex systems, one can prove a theorem similar to the theorem on continuity of transmuta

tion operators. Indeed, if we can construct a hypercomplex system for a measure dcr ( t), then, under certain restrictions imposed on the function h ( t) > 0 (t E E), we can con

struct a hypercomplex system for the measure h(t)dcr(t). Some special systems of orthogonal polynomials were investigated.

If (a) there exists e E E such that

Pr(e) = maxIPr(t)1 (r=O,I, ... ); tEE

418 Hypercomplex Systems and Hypergroups

(b) there is a multiplication formula

Rn(t)Rn(s) = f Rn(z)d~t,s(z) (t, sEE),

E

Supplement

where d~t,s(z) is a nonnegative bounded measure and Rn(t) = Pn(t)/ Pn(e), then, by

using (S.23), we can construct a normal hypercomplex system with the compact basis E.

Here, t* = t and the point e is the basis unity.

2.6. Almost Periodic Functions and Sequences. Let LI (Q, m) be a normal hyper

complex system with locally compact basis and basis unity and let T p be the corres

ponding generalized translation operators (S.12). Following Delsarte and Levitan, a con

tinuous function Q 3 P ~ f (p) E <I: is called almost periodic if the family of func

tions {Tpf} pE Q is precompact in the topology of uniform convergence (i.e., the clo

sures of all these functions are compact). We can construct the general theory of almost periodic functions of this sort. Name

ly, we can define an average M(f) of an arbitrary function f(p) by

M(f) = Lim _1_ f f(p)dm(p), N~oo m(QN)

Qn

where Lim denotes the Banach limit and QN (N = 1,2, ... ) is a sequence of compact

subsets of Q such that QN C Q N + 1 and U QN = Q. Almost periodic functions satisfy the following Parseval equality:

(J,f) = (fa,Ja) + II(f,x)1 2 (X,X)-I, (J,g) = M(fg) , (S.31)

x

where summation is carried out over the countable set of characters of the hypercomplex

system and fa is the projection of f onto a space of "annihilators". We can also estab

lish the conditions under which fa is equal to zero and relation (S.31) turns into the clas

sical Parseval equality. One can also prove a theorem on uniform approximation of almost periodic functions by linear combinations of characters.

This theory is applicable to hypercomplex systems associated with the Sturm-Liouville operator, to hypercomplex systems associated with orthogonal polynomials with

measure h(t)( 1 - t)u(1 + t)~ (t E [-1, 1], a, ~ ~ -1 /2, h(t) > 0), and to almost periodic even functions defined on commutative locally compact groups. For these examples, some results of Levitan and Marchenko were extended and improved.

2.7. References. The results of Subsection 2.2 can be found in [Berl] and [Ber9], For the results of Subsections 2.3 and 2.4, see [Ber2], [Ber9] and [Ber7], respectively.


It is clear that all results in Subsection 2.3 remain valid if we replace a compact group G by a locally compact group. Subsection 2.S contains a presentation of the results obtained in [Ber2], [Ber4], and [Ber9] modified according to the style of [BeKS]. The results of Subsection 2.6 were established in [Ber3], [BerS], [Ber6], [Ber8], and [Ber9].

3. Harmonic Analysis in the Locally Compact Case

In the present section, we describe harmonic analysis for a general noncommutative hypercomplex system LI (Q, m) with locally compact basis Q and basis unity e E Q.

3.1. Fourier Transformation and the Plancherel Theorem. First, we consider the problem of existence of a multiplicative measure in the locally compact case.

Theorem 3.1. Suppose that the structure measure is commutative (i.e., (H3) is sat

isfied) and has the following properties:

(i) c (Q, A, r) E Cb(Q) for any A E 1Jo(Q);

(ii) lim c(Q, A, r) = c(Q, A, 00) exists for all A E 1Jo(Q) and the function

1Jo(Q) 3 A ~ c(Q, A, 00) is a regular Borel volume;

(iii) the inequalities

c(Q,Q,r) >0 (rEQ) and c(Q,Q,oo»O

hold for any open 0 E 1Jo ( Q).

Then, for this structure measure, the multiplicative measure exists (but is not neces

sarily unique).

In the noncommutative case we suppose that the multiplicative measure exists. In normal hypercomplex systems, the multiplicative measure is unique.

All results of harmonic analysis established in the cases of compact and discrete bases can be generalized to the case of noncommutative hypercomplex systems with locally compact bases. Any hypercomplex system of this sort is associated a family of (left)

generalized translation operators given by the formula (Lsi, g*h2 = (j* g)(s) (j, g E

L2). These operators can be extended to uniformly bounded operators acting in Loo. A dense linear subspace s of a Hilbert space H with two pairs of operations: multi

plication a U b and involution a U (first pair) and multiplication a n b and involution

a n (second pair) is called a Hilbert bialgebra if

1) D is a Hilbert algebra for each pair of operations;


2) the operator of connection W in H (8) H defined by the equality

(a (8) b' n, W(a 'u (8) b») H®,! = (a U a', b U b')H (a, a', b, b'),

is continuous.

Theorem 3.2. Let L j (Q, m) be a normal hypercomplex system with basis unity.

Then the linear subspace D = L j n L"" has the structure of a Hilbert bialgebra with

the operations fng=f* g and fU(r) = f(r*) (first pair) and fng(r) =f(r)g(r)

and fn(r) = fer) (second pair).

For the Hilbert bialgebra D = L j n L"" from Theorem 3.2, let Lu be the w* -alge

bra associated with the Hilbert algebra (D, *). By LU we denote the set of positive

elements of Lu and by <p the canonical exact semi finite normal trace on LU' defined

as <p(A)= (a, a)L 2 if A1I2 = Lu(a) a E D, where Lu(a) is the operator of left con

volution with a, i.e., Lu(a)b = a * b Ca, bED). We extend <p to Lu by linearity. From the central decomposition, we obtain

z z

where Z is the quasispectrum of Lu.

The Fourier transform of the function a E D = L j n L"" is defined as an operatorvalued function

a(A) = f a(s)Ls(A)dm(A) (A E Z), (S.32)

Q

where Ls (A) are the components of the central decomposition of the left generalized

translation operator Ls E Lu associated with the hypercomplex system.

Theorem 3.3 (analog of the Plancherel theorem and inversion formula). The Fou

rier transforms (S.32) satisfy the following Planche rei and inversion formulas:

(a, b) = f <p).(L).(b)* L).(a») dp(A), Z

a(s) = f <p).(Ls(A) a(A») dp(A), z

(S.33)

(S.34)


where <pA, and LA,(a) are the components of the central decomposition of the trace

<p and the operator Lu(a) respectively. The Plancherel measure p in (S.33) and (S.34) is unique to within equivalence. It follows from the Gelfand-Naimark-Segal construction that the Fourier transform (S.32) and the inversion formula (S.34) establish a unitary isomorphism between L 2 (Q, m) and the Hilbert space H rp determined

by the trace <p on Lu.

Denote by 9{ the intersection of L1 (Q, m) with the linear span of positive definite

functions on Q. If the hypercomplex system L) (Q, m) is commutative, then the Fourier transform can be written as

a(x) = J a(r)x(r) dm(r) (X E X h)·

Q

Theorem 3.3 yields the following assertion:

Corollary 3.1. There exists a unique regular Borel measure m on the space of

Hermitian characters X h such that, for any function x E 9{, its Fourier transform

x lies in L) (Xh' m) and x(r) = J x(x)x(r)dx. The Fourier transformation can xh

be continuously extended to a unitary operator 'f: L2 (Q, m) -7 L2(Xh , m).

In the commutative case, we have also an analog of the Bochner theorem.

Theorem 3.4. A function <p E C b( Q) is positive definite if and only if there exists

a finite positive regular measure 11 on the space Xh such that

<per) = J x(r) dll(x) (r E Q).

Xh

3.2. Duality of Commutative Hypercomplex Systems. Let L) (Q, m) be a commu

tative hypercomplex system Denote by Q the support of the Plancherel measure m.

For any <1>, 'P E PI... Q) and X E Q, we set

2(<1>, 'P, X) = J x(r) J <per) dm(<p) J ",(r) dm(",) dr. (S.35)

Q III 'P

Our idea is to take the function 2(<1>, 'P, X) as the structure measure of the dual hypercomplex system but, generally speaking, it does not satisfy all axioms (HI)-(H3) (axioms of positivity and finiteness are not true for 2(<1>, 'P, X)). Nevertheless, under certain additional restrictions this function turns into a structure measure.


Theorem 3.5. c(, 'Y, X) is a structure measure if and only if it belongs to

Co (Q) for any fixed <1>, 'Y E 1( Q) and the pointwise product of any two characters

<p, \j1 E Q is a positive definite function.

Under the conditions of Theorem 3.5, we can construct the dual hypercomplex sys

tem Ll (Q, m) associated with the structure measure 2(<1>, 'Y,X). The Plancherel

measure m is multiplicative for c(cp, 'Y, X). The dual hypercomplex system is a nor

mal hypercomplex system with involution X* (r) = x(r) and basis unity e == 1 (r). For

any point rE Q, we can construct an Hermitian character r(x)=x(r) of the dual hy-~

percomplex system. Denote by m the Plancherel measure corresponding to the dual ~

hypercomplex system and let Q = supp Jz. We say that these hypercomplex systems ~

are in duality whenever Q = Q. Let Ls be the left generalized translation operators associated with the hypercomplex

system Ll (Q, m). The hypercomplex system Ll (Q, m) satisfies the condition of sep

arate continuity if, for any fE Co(Q), the function (LsI)(t) is separately continuous.

Theorem 3.6 (duality theorem). Let L 1 (Q, m) be a commutative normal hyper

complex system with basis unity and with the following properties:

(i) e == 1 E Q;

(ii) the pointwise product X (r) \j1(r) of characters x' \j1 E Q is a positive definite function and the support of the measure corresponding to this function by

Theorem 3.4Zies in Q;

(iii) the function c(cp, 'Y, X) lies in Co(Q);

(iv) the hypercomplex system satisfies the condition of separate continuity.

Then the dual hypercomplex system L 1 (Q, m) also satisfies (i)-( iv) and duality

takes place.

Let us now describe an analog of the Fourier algebra for hypercomplex systems. As

sume that a commutative hypercomplex system L 1 (Q, m) satisfies the conditions of

Theorem 3.6. Denote A(Q)=L 2(Q,m)*L 2 (Q,m). It is easy to see that A(Q)C

Co (Q). Moreover, A (Q) is a Banach space with the norm

Theorem 3.7. If a commutative normal hypercomplex system with basis unity satis

fies the conditions of Theorem 4.6, then A (Q) is a Banach algebra with the ordinary


pointwise multiplication of functions. Moreover, the algebra A (Q) is isomorphic to

L1 (Q, m). It is called the Fourier algebra of the hypercomplex system L1 (Q, m).

3.3. Representations of Hypercomplex Systems and Approximation Theorem. Let L1 (Q, m) be a normal (not necessarily commutative) hypercomplex system with basis

unity. A family of bounded operators U = (Up)pE Q in a separable Hilbert space H is

called a *-representation of hypercomplex system if

(a) Ue = 1;

(b) U; = Up' (p E Q);

(c) for any ~ E H, the vector-valued function Q:3 P I---'» Up~ E H is continuous in the weak topology;

Cd) J c(A, B, r) Urdm(r)

Q

= J Updm(p) J Uqdm(q) for any A, B E $o(Q)· A B

A *-representation Up is called bounded if the function Q :3 P I---'» II Up II is bounded. Bounded *-representations of commutative hypercomplex systems coincide with Hermitian characters.

It is easy to see that the formula

Ux = J x(p)Updm(p) (x E L1 (Q, m»

Q

establishes a one-to-one correspondence between bounded *-representations and nonde

generate representations of the Banach algebra L1 (Q, m). If a normal hypercomplex system with basis unity satisfies the condition of separate continuity, then

where Ls are left generalized translations associated with the hypercomplex system, for

any bounded *-representation Ur . It is worth noting that this formula can be used as a

definition of representations of hypercomplex systems. The system of all bounded *-representations of a normal hypercomplex system with

basis unity is complete. There is a bijective correspondence between the set of positive

definite functions on Q and the set of unitary equivalence classes of bounded cyclic *

representations of hypercomplex systems. A positive definite function <p (r) and the

corresponding *-representation Ur are connected by the formula <per) = (Ur~O' ~O)H' where ~o is a cyclic vector.


A positive definite function <per) ;F. 0 is called elementary if all positive definite functions <PI such that <P - <PI is positive definite have the form <PI = A <P (0 ~ A ~ 1). There is a bijective correspondence between the set of elementary positive definite functions on Q and the set of unitary equivalence classes of irreducible bounded *-representations of the hypercomplex system.

Theorem 3.8 (approximation theorem). If a normal hypercomplex system LI (Q, m) with basis unity satisfies the condition of separate continuity, then the linear span of elementary positive definite functions is dense in C ( Q) in the topology of uniform convergence on compact subsets.

Corollary 3.2. The linear span of Hermitian characters of a commutative normal hypercomplex system with basis unity satisfying the condition of separate continuity is dense in C(Q) in the topology of uniform convergence on compact subsets.

3.4. References. Theorem 3.1 for the compact case appeared in [BKr3] and, for the locally compact case, in [Kall]. Theorem 3.3 can be found in [VaK2]. This theorem follows from the results of [VaL]. The other results in Subsections 3.1 and 3.2 are given in the present book (see also [BeKI]). The approximation theorem (Theorem 3.8) for the compact case was proved in [Vrel] and, for the commutative case, in [Ros2]. As far as we know, the general case of this theorem is published in the present book for the first time.

4. Hypergroups and Hypercomplex Systems

In this section, we analyze the relationship between hypercomplex systems and hypergroups.

4.1. Hypergroups. At the beginning of the 70-s, Dunkl [Dun2], Spector [Spel] and Jewett [Jew] independently introduced a concept of hypergroups. For convenience, we now briefly recall the relevant definition.

Let M b( Q) denote the set of bounded Radon measures and let M I (Q) be the set of

probability Radon measures in a locally compact space Q with the weak topology. The

locally compact space Q is called a hypergroup if

(GI) there exists a separately continuous mapping (convolution) Q x Q 3 (p, q) ~

8 p * 8 q E M I (Q); M b( Q) is an associative Banach algebra with convolution

given by the formula

(G2) supp (8p * 8q ) is compact;


(G3) there is an involutive homeomorphism Q :3 P ~ p* E Q such that (Dp * D q)* = o q> * 0 p*' where Jl* is defined by the equality

(Jl*,J) = f f(P*) dJl(p »; Q

(G4) there exists a point e E Q such that De * Dp = Dp * De = Dp (p E Q);

(G5) e E supp (Dp * D q) if and only if p = q* ;

(G6) the mapping (p, q) ~ supp (Dp * Dq) is separately continuous in the Michael to

pology (in the collection of compact subsets 1(Q) of Q, this topology is gen

erated by the subbasis {1(E 1(Q) I K n u"* 0, K C V}, where U and V are

open subsets of Q.

A locally compact space Q is called a weak hypergroup if conditions (Gl)-(G4)

are satisfied. Suppose that the left Haar measure m exists on the hypergroup Q, i.e.,

Op * m = m for all p E Q (in the compact, discrete, and commutative cases, the exis

tence of Haar measure was proved in [Jew] and [Spe4]). If Dp * m = m * Dp = m, (i.e.,

the left Haar measure coincides with the right Haar measure), the hypergroup is called

unimodular. There exists a family of generalized translation operators (L pf) (q) = (D p

* Dq,J) (jE Coo(Q» associated with the hypergroup Q. The Banach space LdQ,m) is a Banach algebra with convolution given by the formula

(j* g)(p) = f (Lpf)(q)g(q*)dm(q). Q

The next theorems is an evident consequence of (G 1)-(G6).

Theorem 4.1. If Q is a unimodular hypergroup, then L 1 (Q, m) is a normal hy

percomplex system with basis unity and condition of separate continuity.

It follows from Proposition 4.1 that the class of normal hypercomp1ex systems is broader than the class of unimodular hypergroups. Hence, all results established for hypercomplex systems of this sort are true for unimodular hypergroups.

The assertion converse to Theorem 4.1 is not true because there is an example of a normal hypercomplex system with basis unity associated with generalized Chebyshev polynomials which is not a hypergroup (the authors are deeply grateful to A. Schwartz for this example). Nevertheless, it follows from Theorem 4.2 that hypergroups and hypercomplex systems are very closely related and, roughly speaking, they are the same algebraic objects.


Theorem 4.2. If the normal hypercomplex system L1 (Q, m) with basis unity satisfies the condition of separate continuity, then Q is a weak hypergroup. If Q is discrete, then the definitions of hype rg ro up and normal hypercomplex system with basis unity coincide.

From the standpoint of applications, hypercomplex systems are useful in duality theory (see Theorems 3.5 and 3.6), while hypergroups are useful for the investigation of more detailed structures (e.g., for the analysis of subhypergroups, etc.). Generally speaking, this distinction is connected with the fact that conditions (G5) and (G6) are not satisfied for hypercomplex systems.

4.2. References. The fundamentals of the theory of hypergroups are exposed in [Jew], [Rosl], and [Hey2]. There is also a forthcoming book by Bloom and Heyer. The relationship between hypercomplex systems and hypergroups was analyzed in [BeKl].

5. Generalizations

In the present section, we briefly discuss the existing generalizations of the concept of hypercomplex systems. One generalization of this sort (hypercomplex systems with complex-valued structure measures) was described in Subsection 1.9. The major part of the results presented in Section 3 are true for hypercomplex systems with complex-valued measures (with the exception of duality theory, because the Plancherel measure is not multiplicative for such hypercomplex systems). Another class of compact and discrete hypercomplex systems with real structure measures, good duality, and meaningful harmonic analysis was proposed by Vainerman [Vai2].

For this class of hypercomplex systems, the unity, in general, does not belong to the basis. In what follows, we describe the so-called "quantized hypercomplex systems".

It is well known that the duality theory of locally compact noncommutative groups can be formulated in terms of Kac algebras [Kac] or, equivalently, in terms of a special class of Hilbert bialgebras. Thus, the functions defined on a group with ordinary mUltiplication (commutative operation) and convolution (noncommutative operation) constitute a bialgebra of the indicated type. The transition from special bialgebras of this sort to general bialgebras with two noncommutative operations may be regarded as "quantization" of the group. In this case, we have the following principle of conformity: Under certain conditions, the commutativity of one operation of the bialgebra enables one to realize this bialgebra as a bialgebra of functions on the group. Quantized hypercomplex systems appear if one wishes to realize a similar construction for hypercomplex systems. The axioms of quantized hypercomplex systems are simpler than the axioms of Kac bialgebras. If one of the operations is commutative, quantized hypercomplex systems can be realized as objects close to hypercomplex systems with locally compact bases.

5.1. Principle of Conformity. Let 1) be a Hilbert bialgebra such that Lu and Ln are left W* -algebras associated with the Hilbert algebras (1), U) and (1), n), let Lu'


and Ln- be the preduals, and let En (E u) be the closure of the linear span of the oper

ators Ln(a Ub) (Lu(a nb») (a,bE lj) in the uniform norm.

A Hilbert bialgebra lj) is called a quantized hypercomplex system if

(A2) Ln- is subalgebra of Eu and Lu. is subalgebra of En.

Theorem 5.1 (principle of conformity). Suppose that the operation n in a quan

tized hypercomplex system lj) is commutative. Then there exist a locally compact

space Q, nonnegative measure m, and involutive homeomorphism Q :3 t ~ t* E Q such that 1) m * = m; 2) En is isomorphic to Coo (Q); 3) this isomorphism can be

extended to isometries of H onto L2 (Q, m) and L n, onto L] (Q, m ) .

It follows from Theorem 5.1 that LJ (Q, m) is endowed with the structure of *-alge

bra from L n_. Thus, L] (Q, m) is an object close to a hypercomplex system with locally compact basis. Theorem 3.3 remains true for this algebra [VaK2].

By interchanging the operations U and n in a quantized hypercomplex system, we

obtain a new quantized hypercomplex system lj)' (the dual quantized hypercomplex

system). It is obvious that lj)" = lj). If one of the operations in the quantized hypercomplex system is commutative, the proposed method allows us to construct an object dual to, in general, a noncommutative hypercomplex system with locally compact basis. If both operations are commutative, by virtue of Theorem 5.1, we can realize this dual

object as an object L J ( Q, m) close to a hypercomplex system with locally compact

basis. Here, Q consists of *-characters of the *-algebra L] (Q, m). As a result, we arrive at a generalization of the Pontryagin duality principle for hypercomplex systems.

5.2. Positivity. If we wish to establish such properties as the existence of the basis unity and multiplicative property for the measure m, it is necessary to introduce the following additional axiom:

(A3) the mappings Lu- :3 00 ~ (1,00) and Ln·:3 (0 ~ (I, (0), where I is the iden

tity operator, can be extended to characters of the C* -algebras En and Eu'

Proposition 5.1. Assume that a quantized hypercomplex system satisfy (A3) and the

operation n is commutative. Then

(i) there exists a point e E Q (basis unity) such that

(j* g)(e) = f f(t)g(t)dm(t) (f, g E L 2(Q, m»;

Q


(ii) the measure m is multiplicative

f (j* g)(t)dm(t) = f f(t)dm(t) f g(s)dm(s) (j, g ELI (Q, m)). Q Q Q

We now introduce an axiom under which (in case where one operation in a quantized hypercomplex system is commutative) the convolution of positive functions in Ll (Q, m) is positive.

An operator A E Lu is called n-positive definite if (A, ill n ill n) ;;:: 0 for all ro E

Lu •. Operators that are U-positive definite can be defined in exactly the same way. We introduce the axioms

(A4) if A, B are n-positive definite, then AB is n-positive definite;

(A4)' if A, B are U-positive definite, then AB is U-positive definite.

Theorem 5.2. Assume that a quantized hypercomplex system satisfy (A3), (A4) and

that the operation n is commutative. Then L 1 (Q, m) is a Banach *-algebra and

the convolution of positive functions is positive.

Hence, the quantized hypercomplex systems satisfying the conditions of Theorem 5.2 have all properties of normal hypercomplex systems with locally compact basis and basis unity, except that the convolution of functions from Co(Q) is not necessarily compactly supported. We need axiom (A4)' in order for a class of such hypercomplex systems to be closed under duality.

The class of quantized hypercomplex systems contains the class of Kac bialgebras:

If the connecting operator W of a quantized hypercomplex system Ij) is unitary and

( W @ /)( / @ W)( W* @ /) = (I @ W)( 0' @ I) (I @ W)( 0' @ I),

then Ij) is a Kac bialgebra (here, 0' is the operator of permutation).

5.3. References. All results of this section can be found in [VaK2] and [Kac]. In [Vai7], the construction of quantized hypercomplex systems was generalized to the nonunimodular case. An object close to quantized hypercomplex systems, namely, a pair of *-algebras in duality was studied in [Veri] and [Ker2]. For related results, see [Kir]. In

the finite-dimensional case, quantized hypercomplex systems were also studied (in another setting) in [Vai 1].

6. Remarks on Terminology

It seems reasonable to standardize the terminology used in the theories of generalized translations, hypercomplex systems, hypergroups, convolution algebras, etc.


We propose to do the following: In our opinion, the terminology should be based on the notion of hypergroups. A (fi

nite) hypergroup is a basis Q of a finite-dimensional algebra (of a hypercomplex sys

tem) with nonnegative structural constants. Its points p, q, r, ... can be multiplied as in a group but, as a result of this multiplication, p * q we obtain not a point of Q but a collection of points with various nonnegative values which can be regarded as a vector with nonnegative coordinates. This vector gives the measure of each point in the product

(S.36)

The meaning of the word "hypergroup" is connected with the fact that we construct something over a group and this is reflected by the prefix "hyper". This term also reflects the opinion of J. Delsarte who suggested to use it in his constructions. It should be noted that the old term "hypercomplex systems" used for the objects of the type (S.36) is somewhat arbitrary because they are quite distant from the system of complex numbers and are cannot be regarded as its direct generalizations.

Special cases of the introduced notion are evident: commutative hypergroups, hyper

groups with unity (lying, clearly, in Q; when the unity of an algebra is not in the basis, we may say that this is a hypergroup with generalized unity), and so forth.

If Q is infinite (countable and discrete), one may say that hypergroups are infinite

(the use of series (S.36) does not cause any difficulties because the coefficients C;q are

nonnegative ).

If Q is infinite and equipped with a nontrivial topology, we can consider compact, locally compact, and other types of hypergroups depending on the type of topological

space Q. In this case, the algebras of measures, functions, or distributions on Q can be used as the original algebra. By analogy with the case of groups, this algebra should be called a hypergroup algebra. In our opinion, there should be no preferences in introducing topologies in hypergroup algebras. This is important because, depending on expected results and the desire to include a great number of examples, the researchers may wish to use different systems of axioms. Thus, to construct duality theory, it is more convenient to use hypercomplex systems. On the other hand, hypergroups are preferable for the investigation of subhypergroups, induced representations, and so on. It seems that the only requirement necessary for a hypergroup algebra is that the product of positive elements of the hypergroup algebra must be positive.

The objects introduced by using different special systems of axioms should be distinguished by prefixes and parentheses. Thus, a hypercomplex system may be called a lo

cally compact L) -hypergroup. This emphasizes the fact that the Banach algebra L) (Q, m) is a hypergroup algebra. Hypergroups may be called M -hypergroups due to the fact that the convolution of measures is defined.

If the structural constants are not positive, one may say that hypergroups are real, complex, etc. Hence, hypercomplex systems with complex structural constants should

be called complex L)-hypergroups.


For finite hypergroups, generalized translation operators acting upon the functions

Q :3 q t---7 f (q) E ([ can be introduced by the formula

(Tpf)(q) = L c;q!(r) (p, q E Q). (S.3?) rEQ

For topological hypergroups, this formula should be appropriately modified (cf. (S.12». Certainly, these objects are very important but it does not seem reasonable to use them as a basic concept. Here, the algebraic side of the construction is more important than the operators (S.3?) associated with this construction (thus, it is not convenient to define the

group 1R by using the ordinary translation operators, although this is, of course, possible).

BIBLIOGRBPHICBL nOTES

We suppose that the reader is familiar with general ideas of the books mentioned in this paragraph. The theory of measure on locally compact spaces is presented in [Bou2] and

the theory of linear topological spaces in [Sch]. Elements of the theory of groups and Lie algebras necessary for our presentation can be found in [Hell], [HeI2], and [Dix3]. For the theory of normalized algebras and harmonic analysis on locally compact groups,

see [Dix2], [Nail], or [HeR]. Necessary facts from the theory of W* -algebras can be found in [Dixl] or [BrR]. The theory of rigged Hilbert spaces and the spectral projection theorem are presented in [BerlO] and [BeKo).

Chapter 1

Section 1. Commutative Banach algebras L} (Q, m) with respect to generalized convolutions of functions constructed for a given class of generalized translation operators were studied in early works by Levitan [Lev2]-[LevS]. For these algebras, he established the Bochner andPlancherel theorems, the inversion formula, and a version of the Pontryagin duality. In a somewhat different form, discrete infinite-dimensional hypercomplex systems appeared in [Haa]. Normal hypercomplex systems with continual basis were introduced by Berezansky and S.Krein [BKrl]-[BKr4]. In these works, an important condition of positivity of the structural measure guaranteeing the possibility of construction of nontrivial harmonic analysis appeared for the first time. Commutative hypercomplex systems with compact and discrete bases were studied by Berezansky [Ber 1][Ber9] and Berezansky and S.Krein [BKrl]-[BKr4]. The case of general locally com

pact bases was only outlined. The results of this series of works were summarized in the survey [BKr4]. The theory of commutative hypercomplex systems with locally compact basis was constructed by Berezansky and Kalyuzhnyi [BeK 1 ]-[BeKS]. Its elements were presented in the monograph [PeKo]. The theorem on existence of a multiplicative

measure in the case of a compact basis was proved by Berezansky and S.Krein [BKr3] and, in the locally compact case, by Kalyuzhnyi {Kall]. The conditions of the indicated theorem in the case of a compact basis were weakened by Bakhtin [Bak]. The concept of Hilbert bialgebras was introduced by Kac [Kac]. All results presented in Section I be

long to the authors except for Theorem 1.7 which was in fact established by Jewett [Jew].

431

432 Bibliographical Notes

Section 2. At present, the Delsarte-Levitan theory of generalized translation operators is very well developed and, hence, it is practically impossible to give a survey of all relevant results within the framework of these notes. We refer the reader to the monographs by Levitan ([Lev9] and [LevlO]) and reviews by Vainerman [VaiS] and Litvinov [Lit3] (In [Lit3], Litvinov, instead of the term "generalized translation operators," uses the term "hypergroup" introduced by J.Delsarte.) Fundamentals of the theory of hypergroups can be found in the work by Jewett [Jew] and surveys by Ross [Rosl] and Heyer [Hey2]. The existence of invariant measures on compact semihypergroups was established by Onipchuk [Oni]. It is worth noting that, parallel with topological hypergroups, serious attention is given to the investigation of abstract hypergroups. Each hypergroup of this sort is a set Q with associative binary operation on the set of its subsets. The theory of abstract hypergroups was developed by Corsini [Corl], [Cor2], McMullen [McMl], McMullen and Price [McPl], [McP2], and other researchers (see also [Rosl]).

All statements presented in Section 2, with the exception of the general properties of generalized translation operators and Theorem 2.2 established for hypergroups by Jewett [Jew], belong to the authors. (Theorem 2.1 proved by Kalyuzhnyi is published for the first time.) The formalization of the process of reduction was proposed by Litvinov [Lit3] .

Let us now give a brief description of other objects related to hypercomplex systems. First of all, one must mention the association schemes studied by P.Delsarte [DeP] (see also [Bal]). Each of these schemes is associated with a finite-dimensional hypercomplex system (a so-called Bose-Mesner algebra). Further, there are numerous works in the theory of probability on semigroups of measures with respect to generalized convolution ([Voll], [VoI2], [Dav], [Ken], [Kenn], [Kin], [Lam], lOst], [OTr], [Trul]-[TruS], [RoU], [Urb], [Cvd], and many others). We want to especially mention the paper [Kin], where, apparently, for the first time, the principal ideas of probability theory on hypergroups were formulated. The relationship between hypercomplex systems and Urbanic algebras [Urb] was analyzed by Olshanetskii [Olsl]. Generalized convolutions were also studied in [Dim], [GhM], [Pyml], and [Pym2].

Section 3. In a somewhat different axiomatics, Theorems 3.1-3.3 were established in terms of generalized translation operators by Levitan [Lev3], [Lev4]. For hypercomplex systems, they were proved by Berezansky and S.Krein [BKr4] and Berezansky and Kalyuzhnyi [BeKl]. The duality of commutative hypercomplex systems in the compact and discrete cases was established by Berezansky [Ber9]. In the locally compact case, it was established by Berezansky and Kalyuzhnyi [BeKI]. For hypergroups, duality of the same type was established by Jewett [Jew] and Spector [Spell. [Spe2] (see also [Bur]) and, for real hypercomplex systems, by Vainerman [Vai2), [Vai3] (see also bibliographical notes to Section S in Chapter 1 below). Unlike hypercomplex systems, in the case of hypergroups, one fails to establish conditions under which the dual object is a hypergroup. Examples of hypergroups whose dual objects are not hypergroups were presented, e.g., in [Jew] (see also Sections 2 and 3 in Chapter 2).

The results obtained by Voit [Voi I] essentially complement Theorem 3.S. Namely, for any commutative hypergroup (Q, *), there exists a unique positive essential charac

ter ao. If ao:t 1, then the law of multiplication in Q can be "corrected" by setting

Bibliographical Notes 433

in such a way that (Q, 0) is also a hypergroup and its unit character is essential. Under

certain conditions imposed on (Q, 0), the support of the Plancherel measure coincides

with the set of all its characters. Thus, by "correcting" the hypergroup SL 2( il) II SU (2)

as indicated above, we obtain the hypergroup /(3) II SO (3), where /(3) is the group of

isometries of IR 3 . In order that the unit character be essential, it is necessary and suffici

ent that there exist a sequence of functions n * ~ (t) (n E Co( Q») approximating

the unit element uniformly on compact sets ([GaG3]). As far as the subsequent development of the technique of positive characters and its applications are concerned, see the works by Voit [Voi3 ]-[Voi8].

The concept of A-systems was introduced by Berezansky in his candidate's degree thesis [Ber9] and, independently, by A.Schwartz [Schw3]. Representations of hypergroups and generalized translation operators were studied by Jewett [Jew], Vrem [Vrel], Bloom and Heyer [BlH2], Levitan [Lev9], [LevlO], Litvinov [Litl]-[Lit3], Vainerman [Vai2]- [Vai3], Berezansky and Kalyuzhnyi [BeK3], [BeK4]. Theorem 3.11 belongs to Vrem [Vrel] and Theorem 3.12 belongs to Jewett [Jew].

The theorem on approximation (Theorem 3.13) for compact hypergroups was proved by Vrem [Vrel]. For commutative hypergroups, under certain additional restrictions, this theorem was proved by Ross [Ros2]. As far as we know, for noncom mutative hypercomplex systems with locally compact basis, the theorem on approximation is published in the present book for the first time. Vainerman generalized this theorem to the case of real hypercomplex systems with compact basis [Vai2], [Vai3]. The space of test and generalized functions were introduced and studied in the language of hypercomplex systems by Berezansky and Kalyuzhnyi [BeK2], [BeK4] (for the analysis of representations of commutative hypercomplex systems). Vrem [Vrel] extended the Peter-Weyl theory to the case of compact hypergroups and Vainerman [Vai2], [Vai3] extended this theory to the case of real hypercomplex systems with compact basis. It is worth noting that elements of the Peter-Weyl theory for compact generalized translation operators were established Levitan [Lev9], [LevIO] (see also [Mat]). Hauenschild, Kaniuth and

Kumar investigated representations of central hypergroups, i.e., hypergroups Q such that

QI Z is a compact set, where Z is the intersection of the maximal subgroup with the center of the hypergroup Q [HKK] (see also Section 4).

Various problems in the theory of functions almost periodic with respect to generalized translation operators were studied in [De13], [Lev5], [Ber3], [Ber5], [Ber6], [Ber8], [Lasl], and [WoI2]. Spectral synthesis on hypergroups and generalized translation operators was analyzed in [Vogl], [Vog2], [KuSI], [Litl], [Lit3], [Chi], [ChKI], [ChK2], and [ChR]. Let us also mention some works devoted to the investigation of various problems of harmonic analysis on hypergroups: [BlH2], [Vre2, Vre3], [GeK], [Dunl][Dun3], [DuRl], [DuR2], [Jew], [Las3], [Las4], [LanWI], [LanW2], [PIal], [Pla2],

[Ras], [Rosl], [Ros2], and [KhiJ. Numerous works are devoted to the analysis of various problems of probability theory on hypergroups. These are [Blo I], [Bl02], [BlH l]-


[BlHS], [Voi2], [Voi3], [Gall], [GaI2], [GaGI]-[GaG3], [GaR], [GebI], [Geb2], [Ze3], [Ze4], [LasS], [Las7], [Letl], [0Is2], and [HeyI]-[Hey6]. For surveys of the results obtained in this field, see [Hey I], [Hey2], and [LeU].

Section 4. The concept of subhypergroups was introduced by Jewett [Jew]. All results in Subsections 4.1 and 4.2, except Theorem 4.2 proved by Vrem [VreI], belong to Jewett. Subhypergroups were also studied by Hauenschild [Hau] and Vrem [Vre4] and

[V reS] . If J.l E Mt (Q) is an idempotent measure on a hypergroup Q, then its support is a compact subhypergroup and Q is its normalized Haar measure [Jew]. Idempotent measures on commutative hypergroups were studied by Bloom [BloI] (see also [B1H2]). The definition of morphisms of hypercomplex systems was given by Vainerman [Vai2], [Vai3]. Jewett [Jew] used a notion of orbital morphisms for the construction of new examples of hypergroups. He also mentioned that orbital morphisms are not necessarily homomorphisms. The major part of results presented in Subsection 4.3 belongs to Kalyuzhnyi. They are published for the first time. The concept of joins of hypergroups was introduced by Jewett [Jew] and studied by Vrem [Vre4] (see also [DuR2]).

Section 5. The results presented in Subsections S .1-5.3 belong to Vainerman and Kalyuzhnyi [VaKI], [VaK2]. Theorem 5.3 was proved by Vainerman and Litvinov [VaL]. In [Kac], it was indicated that Kac algebras admit a generalization in terms of Hilbert bialgebras (in connection with generalized translation operators). At the same time, the relationships between their operations were not analyzed. A similar object, namely, a pair of dual *-algebras, was investigated by Vershik [Veri] Kerov [KerI], [Ker2], and Kirchberg [Kir]. The finite-dimensional case was studied by Vainerman [Vai I], [Vai4]. Harmonic analysis on Kac algebras is developed in the monograph [EnS]. In recent years, many researchers work in the field of quantization of groups in terms of deformations of algebras. The relationship between quantized groups and Kac algebras is discussed in [Dri] and [VailO]. The results presented in SubsectionS.4 belong to Vainerman [Vai2], [Vai3], [Vai5]. Another generalization of locally compact hypergroups was suggested by Gebuhrer [Geb2], [Geb3].

Chapter 2

Section 1. The center of the group algebra of a compact group and the theory of characters belong to the most developed branches of harmonic analysis and the theory of representations (see [HeR], [Nail], [Nai2], etc.). The structure of hypercomplex systems in ZL 1 (G) was discovered by Berezansky [BerI] (see also [Gell] and [BeG]). Iltis [Ilt]

and Hewitt and Ross [HeR] regarded G as an algebraic hypergroup (see also [McMl]

and [Rot]). The structure of a hypergroup on G was studied by Kawada [Kaw] , Berezin and Gelfand [BeG], and Helgason [He13]. Rider [Rid!], [Rid3] used the hypergroup


structure on {; for the characterization of idempotent measures on the classes of conjugate elements (see also [Rag 1]). Gallardo [Gall], Gallardo and Ries [GaR], and Eymard

and Roynette [EyR] studied some problems of probability theory on SU(2). The work

of Blyumin [Bly] is devoted to the application of the hypergroup structure on {; to the problem of digital analysis of signals. Theorem 1.1 belongs to Berezansky [Bed]. The detailed presentation of the theory of duality of Tannaka and M.Krein can be found in [HeR]. The proof of the second part of the Tannaka-Krein duality theorem based on the duality theorem for hypercomplex systems is published in our book for the first time. In Subsection 1.6, we present well-known results from the theory of semisimple groups and Lie algebras (see, e.g., [Nai2], [Hell], and [Hel2]). In Subsections 1.7 and I.S, we present a part of the work by Berezin and Gelfand [BeG].

Section 2. Positive definite functions on a sphere were studied by Schoenberg [Scho]. The commutativity of the algebra of biinvariant functions associated with Riemannian symmetric spaces was established by Gelfand [Gell] (see also [BeG]). M.Krein [Krel] independently investigated another class of Gelfand pairs. Hypercomplex systems connected with these Gelfand pairs were studied by Berezansky [Ber9]. The unimodularity of the group G was discovered by Berg [Ber]. Harmonic analysis on Gelfand pairs was constructed by Godement [God] (see also [Mau]). Theorems 2.1-2.2 can be found in the well-known books by Dieudonne [Die1] and S.Lang [Lan]. Theorem 2.3 belongs to Gelfand [Gell] and Berezansky [Ber9]. Theorem 2.4 was established by Gelfand and Naimark [GeN] (see also [BeG]). Spherical functions on symmetric Riemannian spaces of compact and noncompact types were described by Harish-Chandra [Har]. Note that, for Riemannian symmetric spaces of these types, the explicit form of the Plancherel measure is known (for the rigorous formulation of this result and relevant bibliography, see [HeI2]).

The problems of harmonic analysis on Gelfand pairs were studied in [Gan], [Die1], [Die2], [Kool], [Kril], [Kri2], [Kuc1], [Kuc2], [Kus], [Car], [McDI], [May], [MizIJ, [Miz2], [Nus], [Ras], [Rag2], [Rag3], [RaR], [Rid2], [Far], [FIKl], [Hell], [HeI2], and [Khi]. Dijk [Dij] and Thomas [Tho I] considered generalized Gelfand pairs. Problems of probability theory on Gelfand pairs were studied in [BinI], [Bin2], [Letl], [Let2], [Hey I], [Hey5], and [Hey6] (see also the bibliography contained therein).

Section 3. Generalized translation operators associated with orthogonal polynomials were introduced by Levitan [Lev6] and then studied by many authors (see, e.g., [Zhy], [Rafl]-[Raf6], and [Fed]). Discrete hypercomplex systems constructed in terms of orthogonal polynomials appeared for the first time in the works by Berezansky [Ber4], [Ber5], and [BerS]. He proved Theorem 3.1 , 3.2, and 3 .S-3 .11. Insignificant corrections were made by Berezansky and Kalyuzhnyi [BeK5]. Hypercomplex systems of this sort were also studied by Lasser [Las2]-[Las4], [Las6] and A.Schwartz [Schw3], [Schw4]. Their generalizations were investigated by Vainerman [Vai9]. Theorem 3.3, in fact, belongs to A.Schwartz [Schw4] (see also [BIS]). Relations (3.12)-(3.15) and Theorem 3.5 were established by Lasser [Las2]. Theorem 3.6 was proved by Askey [AskI] (see also


[Kus], [Gasl]-[Gas3], [Hyl], and the survey [Ask2]). The results of Subsection 3.5 are well-known. In addition to the works mentioned in the body of Section 3, examples of hypercomplex systems and generalized translation operators associated with orthogonal polynomials were constructed and studied in [Bav], [Voi2], [Voi4], [YogI], [WoI2], [GaM], [Glal], [Gla2], [GIR], [DuRl], [Dib], [Kenn], [Kool], [McC], [Mar3], [Soa], [AsWl], [AsW2], and [AsG2]. Note that the availability of multiplication formulas in many of these examples is explained by the fact that the corresponding orthogonal polynomials are matrix elements of representations of compact quantum groups (see, e.g., [Ko03], [NoM!], [NoM2], and [VaW]) or algebras with special quadratic relations ([GrZl] and [GrZ2]). Generalized translation operators associated with matrix-valued orthogonal polynomials were studied in by Osilenker [Osi2].

Section 4. An example of generalized translation operators associated with the Bessel differential operator was constructed by J .Delsarte [Dell]. Levitan [Lev 1] constructed generalized translation operators for a differential operator

d2 ±-2 +q(x)

dx

given on a semiaxis in the case where the potential q(x) is analytic. Povzner generalized this result to the case of continuous potentials. The conditions imposed on the potential were then weakened by Leblanc [Leb] and Hutson and Pym [HuP2], [HuP3]. The classification of generalized translation operators corresponding to the Sturm-Liouville operator was given by Gurevich [Gurl] (see also [Leb], [CoS3], [CoS4], [Zeu2], and [Zeu3]). The conditions guaranteeing that generalized translation operators preserve positivity and admit the construction of hypercomplex systems were established by Berezansky [Ber7] (see also [Boc1], [Boc2], and [Wei]). Vainerman [Vai7] studied generalized translation operators constructed for a Hamiltonian system of two differential equations on a semiaxis (in particular, for the Dirac system). Examples of generalized translation operators and convolution algebras associated with partial differential equations can be found in [Tri3], [Hull], [HuI2], [SteI], and [Ste2]. The results presented in Subsections 4.2-4.4 belong to Berezansky [Ber7]. Spectral analysis and synthesis on hypercomplex systems associated with a regular Sturm-Liouville operator are presented in the work by Sinyavskii [Sin].

Chapter 3

Section 1. The fundamentals of Lie theory for generalized translation operators were laid by JDelsarte [Dell]-[DeI4] and Levitan [Lev7], [Lev9], and [LevIO]. In our presentation, we mainly follow the survey by Litvinov [Lit3]. He gave the general definition of infinitesimal objects. All results in Subsections 1.1 and 1.2 and Theorem 1.2


belong to Litvinov. The interpretation of qJj '" Q) as a space tangent to the manifold

qJj (Q) at the point De published here for the first time was suggested by Berezansky. All results in Subsection 1.4 belong to Levitan [Lev9] and [Lev 10]. The results of Subsection 1.5 belong to Gurevich, except Theorem 1.4 proved by Litvinov [Lit3]. A fairly general Lie theory of generalized translation operators was constructed by Karasev [Karl]-[Kar3] by using asymptotic methods (see also [Mas], [KaMl], and [KaM2]). Note that, in the framework of this theory, the axiom of associativity for generalized translation operators and relationships between their right generators hold asymptotically. Among applications of the Lie theory of generalized translation operators, one should especially mention algebraic topology (see [Buk] and [BKh]; concerning the theory of multi-valued groups, see also [Gur3]).

Section 2. Theorems 2.1 and 2.2, in fact, belong to JDelsarte [DeI4]. All other results in Subsection 2.1 belong to Kalyuzhnyi [Ka14]. In Subsection 2.2, we present a part of the paper by Gurevich [Gur2]. The results in Subsection 2.3 belong to Podkolzin [PodlJ-[Pod4]. In these works, Podkolzin also constructed the adjoint action of the hy

pergroup Gil H and its infinitesimal algebra and applied his results to the exact integration of nonlinear differential equations. Note that, in the case Riemannian symmetric spaces, the infinitesimal algebra of the hypercomplex system L j (G, H) was described by Gelfand [Gell], [GeI2] (for a more detailed presentation, see the books [Hell] and [HeI2]).

Section 3. The results of Section 3 belong to Kalyuzhnyi [KaI2], [Ka13]. A close point of view on the generator of the family of commutative generalized translation operators can be found in [HuPl], [Fisl], and [Fis2]. Vainerman applied formulas analogous to (3.2) to the construction of the family of unbounded generalized translation operators for a given Jacobian matrix [Vai9]. Formulas similar to (3.2) were established by Bloom and Heyer [BlH4] for the generator of a contracting semigroup of operators associated with a one-parameter semigroup of probability measures on a hypergroup. For the description of one-dimensional hypergroup, see [Schw5], [Zeu2], and [CoS3].

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Selected Papers Close to the Subject of the Monograph

Which Appeared after the Publication

of the Russian Edition of the Book

Berezansky, Yu. M.

1. Nuclear spaces of test functions connected with hypercomplex systems and representations of such systems, Contemp. Math. 183 (1995),15-20.

2. A connection between the theory of hypergroups and white noise analysis, Rep. Math. Phys. 36 (1995), 215-234.

3. A generalisation of white noise analysis by means of the theory of hypergroups, Rep. Math. Phys. (1996) (to appear).

4. Infinite-dimensional non-Gaussian analysis and generalized translation oper

ators, Funkts. Anal. Prilozh. 30 (1996), No.4, 61-65.

5. Infinite-dimensional analysis associated with generalized translation operators, Ukr. Mat. Zh. (to appear).

478 References

Berezansky, Yu. M., and Kalyuzhnyi, A. A.

1. Hypercomplex systems and hypergroups: connections and distinctions, Contemp. Math. 183 (1995), 21-44.

Berezansky, Yu. M., and Kondrat'ev, Yu. G.

1. Non-Gaussian analysis and hypergroups, Func. Anal. Appl. 29 (1995), N03, 51-55.

2. Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis, Meth. Func. Anal. Topol. 2 (1996), No.2.

Bloom, W. R., and Heyer, H.

1. Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin-New York, 1995.

Chapovsky, Yu. A.

1. Gelfand pair associated with a Hopf algebra and a coideal, Ukr. Mat. Zh. 46 (1994),No.8,1055-1067.

Chapovsky, Yu. A., and Vainerman, L. I.

1. Gelfand pair of compact quantum groups, Func. Anal. Prilozh. 29 (1995), No.2,67-71.

Kalyuzhnyi, A. A.

1. Algebras with non-quadratic relations associated with Bessel hypergroups, Meth. Func. Anal. Topol. (to appear).

Podkolzin, G. B.

1. An infinitesimal algebra of the hypergroup generated by double cosets and

nonlinear differential equations, Contemp. Math. 183 (1995), 287-297.

Podkolzin, G. B., and Vainerman, L. I.

I. Quantum Stiefel Manyfold and Double Cosets of Quantum Unitary Group, Preprint Universite d'Orleans, 1996.

References 479

Vainerman, L. I.

1. Gelfand pairs of quantum groups, hypergroups and q-special special functions, Contemp. Math. 183 (1995), 373-394.

2. Hypergroup structures associated with Gelfand pairs of compact quantum

groups, Asterisque 1995 (1995),231-242.

3. On the Gelfand pair associated with the quantum groups of motions of the

plane and q-Besselfunctions, Rep. Math. Phys. (to appear).

SUBJECT InDEX

Algebra convolution, 67 Fourier, of a hypercomplex system, 97 Hilbert, 37

perfect, 158 hypergroup, 321

of reduced generalized translation operators, 321

infinitesimal, 325 of reduced generalized translation operators, 325

Krein, 181 complete, 183

Lie, 189 compact, 194 generalized, 365 real form of, 194 semisimple, 190

of infinitesimal left translations, 326 of infinitesimal right translations, 325 of Poincare-Birkhoff-Witt type, 349 Poincare-Birkhoff-Witt, 349

Annihilator, 27 Approximative unit, 28 A-system of functions, 100

Basis of hypercomplex subsystem, 121 of hypercomplex system, 7,9 Weyl, 192

Basis unity, 28 strong, 28

Bialgebra

Hilbert, 37 topological, 322

dual, 323 Binding operator, 38

Canonical commutation relations, 396 Cartan matrix, 192

481

Cartan numbers, 192 Cartan subalgebra, 190 Center of a group algebra, 167 Character

of hyper complex system, 13 essential, 84 generalized, 13 Hermitian, 25

of representation of a group, 169 Complex structure, 194 Complexification of a real Lie algebra, 194 Comultiplication, 323 Condition

finiteness, 46 of separate continuity, 55

Convolution of a measure with a function, 88 of measures, 60 of functions, 10, II

Counit, 324

Decomposition Cartan, 194,195 Iwasava, 195

Direct product of hypercomplex systems, 133 Duality

of commutative hypercomplex systems, 87

of quantized hypercomplex systems, 152

Even subsystem, 33

Fourier transformation, 72,89, 120,159 Formula

Freudenthal, 197 inversion, 77, 159 of multiplication for orthogonal

polynomials, 266 Plancherel, 159

W ey I, for a character, 197

482

Function biinvariant, 223 central, 167 modular, 126 positive definite, 69

elementary, 115 spherical, 227

Gelfand pair, 225 symmetric, 227

Generalized translation operators, 43 adjoint, 46 commutative, 44 Delsarte, 247 generator of, 325

left, 325 right, 325

infinitely differentiable, 317 involutive, 45 preserving positivity, 45 preserving unit element, 47 real, 45 reduced, 44 reduction of, 44 weakly continuous, 45

Group Lie, 189

semisimple, 190 reductive, 348 representing, 185 Weyl, 193

Homomorphism of hypercomplex systems, 128

Hypercomp1ex subsystem, 121 commutative, 122 normal, 128 supernormal, 128 unimodular, 126

Hypercomplex system, II commutative, 12 Hermitian, 20 normal, 20 quantized, 144 real, 161,162

Hypergroup, 60 commutative, 60 Delsarte, 246 generalized, 161 quotient, 128 weak, 60

Subject Index

Joint analytic vector, 396 Join of hypergroups, 136

Killing form, 190

Maximal subgroup, 127 Maximal torus, 196 Measure

left-in variant, 48 strongly, 46

multiplicative, 10 Plancherel, 78,161,162 right-invariant, 48

strongly, 46 structure, 9 unimodular, 45,126

Multiplicity of weight, 193

Orthogonality relations, 172

Quasispectrum, 159

Representation of hypercomplex system, 105

bounded, 108 by unbounded operators, 106 irreducible, 112 regular, 105, 106

of group, 168 adjoint, 190 completely reducible, 170 finite-dimensional, 168 irreducible, 170 of Class I, 232,233 unitary, 168

SchrOdinger, of canonical commutation relations, 397

weight of, 193 dominant, 194

Root, 190 simple, 192

Semidirect product, 134 Space

Frechet, 339 Mackey, 339 principal, 44 reflexive, 339 Riemannian, symmetric, 226

Subhypergroup, 123

Symmetry, 366

Subject Index

Theorem

on duality, 90

on realization of a quantized hypercomplex

system, 147

Poincare-Birkhoff-Witt, 326

Trace, 158

canonical, 159

faithful, 158

normal, 158

semi finite, 158

Topology in the space of Radon measures

vague, 9 weak, 9

Mackey, 339 Michael, 60 projective, 339 strong, 339 weak, 338

WeyI chamber, 193 dominant, 193

483

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A.J. Jerri: Linear Difference Equations with Discrete Transform Methods. 1996, 462 pp. ISBN 0-7923-3940-1

I. Novikov and E. Semenov: Haar Series and Linear Operators. 1997,234 pp. ISBN 0-7923-4006-X

L. Zhizhiashvili: Trigonometric Fourier Series and Their Conjugates. 1996, 312 pp. ISBN 0-7923-4088-4

R.G. Buschman: Integral Transformation, Operational Calculus, and Generalized Functions. 1996,246 pp. ISBN 0-7923-4183-X

V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan: Dynamic Systems on Measure Chains. 1996,296 pp. ISBN 0-7923-4116-3

D. Guo, V. Lakshmikantham and X. Liu: Nonlinear Integral Equations in Abstract Spaces. 1996,350 pp. ISBN 0-7923-4144-9

Y. Roitberg: Elliptic Boundary Value Problems in the Spaces of Distributions. 1996, 427 pp. ISBN 0-7923-4303-4

Y. Komatu: Distortion Theorems in Relation to Linear Integral Operators. 1996, 313 pp. ISBN 0-7923-4304-2

A.G. Chentsov: Asymptotic Attainability. 1997,336 pp. ISBN 0-7923-4302-6

S.T. Zavalishchin and A.N. Sesekin: Dynamic Impulse Systems. Theory and Applications. 1997,268 pp. ISBN 0-7923-4394-8

U. Elias: Oscillation Theory of Two-Term Differential Equations. 1997,226 pp. ISBN 0-7923-4447-2

D. O'Regan: Existence Theory for Nonlinear Ordinary Differential Equations. 1997, 204 pp. ISBN 0-7923-4511-8

Yu. Mitropolskii, G. Khoma and M. Gromyak: Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type. 1997,418 pp. ISBN 0-7923-4529-0

R.P. Agarwal and P.J.Y. Wong: Advanced Topics in Difference Equations. 1997, 518 pp. ISBN 0-7923-4521-5

N.N. Tarkhanov: The Analysis of Solutions of Elliptic Equations. 1997,406 pp. ISBN 0-7923-4531-2

B. Rieean and T. Neubrunn: Integral, Measure, and Ordering. 1997,376 pp. ISBN 0-7923-4566-5

N.L. Gol'dman: Inverse Stefan Problems. 1997,258 pp. ISBN 0-7923-4588-6

S. Singh, B. Watson and P. Srivastava: Fixed Point Theory and Best Approximation: The KKM-map Principle. 1997,230 pp. ISBN 0-7923-4758-7

A. Pankov: G-Convergence and Homogenization of Nonlinear Partial Differential Operators. 1997,263 pp. ISBN 0-7923-4720-X

Other Mathematics and Its Applications titles of interest:

S. Hu and N.S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. 1997, 980 pp. ISBN 0-7923-4682-3 (Set of 2 volumes: 0-7923-4683-1)

L.A. Sakhnovich: Interpolation Theory and Its Applications. 1997,216 pp. ISBN 0-7923-4830-0

G.V. Milovanovic: Recent Progress in Inequalities. 1998,531 pp. ISBN 0-7923-4845-1

V.V. Filippov: Basic Topological Structures of Ordinary Differential Equations. 1998,530 pp. ISBN 0-7293-4951-2

S. Gong: Convex and Starlike Mappings in Several Complex Variables. 1998, 208 pp. ISBN 0-7923-4964-4

A.B. Kharazishvili: Applications of Point Set Theory in Real Analysis. 1998, 244 pp. ISBN 0-7923-4979-2

R.P. Agarwal: Focal Boundary Value Problems for Differential and Difference Equations. 1998,300 pp. ISBN 0-7923-4978-4

D. Przeworska-Rolewicz: Logarithms and Antilogarithms. An Algebraic Analysis Approach. 1998,358 pp. ISBN 0-7923-4974-1

Yu. M. Berezansky and A.A. Kalyuzhnyi: Harmonic Analysis in Hypercomplex Systems. 1998,493 pp. ISBN 0-7923-5029-4

Documents

Harmonic Analysis in Hypercomplex Systems