Coxeter Graphs and Towers L

of Algebras

Frederick M. Goodman Pierre de la Harpe Vaughan F. R. Jones

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

i, . .

Frederick M. Goodman Department of Mathematics University of Iowa Iowa City, Iowa 52240 USA

Vaughan F.R. Jones Department d Mathematics University of California - Berkeley Berkeley, Cal~fomia 94720 USA

Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, California 94720 USA

Pierre de la Harpe Section de Mathimatiques Universiti de Genkve CH- 121 1 Genkve 24 Switzerland

Mathematical Subject Classification (1980): 46L10, DSC50, 16A40 ,

Library of Congress Cataloging-in-Publication Data Goodman, Frederick M.

Coxeter graphs and towers of algebras. (MathematicaI, Sciences Research Institute publications ; 14) Bibliography: p. 1. Class field towers. 2. Coxeter graphs.

I. La Harpe, Pierre de. 11. Jones, Vaughan F.R., 1952- . III. Title. IV. Series. QA247.G68 1989 512'.55 89-5991

Printed on acid-free paper.

O 1989by Springer-Verlag New York Inc. . All rights reserved. This work may not be translated or copied in whde or in part withoutthe written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, bJY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with

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any form of information-storage and retrieval, electronic adaptation, computer software, or b y , W a r or dissimilar metliodol6gy now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used M y by anyone.

Camera-ready copy prepared by the authors. Printed and bound by R. R. DonneUey & Sons, Harrisonburg, Virginia. Printed in the United States of America.

9 8 7 6 5 4 3 2 1

PREFACE

The paper [Jol] on subfactors of von Neumam factors has stimulated much research in von Neumann algebras. Quite unexpectedly, it was discovered soon after the appearance of [Jol] that certain algebras which are used there for the analysis of subfactors could also be used to define a new polynomial invariant for links [Jo~]. The period of activity following this discovery saw the creation of a number of related invariants as well as the successful use of these invariants in knot theory. Furthermore, rece11t effort to understand the fundamental nature of the the new link invariants has l& to connections with invariant theory, statistical mechanics and quantum field theory. In turn the link invariants, the notion of a quantum group, and the quantum Yang-Baxter equation have had great impact on the study of subfactors.

It is not yet the time to give a comprehe~sive account of these developments, and we make no attempt to do so here. Our subject is certain algebraic and von Neurnam algebraic topics closely related to the original paper [Jol]. Hbwever, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expositoG material.

We give here a brief preview of the book. Each of the four chapters has its own introduction, with a more thorough description of the contents.

Chapter 1 begins with a (slightly new guise of) a familiar combinatorial problem: to classify finite matrices over the non-negative integers which have Euclidean norm no greater than 2. These are classified by the ubiquitous Cozeter graphs of type A, D, or E (see [HHSV] for other occurrences of these graphs) and the set of possible nomu is (2) u {2ws7r/q : q 2 2).

The central theme of the book - the discussion of which begins in Chapter 2 -is the tower of algebras MO c M1 c . -, c Mk c . . . determined by a pair MO C Ml of algebras

(with the same identity). The tower can be used to define various invariants of the pair, including the index [M1:MO]

In Chapters 2 and 3, we study two cases of the tower construction in detail. In Chapter 2, the algebras are finite direct sums of full matrix algebras over some field.

A pair Mo c M1 is described, up to isomorphism, by an inclusion matrix A with . non-negative integer entries. This matrix may be encoded as a graph, known as the Bratteli diagram of the pair. It turns out that the index [M1:Md equals 1 1 ~ ~ ~ ; thus it

follows from Chapter 1 that [M1:Md 5 4 if and only if the Bratteli diagram is a Coxeter . graph of type A, D, or E.

ISBN 0-387-96979-9 Springer-Yerlag New York Berlin Heidelberg ISBN 3-540-%979-9 Springer-Verlag Berlin Heidelberg New York

vi Preface

In Chapter 3, the algebras are finite von Neumann algebras with finite dimensional centers. Somewhat surprisingly, the results of Chapter 2 essentially extend to this setting. But now a pair Eho c M1 is (partially) described by an inclusion matrix A with entries in

{2cosn/q : q 2 2) U {I : r 2 21, and pairs with index no greater than 4 are associated to Coxeter graphs of arbitrary type, including types B,F,G,H,I. Finally, Chapter 4 is a M h e r analysis of pairs N c M of finite factors of finite index.

There are two main themes. The first is the notion of a commuting square, due to Popa [Popl], and its use in approximating pairs of hyperfinite 111 factors simultaneously by

finite dimensional von Neumann algebras. The second theme is the derived tows of a pair of II, factors, which is the c h i n of (necessarily finite dimensional) rdative cornmutants

Mo' 17 Mk in the tower. All the information in the derived tower can be encoded in a

(possibly infinite) graph, the principal graph of the pair. When the index is less than 4, the graph is a Coxeter graph of type A, D, or E.

In Chapter 2, we also describe how a certain quotients of the Hecke algebra of type A appear in the tower construction associated to a pair Mo c M1 of, say, finite dimensional

semi-simple algebras over the field C. For each choice of a positive faithful trace t r on / MI, there is a unique trace preserving conditional expectation El : M1 --, MO, and it

twm out that Ma is naturally generated by M1 and El. Now if the trace t r satisfies

the so-called Markov condition, then the situation propagates up the tower, and each algebra Mk+l is naturally generated by Mk and a conditional expectation

Ek : Mk --I Mk-l, for aU k 2 1. Moreover the Ekl s are idempotents which satisfy the

"braiding" relations

pEiEjEi=Ei i f l i - j ( = l , a n d

EiEj = EjEi if J i - j J 22,

where 0 = [M1:MO]. The abstract algebra APsk presented by generators

{l,E1,. 9 Ek-l) and relations as above is a quotient of the Hecke algebra Hk(d), where

q E E satisfies p ='"2 + q + q-l. Although we do not discuss this in the text, we might remark here that the milp

C 3 Hm(q) 4 alg {l,El,. . ) A c

where Em is the inductive limit of the braid groups Bk, is, up to a normalization, the

Jones link invariant [Jo~]. Also let us point out that to obtain the Jones invariant in this way, it is necessary to deal only with finite dimensional algebras, not the less familiar intinite dimensional von Neumann factors.

_--,

Preface vii

There are several appendices. Appendix I extends the cornputations of Chapter 1.

Appendix ILa relates complex semi-~imple algebras and finite dimensional c*-algebras: Appendix X b explains one appearance of the algebras AD,k in s tat is t id

m-w. Appendix 1I.c is a further discussion of ADik for special values of jl.

Appendix 111 is an exposition of Hecke subgroups in PSL2(R), and thus another famous

occurrence of the sequence (2cos(~/k))~,~. -

It is a pleasure to record our gratitude to numerous fiends and colleagues for their generous help, including: ,

R. Baldi, D. Bichsel, H. Dherete, M. Kewaire, A. Ocneanu, M. Pimsner, S. Popa, G. Skandalis, C. Skau, R. Steinberg, V. Sunder, A. Valette, and H. Wenzl.

We-gatefully acknowledge support from the MSRI in Berkeley, the IHES in Bures, the United States NSF, ttie IMA in Minneapolis, and our home institutions during our work on this project.

CONTENTS

Preface <

Chapter 1. Matrices over the natural numbers: values of the norm, classification, and variations.

1.1. Introduction. 1.2. Proof of Kronecker's theorem. 1.3. Decomposability and pseudo-equivalence. 1.4. Graphs with n o m no largef than 2. 1.5. The set E of norms of graphs and integral matrices.

J Chapter 2. Towers of multi-matrix algebras. 2.1. Introduction. 2.2. Cornmutant and bicommutant 2.3. Inclusion matrix and Bratteli diagram for inclusions of

multi-matrix algebras. 2.4. The fundamental construction and towers for multi-matrix algebras. 2.5. Traces. 2.6. Conditional expectations. 2.7. Markov traces on pairs of multi-matrix algebras. 2.8. The algebras for generic P.

1 2.9. An approach to the non-generic case. 1 2.10. A digression on ~ e & e algebras.

2.10.a. The complex Hecke algebra defined by GLn(q) and its Bore1

subgroup. 2.10.b. The Hecke algebras H

4,n' 2.10.c. Complex repreeentations of the symmetric group. 2.10.d. Irreducible representations of H for q @ R.

qln 2.11. The relationship between A aind the Hecke algebras.

P,n

Chapter 3. Finite von Neumann algebras with finite dimensional centers. 3.1. Introduction. 3.2. The coupling constant: definition. 3.3. The coupling constant: examples.

3.3.a. Discrete series. 3.3.b. Factors defined by icc groups. 3.3.c. w*(r)-modules associated to subrepresentations of ,IG.

Contents

3.3.d. The formula dimr(H) = covol(I') dr. 145

3.3.e. A digression on the Peterson inner product. 148 3.4. hdex for subfactors of nl factors. 149

3.5. Inclusions of finite von Neumann algebras with finite dimensional centers 156

3.6. The fundamental construction. 161 3.7. Markov traces on EndN(M), a generalization of index. 172

Chapter 4. Commuting squares, subfactors, and the derived tower. 182 4.1. Introduction. 182 4.2. Commuting squares. 188 4.3. Wenzl' s index formula. 199 / 4.4. , Examples of irreducible pairs of factors of index less than 4,

and a lemma of C. Skau. 203 4.5. More examples of irreducible paris of factors, and the index

value 3 + 3lI2. 207 4.6. The derived tower and the Coxeter invariant. 212 4.7. Examples of derived towers 219

4.7.a. Finite goup actions. 219 4.7.b. The An Coxeter graphs. 220

4.7.c. A general method. 221 4.7.d. Some examples of derived towers for index 4 subfactors. 222 4.7.e. The tunnel construction. 224 4.7.f. The derived tower for R 3 R when /3 > 4. P 225

Appendix I. Classification of Coxeter graphs with spectral radius just beyond the Kronecker range. 232

1.1. The results. 232 1.2. Computations of characteristic polynomials for ordinary graphs. 235 1.3. Proofs of theorems 1.1.2 and 1.1.3. 243 Appendix 1I.a. Complex semisimple algebras and finite dimensional

c*-algebras 253 Appendix 1I.b. The algebras A in statistical mechanics. P,k 259

Appendix I1.c. More on the algebras for non-generic B. 266

Appendix 111. Hecke groups and other subgroups of PSL(2,R) generated by parabolic pairs. 274

References. 281 Index. 287

CHAPTER 1 Matrim over the natural numbem

. Valpea ?f the norms, dassification, and variations . .

1.1. Introduction.

As already mentioned, the initial problem for this chapter is combinatorial: it is the classification of finite matrices over the nonnegative integers 01 = {0,1,2,. . . .} which have Euclidean operator norma no larger tban 2. The reader should be aware from the start that most matrices below are not square.

We establish first some notation. For m 2 1 the real vector space Illm has the

stanerd basis {el,. . .,em}, the standard inner product (f 1 q) = C fiqi, and the l< i ~ m

associated norm llCll = (<I o1I2. For m,n 2 1 and for a subset S c R, we denote by Matmln(S) the set of m-by-n matrices with entries in S; we write Matm@) for

Matmlm(S), and Matfin@) for the disjoint union of the Mat (S)'s over positive m,n

integers m,n. For X E Matm,n(R), the transpose of X is xt and X is the entry of llj X in row i and column j. We think of vectors in I R ~ as column vectors, and consequently r e identify X fi at^,^(^) with a linear map P - Rm. The Euclidean

operator norm of X is defined to be

Ilxll = Max{llxEll : E E i~~ and IItlI s 1). t

For S C IR we set

K(S) = {t E R+ : t = IlYll for some Y E Math(S)).

Our first result is essentially due to Kronecker [Kro]; it shows that K(Z) n [0,2] is infinite and easy to describe. Though not necessary for the logical understanding of the classification of integer matrices with small norm, the result helps us to believe that the matrices under study admit a comprehensible classification.

Theorem 1.1.1. Let X be o fnite mat& over I. Then either IJXIl = 2 cos g for

some natural nu@er q 2 2 or llxll 2 2. Moreover, for any integer q 2 3, the& ezists a pby-p m a t h X over {0,1) with

llXll = 2 cos $, where p ts the largest integer with p I q/2.

This theorem is. proved in section 1.2. Let us now describe the classification of matrices

2 (-' , Chapter 1: Matrices over the natural numbers '!?".,.

X E Matfin@) with llXll i 2. Obviously, we need only consider X E Matfin({0,1,2)).

The first stgp is to encode the matrix as a bicolored labelled graph, some edges being marked with a sign CO. (The choice of co fits with the marking conventions for Coxeter graphs, as will be explained below. We distinguish labelled graphs, in which each vertex has a well-defined name, from marked graph, in which some edges are decorated with various signs.) More generally, a matrix X E Mat (R) is encoded in a labelled bicolored

m,n graph f(X) which has m black v a t i m bl,...,bm and n white vertices wl,...,wn;

there is no edge joining either two b's or two w's; there is an edge joining bi and w. J

if and only if X i j # 0; W y an edge corresponding to XiSj f {0,1} is marked with some

sign conveying information about the value of X. In particular, an edge corresponding 1,j'

to X. . = 2 is marked with co and thus looks like o a . For example: 14 , 1

As one may expect for any classification, the next step is to establish appropriate notions of indecomposability and equivalence. We say that X is indecom~osable if f (x) is connected; two matrices X and X' are pseudo-eauivalent if f (x) and f (x') are isomorphic as (unlabeled) marked bicolored graphs. These notions will be discussed in greater detail in section 1.3, which also contains a proposition reducing the classification to the indecomposable case. Now comes the solution to the initial problem.

Theorem 1.1.2. The encoding described above set% q a bijection between: (i) . Indecomposabk - .matrices in Matfin({O,l)) of nonns smaller than 2, up to

pseudo-equivalence, and (ii) Irreducibk Cozeter graphs from the following list, together with a ~bicoloration, up to

n isomorphism ofbicolored graphs. The list is \..

I 1 A! (1 2 21, DL (1 t 4), El (k6,7,8).

Also it sets up a bijection between (iii) Indecomposable matrices in Matfin({0,1,2)) of norm equal to 2, up to

pseudo-equivalence, and (iv) Irreducible Cozeter graphs from the following list, together with a bicoloration, up

to isomorphism of bicolored graphs. The list is I

!j 1.1. Introduction

This theorem is proved in section 1.4, which also contains pictures of the graphs and tabla of matrices. The theorem is also essentidly proved in numerous sources, including, for example, [Sd and [CGSS]. One point should be stressed: the oombinatorics underlying Theorem 1.1.2 is the same as that which enters into the classification of simple Lie algebrai or of reflection groups of the comespondix types, but the theory of Lie algebras is not needed, nor that of reflection groups. On the other hand, these theories immediately suggest both statement and proof of the next result.

We set

so that K = [0,2] n K ( H ) by Theorem 1.1.1. To encode X E Mat (I(), we mark an edge m,n

with m if it corresponds to Xi . = 2, with q if it corresponds to X. . = 2 COST for some J 1 J q integer q 2 4, and not at all if X. . = 2 cos $ = 1. For example:

W

Such a marked graph is known as a Coxetec a. Theorem 1.1.2 may now be generalized as

Theorem 1.1.3. The encoding above sets up a bijection between: [i) Indecomposable matrices in Math(K) 4 nonn smaUer fhan 2, up to

pseudo-equivalence, and (ii) Irreducible Cozeter graphs j-om the' following list, together with a bicoloration, wp to

isomorphism of bicolored graphs. The list is

Also it sets up a bfjection between: (iii) Indecomposable matrices in Matfin@) of norm equal to 2, up to

pseudo-equivalence, and (iv) Irreducible Cozeter graphg from the following list, together with a bicoloration, t ~ p

to isomorphism of bicolored graphs. The list is

A , e 2 1 , BY) (e t 21, C P ) (I t 31,

4 Chapter 1: Matrices over the natural numbers

This is ad explained in Section 1.4. Theorems 1.1.2 and 1.1.3 imply m interesting fact:

?- -.

This generalizes to the following theorem, which was shown to us by G. Skandalis.

,Theorem 1.1.4. For any S c R one has. 4 4 s ) ) = 4s).

This is proved in Section 1.5. The set 4IN) is.also the set K({0,1)) of possible spectral radii of graphs, as has been observed by Hoffma~ [Hor]. This K ( M ) is an interesting set of totally real algebraic integers. Here is a sampling of some known facts / regarding K(N).

t '3 Pro~osition 1.1.5. The set E = K ( N ) has the following properties:

2 ,2 112 (i) If s,t 6 E then s+t, st, and [t -1 + are in E.

(ii) If s E El s > 1, then s+s-l, are in the limit set of E. (iii) The smallest limit point of E is 2 = l i m 2 cos(z); there ezists d > 2 with

q+m Q

E n ]2,d[ = 4. (iv) The smallest limit point of limit points of E is a = b? 8 2,058171. Limit

points of E in the interval ]2,AJ c o ~ t m t b an increasing injnite sequence

furthermore for each q 2 3, there are increasing sequences and decreasing sequences in E converging towards X

(1' (v) The closure of E contains [X,O , m [ . . (vi) E is not closed.

Statement (i) is proved in Section 1.5. See also [HW] for other operations defined i inside E. statements (ii) to (iv) are proved in Appendix 1. Statement (v) is a recent

I result of J. Shearer [She]. For (vi), see Appendix I, after 1.3.6. What is known of E . , I makes it look somehow similar to another set recently studied by [Smy]; see also Lehmer'e -

problem, referred to below in remark (3) following Proposition 1.3.4, in Appendix I. I I

I ;

1.2. Proof of Kronecker'a themem. I

We repeat firat the two results of [Kro], with their original proof.

5 1.2. - Kronecker's theorem . . 5

pro~osition 1.2.t Let X be a nonzero complez number which is a root of a monic polynomial P E H[T]. If all the roots of P are in the disc {z E C : lzl r 11, then X is a root of unity.

&&, Let Z denote the degree of P. The set of all monic polynomials in HIT] of degree Z having all their roots in the unit disc is finite. Indeed, such a polynomial is of the form

with lull < l,...,lvel s 1, and consequently with

As the aj' s are in 2, the claim is clear.

%ti P(T) = P1(T) = n y - p j ) mi, for any k iL, set = n L ~ - p i ) . 1s JS 1r J L

Then Pk E H[T] by Newton's formulas for symmetric polynomials (see, e.g., [BA4], page

A.IV.57). It follows from the begiining of the proof that P . = Pk for some j,k with J

1 r j < k, and thus that there exists a prmutation a ,of {1;2,,. . .iZ) with

I£ d denotes a iterated i t i e s (thus dn(1) = 1 for some m < 9 and if notations are such that y = A, one obtains after elimination

In other words, X is a (km-jm)th root of unity. #

ProDosition 1.2.2- Let be a n o w e m real number which' is a root of a manic polyomial P E H[T]. If all the roots of P are real and in [-2,2], then X = 2 cos(2n r) for some rational number r.

6 ,,* '- . +"

i Chapter 1: Matrices over the natural numbers L. ,,

w. Let L be the degree of P and set

Denote' by X = 2 cos el, 2 cos 02,' , 2 cos el the roots of P. Then

P(T) = n$T-2 cos Oj) l s j s

Q(T) = njT2-2T cos li js

iel The previous proposition shows that e is a root of 1, namely, that $ 1 2 ~ is rational.

# /

Corollarv 1.2.5 Let P E H[T] be a monic polynomial of degree 4 of which the roots e A1,- + ',Ae are real and in 1-2,2[; suppose P(T) # T . Then

(-42)

max{IAdI :1 i j s ~ = 2 c o s ~ J 9

for some natural number q 2 3.

*f. By the previous proposition, there exists pl,. . . ,pe, ql,- ,qe E H, with

(p., q.) = 1, such that A. = 2 cos(2rp./q.). But 2 cos(2rp./q.) and 2 cos(27r/q.) are J J J J J J J J

conjugate in the field Q(exp(iPrp./q.)), and as P has coefficients in H, the number J J

2 cos(27r/qj) is also a root of P. In case q - 2k.+l is odd, 2 cos(27rk.lq.) is also a root j - J J J

- of P, and 12 rm(27rkj/qj)l = 12 w ( r - 7r/qj)l = 12 cos(dqj)l. Thus ,

rnax{lAjl : 1 i j s =2lcos(r/q)I with

- 9 = ={q' A') q' = max{ 1 qj 1 : q is odd)

1 q' = max{ I qj I : qj is even}.

Finally q i 3 because 2lcos 7rI = 2 and 2lcos;l = O . #

The last ingredient in the proof of Theorem 1.1.1 is a well known exercise in matrix algebra:

$1.2. Kronecker's theorem

Lemma 1.2.4. Consider integers 4m,n 2 1 vrifi = m+n and a matriz X E Matman(R). Set

Then llxll = llxtll = IlYll = ~lx~xlll/~ = llxxtl11/2.

w. For any 6 E kRn one has

llxt112 = ( x t x t l o i llxtxlllld12 s llxtllllxllll~12. (*I

It follows that llxll i 1 1 ~ ~ 1 1 . As this holds with X and xt exchanged, llXll = 1lxtll. Now (*) implies 1 1 ~ 1 1 ~ = l l ~ ~ x l l , and then l l ~ 1 1 ~ = llxt112 = 1 1 ~ ~ ~ 1 1 , FinaUy

Proof of Theorem 1.1.1. Let X be a finite matrix over Z with llXll < 2 and let Y E Mat@ be as above. As Y is symmetric, its eigenvalues A1,. , . ,Ae are real and

llXll = IlYll = max{ I Al I ,, * ., 1 All}. These eigenvalues being the roots of the characteristic

polynomial of Y, one has Y = 0 or IlYll = 2 cos for some q > 3 by corollary 1.2.3. 9

The last daim in the theorem follows from the example below. #

l3xam~le 1.2.5, Given an integer l 2 2, consider >

For j = 1,- ..,e, one checks that Y[ - A.t. with j - J J

As Y is symmetric one has

8 Chapter 1: Matrices over the natural numbers

One may rewrite the rows and columns of Y in the order

2,4,...,1,1,3,...,1-1 if 1 iseven 2,4,...,I-1,1,3,,..,1 if I i sodd ,

0 X and obtain a matrix of the form [ ] with

xt 0

In both cases, llXll = 2 cos &.

Remarks. (1) The ideas of Kronecker in [Kro] have motivated much further research; see, for

example, [Boy] and [Rob]. (2) The matrix in the example above is "well known". Indeed, it could well be the

first matrix in the history of mathematics of which the eigenvalues have ever been computed! This occmed in 1759, when Lagrange was studying a system of ordinary differential equations which "approa~h~~ the equation for a vibrating string ([Lag], nos 14-20). We owe this observation to G. Wanner.

(3) The set of positive algebraic integers which arise as spectral radii of (not necessarily.symmetric) integral matrices is considerably larger than the set qn). Indeed, the former is dense in [I,@[, even if one considers only aperiodic matrices with entries in IN [Lin]; but the latter is not, as shown by Theorem 1.1.1.

(4) It is h o r n that the degree of the algebraic integer 2 ms is f dq) , w h m p '4 is Euler's function [Le2]. It is easy to see that the estimate p 5 q/2 in Theorem 1.1.1 is not sharp.

1.3. Decomposabiity and paeudo-quidenoe.

We consider first matrices in Mat (IS) for some m,n 2 1. Any permutation CY of m,n

{1,2,. .+,m} will be identified with the m-by-m matrix of the linear automorphism of LRm mapping e, to edi1; .with this wnvention at = oil. Similarly any permutation

1.3. Decomposability and pseud~quivalence 9

E 6, is identified with a matrix in Mat,((O,l)). Matrices X1 . and X2 in Matm,,@)

are defined to be & wiva les t if there exist CY E em and P E 6, with $ = d l P , namely, if appropriate exchanges of rows and columns convert X1 to X2. A matrix is

irredundar& 'if none of its rows and columns is zero. An irredundant matrix X E Matm,,(R) is if there exist integers m' ,m',nl ,mn 2 1 with

m = ml+m' and n = nl+n', as well as X' E Matm, ,+,, (R) and X' E Matm.,n.(tR)

such that X is pseudmuivalent to . A matrix is indecom~osab P' 081 la if it is

irredundant and not decomposable. For example, it is not difficult to check that

are indecomposable, but that

is decomposable and pseudo-equivalent to

These notion$ should be distinguished from those defined by Frobenius [Fro] for square matrices. Recall that Z E Matm@) is reducible if there q s t s permutation -y € em with 7Z7-' of the form

(I Z. , where ZZ. and Z' are square blocs of siaes ["' 7

m' ,m' > 1 with m' +m' = m. Matrices such as Z and $37-I are called eauivalent. or example, the matrix

already considered is irreducible. (Frobenius uses "zerfallend" and llzerlegbar" for "reducible," and no specific word for llequivalent.l')

l e t X € Matm,n(R). Denote by xC E Matm,({O,l}) the matrix defined by x:,~ = 1

if X i j # 0 and X! . = 0 for' Xi . = (I; the w p e m i p t c stands for Ucombinatorial". We 1 J J

have explained in section 1.1 how xC caabe encoded in a bicolored labelled graph f (xC) containing m black vertices bl,. . .,bm and n white vertices wl,. . .,wn. Now set

4 = m+n and wnsider the square matrices

10 7- Chapter 1: Matrices ove~ the natural numbers (, -

Denote by f (xC) the labelled graph underlying p(XC): it has uncolored vertices

and is Caned its *cv matrix in graph theory. Here is a list of easily established observations.

(a) The following are equivalent: (i) one of the matrices X, xC, Y, yC is irredundant; (ii) all these matrices are irredundant; (iii) the graph ~ ( x ~ J has no isolated vertex.

(b) The following are equivalent: (i) one of (hence both) the matrices X, xC is fndecomposable; (ii) the graph I'(xC) is connected; (iii) one of (hence both) the matrices Y, yC is irreducible.

(c) The following are equivalent: (i) X: and X: are pseudo-equivalent;

(ii) f ($1 and f (xi) are isomorphic aa bicolored graphs.

(d) The hllowing are equivalent: (i) Y: and Y; are equivalent;

(ii) I ' ( x ~ ) and r($) are isomorphic as graphs.

Pro~osltlon . . 1.3.1. For any irredundant matrix X E Mat (R), there azist an bteger

m,n a z 1, partitions

and irredundant indecomposable matrices Xr E Matm (R), for r = 1,. . .,a, such that '

r' r X is pseudo-equivalent to the mat*

,Fh- '

5 1.3. ~&omposability and. pseudwquivalence.

In this case one has 11X11 = max{llXlll,. .,llXall).

Uniqueness holds as follows: if X is also pseudo-equivalent to a mat& with indecomposable blocs Xb E Matm, ,n, (W), for s = 1,-, .,b, then b = a and, after some

8 8

renumbering of the triples (m;l,ni,X;1), one has

m i = mr , n' = nr, and r

Xi pseudo-equivalent to 5 for r = 1,. . ., a.

b f . For the existence part, consider the connected components f (xC), of f (xC),

where r = 1,. . .,a. Denote by m, [respectively, q] the number of black [respectively,

white] points in f (x')~. Permute the rows of xC and obtain a matrix X' in such a way

that the black point bi belongs to f XI)^ if

Similarly, permute the columns of X' and obtain a matrix X' E Mat ((0,l)) in such m,n

a way that the white point w. belongs to f (x'), if J

7

nl+.. ,+nr-l<jsnl+.. .Snr (r=l,...,a) .

The matrix X' has the desired form. Uniqueness follows from the fact that f (xC) has well defined connected components.

#

Remarks. (1) If X corresponds to the bicolored labelled graph f (x), then xt corresponds to

a graph with colors exchanged. The example at the end of section 1.2 is a Jordan block, say

for m = 4. As f (x) is clearly isomorphic (as unlabeled bicolored graph) to the graph with colors exchanged, a Jordan block is pseudwuivalent to its transpose. But

12 Chapter 1: Matrices over the natural numbers 3 1.4. Graphs with norm no larger than 2 13

are not peudo-equivalent to each other, because the graph corresponding to the former has a white vertex of degree 3.

(2) Consider X,X1 E Matmln(R) and the cohesponding Y,Y1 E Mate (R), with

e= m+n. If X' is pseudo-equivalent to X, say X' = &/I. then Y' is equivalent to Y, because Yf = 7 ~ 7 " with , 7 = (' 1. But one may have Yf equivalent to Y -

0 r1 without X' being pseudo-equivalent to X. The reader can check this with X and X' the two 3-by-3 matrices of the previous remark, or with X E Mat4({0,1)) and

X' E Matg,2({0,1)) corresponding to

(3) There is of course a proposition analogous to 1.3.1 in terms of irreducible square matrices and equivalence ia kobenius. See [Gan], 3 XIII.4.

For future reference, we also note the following fact. A square matrix X E Matm@+)

. with nonnegative entries is irreducible if ad only if, for each ilj E (1,. . ,m), there exists an integer p (depending on i and j) such that the (ij)@ entry of xP is strictly positive (see [Gan], 3 XIII.1). It is a~eriodic ("primitive" is also used) if there exists an integer p such that all entries of xP are strictly positive.

Lemma 1.3.2. (a) For X E Matmln(R), the following are eqdvalent:

(i) X is irredundant; (ii) X ~ X E Matrn@) and XXt E Matn(R) are irredundant.

(b) For X e"Matmln([R+), the following are equivalent:

(i) X is indecomposable; (ii) XXt is irreducible; (iii) XXt is aperiodic; (iv) xtx is irreducible; (v) X ~ X is aperiodic.

m. Claim (a) is immediate from multiplication rules. For (b), if bl,. . ,bm,wl,. . . ,wn are the vertices of f (x), then f (XXt) has m

black vertices bl,.-.,bm and m whitevertices bi,.w.,b&. There exists a line between

bi and bi, if and only if there exist a line in f ( ~ ) between bi and w. as well as J

between w. and %, for some j E (1,. . .,n). This shows equivalence of (i) and (ii); the J

same works'for (i) and (iv). Now a matrix such as X ~ X , which is positive both in the sense of Perron-Frobenius

theory and in the sense of operator theory, is irreducible if and only if it is aperiodic. This follows from Theorem 8 in 3 XIII.5 of [Gan]. #

About (a), obserke that X = [: :] is redundqt, with XXt imdundant and xtx redundant.

Let r be a finite &aph with e vertices. The 8- of I' is (the equivalence class of) a symmetric matrix Y = Yr E Mate ({0,1)), with Y. . = 1 if there

1lJ

is an edge between vertices i and j and with Y. - 0 if not. The ~haracterlstlq . . llj -

plvnomial of I' is that of Yr, and the s w t r a l @&Q or of I' is

lll'll = max{ I A 1 : A is an eigenvalue of Yr).

From Proposition 1.3.1, it is clear that the following program will solve the problem of classifying matrices X E Matfin({Oll)) satisfying ((X(( < 2:

(1) classify finite connected graphs satisfying JJr)J s 2; , (2) classify bicolorations on these.

Consider now an indecomposable matrix X E Matfin({0,1,2)) and assume that 2 is an

entry of X. Then one has either X = 2 (as 1-by-1 matrix) or llXll 2 51t2: this is clear from the definition of IlXll, and follows also from Lemma 2 in XII1.2 of [Gan]. Consequently the program above will also solve our initial problem of classifying X E Matfin({0,1,2)) with IlXll s 2.

1.4. Graphs with norma no larger than 2.

The elementary classification of the present section is based on two fundamental facts of Perron-Frobenius theory, for the proof of which we refer $0 [Gan].

The first one is about ah irreducible square matrix Y E Matk([R+) with nonnegative j.

entries. A Penon-Frob- yBEtpI for Y is an eigenvector (ER: of Y with

nonnegative entries. The fact is that such a vector (i) always exist;

(ii) is unique up to multiplication by a positive scalar;

14 jrc"' #+.:,-

Chapter 1: Matrim over the natural numbers -

(iii) corresponds to an eigenvalue which is simple and which is the spectral radius of Y.

In partic&ar, if I' is a finite connected graph, and Y € Matk({O,l)) is its adjacency

matrix, then Y is irreducible, although in general not aperiodic; a Perron- Frobenius vector C for I' (by which we mean for Y) satisfies

Y t = IlI'll C.

As the vector space where Y acts has its natural basis indexed by the vertices of 1', it is convenient to represent such a ( on the graph I', with the value of each coordinate of wlitteh near the corresponding vertex of r. For example, if Y = 8, then [[I E !U2

1 1 is represented by -.

We consider the following graphs, each given with a Perron-Frobenius vector. (Note that the subscript .! means that the graph has kl vertices (sic).)

Lemma 1.4.1. Each graph in the list above has norm 2.

u f . Check that the indicated vector is a Perron-Frobenius vector corresponding to ,

the eigenvalue 2. #

The second fact from Perron-Frobenius theory concern two matrices Y' ,Y E Matk(nt+) with Y irreducible, Y' s Y (this means (Y-Y')ilj 2 0 for

I F~'\,

$1.4. Graphs with norm no larger than 2 15

i,j = 1, . . ,k) and Y' # Y. Then the fact is that the spectral radius of Y' is strictly' smaller than that of Y. In particular, this implies the following lemma.

Lemma 1.4.2. Let I' be a frnite connected graph and let r: be a proper subgraph of

r; then llr' I I < Ilrll.

(By a subgaph of I', we mean one obtained by erasing some edges and some vertices together with all edges emanating from these vertices; for example, - o is a subgraph of -.)

The graphs list& in the next theorem are the ones above together with the following (Now L is the number of vertices!):

Given integers p,q,r with 2 < p 5 q 5 r, we consider also

q-1 - 4'. -0-

(p+q+r-2 vertices). - --0"'0--0

p-1 - r-1

One has in particular DL = T2,2,e-2 and Ee = T2 1 1

Theorem 1.4.3. Let I? be a nonempty finite connected graph. ( a ) l l r l l < 2 i f a n d o n l y i f r i s o n e o f .

In this case, llI'll = 2 cos(lr/hr) where hr is the so-called Cozeter number of I' giuen by - e+i for A!, 2e-2 for D ~ , 12,18,30 for E~,E,,E*.

(b) IlI'll = 2 if and onlg if I' is one of

AP) (e 2 21, D P ) (e I 41, EY) (e = 6,7,8).

16 . Chapter 1: Matrices over the natural numbers

M. Assume first that llI'll < 2. By Lemma 1.4.2, the graph I' contains no cycle (because l l ~ j ' $ = 2 for l r 2), no point of degee 4 or more (because 1 1 ~ ~ ~ 1 1 = 2),

and at most one point of degree 3 (because llDf 111 = 2 for L 2 5). Consequently I' is

either a segment, namely Ae for some L 2 1, or one of T2 e-2 = DL for some l 2 4, or 1 ,

some graph T with q > 3. In this last case one cannot have p 2 3 (because p,q,r = T3,3,3 is of norm 2), nor r ? 6 (because E&') = TZIt6 is of norm 2). Thus

one is left with the list of (a). Assume now IlI'll s 2. The same argument shows that I' does not properly contain a

cycle, a vertex of degee 4 together with its nearat neighbors, or one of the DV) for

L 2 5. If, moreover, llrll = 2, it follows readily that I' is in the list of (b). If I' = Al then llrll = 2 cos(lr/(kl)) by the example of section 1.2. For the other

value8 of h , see the literature,or Proposition 1.2.5 of Appendix I. For various meanings /

of the number hr, see [Stel]. #

Define the bicoloratio~ number of a graph I' to be the number of possible partitions of its vertices in two nonempty sets, with no pair of adjacent vertices in one of the parts. It is both elementary and well known that this number is 0 if and only if I' is a point or contains an odd cycle. For the graphs of the previous theorem, one has

Table 1.4.4. The bicoloration numbers of the graphs listed in 1.4.3 are as follows:

A ~ : o A P ) with I 2 2 even: o A! with l even: 1 A?) with ( 2 3 odd: 1

A~ with l r 3 odd: 2 DP) with t 4 even: 2

Dewith Lr 4: 2 Df) with l> 5 odd: 1

El with l = 6,7,8: 2 E ~ I ) with != 69,s: 2

This completes the program set up at the end of Section 1.3, and consequently the proof of Theorem 1.1.2. The matrices listed in the following tables (Tables 1.4.5 and 1.4.6) correspond to the graphs of Theorem 1.4.3; the one additional matrix [2] E Matl@) is

conveniently made to cornrespond with the marked graph AP), pictured as A. The I

tables list Coxeter exDonen& of the graphs l'; these are integers m. such that the , J

adjacency matrix of the graph I' has spectrum (2 cos [m. lr I}, with j = 1,. . -,m+n. ~a;: More about Coxeter exponents can be found in section 2 of [Cox] and in exercise V.6.4 of [BLie] .

$ 1.4. Graphs &th norm no larger than 2 17

Although the theory above is independent of that of semisimple Lie algebras, root systems are very close at hand. In fact, consider X E Mat ((0,l)) with llXll s 2, and m* set Y = [Ot E MatL((0,l)) as above. Then 2-Y is a real symmetric matrix with/

X 0 2 positive eigenvalues; as &ch it can be written uniquely as 2-Y = Z , with Z E Mat@) a

symmetric matrix with non-negative eigenvalues. Let orl,, . .,ort denote the oolumns (or 2 rows) of Z. Then (Z )i,j is the scalar product (a.10.); since it is 2 if i = j and

J -Yij E {O,-1) if i # j, the vectors 5,. . .,orl are of equal length 8 and their mutual

angles are lr/2 or 243. Assume moreover IIXII < 2, so that Y and Z are invertible, and thus 4, . . ,?

linearly independent. Then, {al,. , .,eye) is what is called a reduced system of roots. It is

clearly irreducible if and only if X is indecomposable; in this case it is of one of the types A,D,E because all roots have the same length. See [CGSS] for more along these lines.

For the remainder of this section, we assume that the reader has some familiarity with Coxeter graphs. Recall from section 1.1 that

and that a matrix X E Mat (K) is encoded as a Coxeter graph, say I', plus a m,n

0 X bicoloration. In fact, 2 - [ ] is twice the matrix of the cmonical bilin* form

xt 0 associated to r (the form denoted by BM in [BLie], chap. V, $4, no 1). Any Coxeter

graph r corresponds to such a matrix X, and llrll = llXll < 2 [respectively [lXll= 21 if 0 X

and only if 2 - [ ] is positive ddni te [rap., positive semiddnite]. Consequently the xt 0

classification of Theorem 1.1.3 is nothing but another phrasing of the classification of irreducible Cozeter systems which are of fhe finite type [resp., the afbne type]. We refer to

[BLie] for further details. It is again true that an indecomposable matrix X E Mat (K) m,n corresponding to a Coxeter graph I' has norm 2 cos(r/hr), where hr is either as in

Theorem 1.4.3 or as in the following list:

2L for Be 12 for F4 6 %r G2

10 for H3 30 for H4 p for $(p). ,

Table 1.4.5. w 00

Type of Y .size . matrix Y bicolored graph Coxeter number exponents *Y figures in table ,*--

.- , 1; 4 , ! ? = 2 m m > l m-by-m ( ) -...- !?+I 1,2, . . ., !? no I i

0 D,, !? = 2m m 2 2 m - 1 ) b + 1 ( ) Q--p

22 - 2 and 1,3, 2- 5 , . . 1 .,2P-3, yes

'1 '1 1

w 1 0 0 0

E8 (i i i r ) 7 30 17,19,23,29 1, 7,11,13, yes B er.

N.B.: type Al doesnot appear; it would correspond to the empty matrix, identified to the (linear!) map Ro + Ro.

E -B i2

Type of Y

Table 1.4.6.

size

1-by-l

mat" Y ' bicolored graph

2 t-cU

.*Y figures in table

no P £2'

yes

NB.: !? * m + n - 1 in this Table, but P = m + n in Table, 1.4.5.

Chapter 1: Matrices over the na;tural numbers. 3 1.4. Graphs with norm no 1arger.than 2

Table 1.4.7. List of Coxeter graphs which satisfy llrll s 2 '

and which do not appear in Tables 1.4.5 and 1.4.6.

4 Be o---. . .-o-- 1 2 2 L vertices h = 2t

F~ o c 4 0 - 0 4 vertices h = 12 6 G2 6-0 2 vertices h = 6 5 H3 0-- 3 vertices h = 10 5 H4 0-0-0-0 4 vertices h = 30

12(p) o-E, P = 5 or p 2 7 2 vertices h = p

For these IlI'll = 2 cos(r/h)

4 03- - - -0---o L 2 3 kl vatices

J z 2 2 2 2 8 L 2 2 t+l vertices

5 vertices

3 vertices

For these, llI'll = 2 (a Perron-Frobenius vector is indicated)

Table 1.4.8. Perron-Frobenius eig&vectors for Coxeter gaphs of finite type

Type Ae (1 L 2). . Eigenvalue: 2 cos[lr/(t+ I)] ,

sin[./(! +I)] ?in[2lr/(l +l)] sin[(!-l)r/(t +I)] , sin[Llr/(t +I)] *~-*- . . . -

. . Type Be (1 L 2). Eigenvalue: 2cos[lr/24

sin[r/2lJ sin[2,~/2&l sin[(t-2)lr/2lJ sin[(t -l)lr/24 i/p *-*-. . .--* - a

Type ( t L 4). Eigenvalue: ~ c o s [ T / ( ~ ~ -2)]

Type EEg' Eigenvalue: 2cos[lr/12] = (13 + 1 ) / p

, sin[lr/l2] sin[2lr/l2] sin[3lr/12] s in[2~/12] sin[lr/l2] - i-

b I

111 Lemma 1.4.1 and in Table 1.4.7, we have indicated a Perron-Frobenius eigenvector ; for the Coxeter graphs of affine type. For Section 4.5 we will also need to compute the Permn-Frobenius vectors for the connected Coxeter graphs of finite type in the classes A, D, and E. For completeness, we repeat the case of the gaphs At, already dealt with in

I Section 1.2, and also give the results for the classes B, F, G, H, and I. Recall that in the , standard notation for Coxeter graphs of finite type, the subscript gives the number of vertices. The details of the verifications are straightforward and are left to the reader. 1 ,

I

!

Note: sin[lr/12] = ( - 1 2 sin[2dl2] = 112 sin[3rr/l2] = 1/@ sin[3lr/12]/2cos[lr/i2] = ($3-1)/2

Type E7' Eigenvalue: 2cos[lr/18]

F--\ Chapter 1: Matrices over the natural numbers

Type Es. Eigenvalue: 2cos[d30] = 21- J3 + 1 1

5 1.4. Graphs with norm no larger than 2

Let J? be a connected Coxeter graph of finite type A, Dl or E with e l 2 vertices. Choose a bicoloration of I?. with m black and n white v e r t j ~ s &y+n = 9, and let

X E Matmln({O,1}) be the corresponding matrix, so that Y = [ i t is the adjacency

matrix of r. Let x be the row vector defined by the n white co-ordinates of the Perron-Frobenius vector for Y (as listed in Table 1.4.8); then x is a Perron-F'robenius row vector for X ~ X . For use in Section 4.5, we need to know the square ( ( ~ ( 1 ~ of the Euclidean norm of X. (In case I' has two bicolorations, there are two distinct choices for X, but it follows from the eigenvalue equation for Y that they have the same norm.)

Type F4. Eigenvalue: 2cos[lr/12] Pro~osition 1.4.9, With the notation as above, the values of the square 11x1(2 of the norna of the Perron-Frobenius eigenvector for X ~ X are as follows:

P J 3 + 1 C 3 + 1 P 0-0 = .----. Type At: (e + 1)/4 Type Dl : (e- 1)/4

Type E6: (3 - $3)/2 Type E7: ca. 0,57999 Type Ga. Eigenvalue: 2cos[lr/6] = /

Type E8: ' Ca. 0,38502 10-1

Type Hq. Eigenvalue: 2cos(lr/30]

Type 12(p). Eigenvalue: 2cos[lr/p] (PI

1 .- 1

Remarks.

(1) Let be a graph with .t vertices and let X1 5 , -. s Xe be the ordered sequence

of its eigenvalues. The s~ectral s ~ r e d s(r) of r is Xe-XI. Of course llrll < 2 implies

s(r) s 4; the converse happens to hold with finitely many exceptions which have been classified by Petrovit [Pet].

(2) It has been pointed out to us by D. Cetkovit and C. Godsil that it may also be possible to classify' indecomposable matrices X E Matfin({-l,O,l)) with [[XI[ i 2. One can

0 X mite 2- [ t ] = ((41 7)IlriIl as before, so that the problem is equivalent to the

X 0 e classification of irreducible sets of vectors {al,. ',ae} in R , all of the same length and

with mutual angles in {lr/3,x/2,2lr/3}. The possible sets of lines spanned by such sets of vectors are classified in [CGSS]. Once this is worked out, the next cases would be X E Matfin(Ku(-K)) with l\Xll < 2 as well as X E Matfin@) with IIXII > 2 but IlXll close

to 2. (3) The subject of spectra of graphs has been extensively investigated. We refer to the

excellent early monograph by Biggs [Big], to more recent books by "CGetkovi~, Doob, Gutman, Sachs, and TorgaSev [CDS] and [CDGT], as well as to the reviews [CD], [GHM] and [Schl].

24 Chapter 1: Matrices over the natural numbers 3 1.5. Norms of graphs and integral matrices 25

1.5. The set E of norms of graphs and integral matrim.

The assignment to a subset S of !It of the set K(S) of norms of matrices on S has interesting properties. Observe that, obviously,

s na+ c 4 s )

and also that s,t E 4 s ) 9 st E 4 s )

Ilrpef. It is obvious from considering 1-by-1 matrices that 4 s ) r U(K(S)). To show the theorem, set T = 4 s ) and let X E Matm,,(T); one has to find some Z E Matfin(S)

with IIZII = IIXII. . For any pair (i,j) with 1 i i i m and 1 i j j n, we choose integer p. . 2 1 and a

1 J

symmetric matrix Y. of size p. over S with IIY. .I1 = Xi,f Let p be the product of l,j 1,j 1 , ~

the p. .Is. Write Z. for 1 e ... e Y. . e ... e 1 E Matp(S), with the factor Y i j at 1 , ~ 1,j w

th the (i,j)- place. Consider the matrix Z E Matpm,pn(S) with the Z. .'s as blocks.. 1,J - -

Choose for each pair (ij) some vector t. . # 0 with p. coordinates such that because st = llX @ Y11 if s = llXll and t = ((YII. (The inequality (JX e YII < IIXII((YI( l j follows from X @ Y = (X @ 1)(1@ Y); the converse inequality follows from the existence of YiYjGJ = X. .& Set t = @$,j E R ~ , so that Z i j (= X. .(. C h m also qeRn with

Id 1,f 1,J vectors t,q of norm 1 with IlXtll = IlXll and IIYqll= llyll, hence with q # 0 and IIXqll = IIXllllqll. With C = t @ 11 one obtains IlzCll = IltllllXllllrlll = 11x1111'41. Il(x @ y ) ( l @ I1)I = IlxllllyII.) it follows that llZll 2 IlXll.

Given So c S c R+ such that every number in S is a sum of numbers in So, one has As JJZll j llXll by the next lemma, this proves the theorem. # also / The integer p 2 1 being given, let B % the algebra ~nd(@), considered together

K(S0) = 4 s ) . with the Euclidean operator norm (a "real C -algebrau). We identify Matm,,(B) with

I , the space of linear maps from en to RPm

Indeed, the following nice argument of Hoffman shows that 4 s ) c 4SO). Let Lemma 1.5.1. Let Z E Matmp(B), a d let X E Matmp(R+) be defined by X E M a t ( S ) and let Y = [O X] E MatAS). We have to find some X' E Matfin(So)

xt 0 Xilj = llZi,jll for i = 1,. . .,m and j = 1,. . .,n. Then llZll s IlXll. with IIX' 1) = 11Y1); we may assume that Y is irreducible. Now there exists a decompoai-

tion Y = Yl++ .. +Yk with Y. symmetric matrices in Mate (So) for j = l,, ... ,k. Set , J M. For any Y E Matn(B), set

e kt Let t E W+ be a Paron-Frobenius vector for Y and define 6' = (t,t,. .. ,() E R+ . Then X' t' = I(YIIt', so that t' is a Penon-Frobenius vector for X' . Consequently Ilx' 11 = IlXll. Ib particular one has

which is Proposition 2.1 of [Hofl. Now we state again the main result of this section, due to G. Skandalis.

Theorem 1.1.4. For any S C IR one hos 4 s ) = 4 4 s ) ) .

llMll= sup{((Y. .)) : 1 i i i m and 1 3 j r n). 11J

one has

Chapter 1: Matrices over the natural numbers ''=>?\

5 1.5. Norms of graphs and integral matrices

Now we particuladze to Y = ztz E Matn(B). For any integer k t 1, the entries of Y k Remarks about E = 1/(f0,1)1 are positive sums of products of entries of zt and Z; it follows that

! (1) Given s,t E E, one may look for an explicit graph with spectral radius -s+t

2 112 (respectively at, and hs+(s2+4kt ) 1). Some solutions can be found in [Schl] III(Z~Z)'III r III(X~X)~II.

Consequently, (respectively [We], and Theorem 2.13 of [CDS]).

r n 1 4 ~ ( ~ t ~ ) q ~ f / k r nl /k l l (~ t~)k l l l lk = nl/k11x112

for all k 2 1. The lemma follows. #

One may deduce from Theorem 1.1.4 numerous properties of the set K(S). The following is a sample.

Corollarv 1.5.2. Let s,t E 4 s ) . The following numbers are also in K(S): f

a+t, (s2+t2)'l2, $s+(s2+at2)1/2) for evety L E I.

If. s # 0 and s # 1, the numbers

s+s-l, s2(s2-1)-1/2 are in the derived set of 4 s ) .

&&. The first claim is a consequence of the following equalities for Perron- Frobenius eigenvectors:

3,=,,, with ,= [;I E R ~

1 i i] ( = ( ~ ~ + t ~ ) ' / ~ < with ( =

1

s+(s2+4kt2)lI2 2 112 4 = $s+(s2+at ) }( with ( = [ ] uk+l.

The second claim is proved in Lemma 1.3.7 of Appendix I. #

(2) Say s E E is irreducible if a # 0,l and if s = sls2 with sl,s2 E E implies sl = 1

or s2 = 1. Any number in E can be factored as a product of finitely many irreducible.

Are there only finitely many factorizations? (The answer is obviously yes for s r 4.) If

1 yes, does the number of factorizations relate simply to the minimal t E I for which there I exists X E Matl ( I ) 4 t h s = llXll?

(3) Is it true that K(M) = f lu)?

i 5 2.1. Introduction 29 I

i

CHAPTER 2 TOW- of multi-niatrix algebra5

2.1. Introduction.

The first purpose of this chapter is to study inclusions of one finite dimensional semi-simple algebra in another. Following [Jo~], we introduce a real-valued invariant, called w, for a pair 1 6 N c M of axbitrary -not necessarily semi-simple or even finite dimensional - algebras over a field K, as follows:

First, the fundamental construction associates to N c M the pair M c L where L = ~ n d i ( ~ ) is the algebra of endomorphisms of M viewed as a right N-module; M is

identified with a subalgebra of L, each x E M being identified with the left multiplication operator (y w xy) E L. Second, the induced by the pair N c M is the nested sequence

I E M ~ = N C M ~ = M C s e e CMkCMkSIC"-

/'

of K-algebras, where Mk c Mk+l is obtained from Mk-l C Mk by the fundamental

restrict our attention to semhimple algebras for which the simple components are central ,

(have center equal to K). In fact we can even restrict attention to multi-matrig I 1 over K, semi-simple algebras whose simple components are isomorphic to matrix algebras

over the ground field K Note that if K is algebraically closed, then every semi%imple I

I K-algebra is a multi-matrix algebra, since K has no proper finite dimensional division

i algebras. We will call an algebra which is isomorphic to Mat (K) for some p > 1 a

P factor. Some authors refer to multi-matrix algebras as "split semi-simple algebras".

I I

The reason why it sufGces to study multi-matrix algebras is that index is stable under

! &ggg Qf f& m d field, Let K be a perfect field and E an extension field; for any E [ K-algebra M, let M denote M % E, an algebra over E. If 1 E N c M is a pair of

/ finite dimensional K-algebras, then [M:N] = [ M ~ : N ~ ] (Proposition 2.4.4). If M and N I E are in addition semi-simple over K, then it is possible to choose E so that M and N E

i are multi-matrix algebras ov& E. Taking E to be an algebraic closure of K will do, but one can also accomplish this with a finite dimensional field extension.

i \ We now come to the definition of the index m a t r i ~ A: for a pair of semi-simpb

I K-algebras 1 E N c M. First for a pair of 1 E N c M, the reader can easily check i I (after looking at Section 2.2) that the index [M:N] is just the ratio of dimensions,

construction (k 2 1). Third, the & rk(MklMO) of Mk over Mo is defined to be the I I

smallest possible number of generators of Mk viewed as a right Mo-module (this rank lies ; ..

in DI U {a)). And finally the index of N h M is the growth rate

[M:N] = lim sup[rk(Mk!Mo)] l/k. k-ko

I

i ' ;

In this case A: is the l-by-1 matrix withaole entry [M:N]'/~. Next mnsider a pair of

I I

multi-matrix m. Let {pi : 1 i i i m) be the minimal central idempotents in M, so

i that M = @ Mpi and each Mpi is a factor. Similarly let {q. : 1 i j b n) be the i=l J

Two comments on this definition: First, we could exchange the words left and right and ' minimal central idempotents in N. For each pair i j (1 5 i 5 m, 1 5 j 5 n) set

obtain a rank and an index "from the other side"; but we shall not study this variation I

results of Chapter 1 yield restrictions on the possible values of the index. 1 Recall that a semi-simple algebra over a field K is the direct sum of its minimal two

sided ideals, and each of these is isomorphic to a matrix algebra Mat (A) for some p 1 a 1

'here. Secondly, a more interesting variation comes from using tensor products M % M aN . (a Tor-like idea) instead of endomorphisms . , ~ n d ; ( E n d i ( ~ ) ) (an

Ext-like idea); we refer to [Jo4] for this. One could check that these variations give the ,

same index for semi-aimple pairs, but more general examples may have several indices. The conination of this subject with Chapter 1 is this: For inclusions 1 E N c M of

semi-aimple algebras over a perfect field, the index turns out to be the square of the norm M of a certain matrix of natural numbers AN associated with the pair of algebras. Thu's the

X i j = 0 if piqj = 0, and

. = [M. .:N. .l1I2 if piqj j 0. +,I 191 U

M. l r j - -p.q.Mp.q. 1 J 1 J and N. 1,J .=p.q.Npiqj. 1 J

Since pi is central in M, the product p.q. is an idempotent in Mi. If piqj # 0 then 1 J

M. . is a factor (Proposition 2.2.3); and, since the map x n pixpi from Nq. to N. is a ' 1 J J 1,j

non-zero homomorphism, N. is a h a factor with identity element p.q.. Define A: to l j 1 J

be the m-by-n matrix with entries LI

28 Mij P Nij @MatA. JK), by Proposition 2.2.2. As the set of pi's is well defined by M Id

and some K-division algebra A. However, for studying index, we can for the most part [Mi,fNi,j] is a square integer, so A , is a natural number; in fact

3 2.1. Introduction

CHAPTER 2 Towers of multi-matrix algebras

2.1. Introduction. f

The first purpose of this chapter is to study inclusions of one finite dimensional semi-simple algebra in another. Following [Jo~], we introduce a real-valued invariant, called m, for a pair 1 E N c M of arbitrary - not necessarily semi-simple or even

finite dimensional - algebras over a field K, as follows: First, the fundamental construction associates to N c M the pair M c L where

L = End;(~) is the algebra of endomorphisms of M viewed as a right N-module; M is

identified with a subalgebra of L, each x E M being identified with the left mdtiplication operator (y n xy) E L. Second, the tower induced by the pair N c M is the nested sequence

~ E M ~ = N C M ~ = M C . . - cMkCMkSIC * " '

of K-algebra, where Mk c Mk+l is obtained from MkVl C Mk by the $damental

construction (k 1 1). Third, the & rk(MklMo) of Mk over Mo is defined to be the

smallest possible number of generators of Mk viewed as a right Mo-module (this rank lies

in M U {w}). And finally the index of N & M is the growth rate

Two comments on this definition: First, we could exchange the words left and right and . obtain a rank and an index "from the other side"; but we shall not study this variation

here. Secondly, a more interesting variation comes from using tensor products M M eN . . - (a Tor-like idea) instead of endomorphisma . . E n d L ( ~ n d i ( ~ ) ) (an

Ext-like idea); we refer to [Jo4] for this. One could check that these variations give the same index for semi-simple pairs, but more general examples may have several indices.

The connection of this subject with Chapter 1 is this: For inclusions 1 E N C M of semi-simple algebras over a perfect field, the index turns out to be the square of the norm ,

M of a certain matrix of natural numbers AN associated with the pair of algebras. Thus the

results of Chapter 1 yield restrictions on the possible values of the index. $Recall that a semi-simple algebra over a field K is the direct sum of its minimal two

sided ideals, and each of these is isomorphic to a matrix algebra Matp(A) for some p 1 1

and some K-division algebra A. However, for studying index, we can for the most part

28

restrict our attention to semi-simple algebras for which the simple components are central (have center equal to K). In fact we can even restrict attention to multi-matrix over K, semi-simple algebras whose simple components are isomorphic to matrix algebras o m the ground field K. Note that if K is algebraically closed, then every semi-simple K-algebra is a multi-matrix algebra, since K has no proper finite dimensional division algebras. We will call an algebra which is isomorphic to Matu(K) for some p 1 1 a

m. Some authors refer to multi-matrix algebras as "split semi-simple algebras". The reason why it suffices to study multi-matrix algebras is that index is stable under

&ggg pf pound f i e . Let K be a perfect field and E an extension field; for any K-algebra M, let M' denote M % E, an algebra over E. If 1 E N c M is a pair of

E E finite dimensional K-algebras, then [M:N] = [M :N ] (Proposition 2.4.4). If M and N E are in addition semi-simple over K, then it is possible to choose E so that M and N E

are multi-matrix algebras over E. Taking E to be an algebraic closure of K will do, but one can also accomplish this with a finite dimensional field extension.

We now come to the definition of the matrix A: for a pair of semi-simple

K-algebras 1 E N c M, First for a pair of a 1 E N c M, the reader can easily check (after looking at Section 2.2) that the index [M:N] is just the ratio of dimensions,

In this ease A: is the 1-by1 matrix with sole entry [M:N]'~. Next consider a pair of

multi-matrix &g&ug. Let {pi : 1 i i i m} be the minimal central idempotents in M, so

that M = @ M q and each Mpi is a factor. Simila~ly let {qj : 1 i j i 4 be the i=l

minimal central idempot~ ts in N. For each pair i,j (1 i i 5 m, 1 i j i n) set

Me - p.q.Mp.q. and Ni,j = p.q.Npiqj. 1 , - J 1 3 1 J

Since pi is central in M, the product p.q. is an idempotent in Mi. If piqj # 0 then 1 J

Mi,j is a factor (Proposition 2.2.3); and, since the map x n p.xp. from Nq. to N. . is a 1 1 J 1,J

nonzuo homomorphism, N. is also a factor with identity element piqj. Define A: to 1,j

be the m-by-n matrix with entries

Xilj = 0 if p.q = 0, and 1 J

1. . = [ M ~ .:N. .11/2 if p.q. / 0. W J 171 I J

[Mi .:Ni .] is a square integer, so X i j >J J is a natural number; in fact

M. p Nij @ MatX. .(K), by Proposition 2.2.2. As the set of pi's is well defined by M 14 1,J

Chapter 2: Towers of multi-matrix algebras 3 2.1. Introduction

and the set of q.'s by N, the inclusion matrix is well aefined by the pair N C M up to J

pseudo-equivalence. Obviously one has A pseudc+equivalent to A: for any 9 (N)

automorphism 9 of M. (See also the discussion following Corollary 2.3.2.) Next consider an arbitrary semi-~imple pair 1 E N c M. The obvious thing would be

M to define AN just as before, using the simp19 components of M and N, but t p s does

not suit our purpose. Instead let E be an algebraic closure of K and set

M Again A N is well defined up to pseudoequivalence by the pair N c M, as the set of E E factors in N u d M are determined by N c M. (In case the simple components of M

and N are central, we would in fact obtain the same result without extending the ground field: thus an alternative definition of A! is this: let E be any field extension of K such

E E that the simple components of N and M are central. Then A: = A I.) N ,

For pairs of multi-matrix algebras, I E N c M, the index matrix, togethei with the dimensions of the minimal ideals of N and M, determine the inclusion up to an inner automorphism of M (Proposition 2.3.3). This is not true for arbitrary semi-Bimple pairs. We will also refer to the index matrix A! as the inclusion a, when N c M is a pair

of multi-matrix algebras. ,

For a semi-~imple pair, the index is related to the i&l&ion matrix in the following simple fashion (see Section 2.4):

-

Theorem 2.1.1. Let N c M be a semi-simple pair. Then the indez of N in M is

given by

~oreovek, for any irredundant mat* A E Matfin@)' there ezists a multi-rn0tr-z pair M ,

NCM vith A=AN.

Corollary 2.1.2. Let N c M be a pair of semi-simple algebras. Then either 2 [M:N] = 4 cos (~r/q) for some integer q 2 3, or [M:w > 4.

Let N c M be a multi-matrix pair and let M c L = ~ n d i ( ~ ) be the pair obtained

by t$e fundamental construction. We will show that M c L is again a multi-matrix pair L M and that AM is the transpose of AN. Consequently the inclusion matrices of the tow^

are immediately determined by that of N c M. These towers have a rich structure, the further study of which requires the introduction of traces.

A on M is a linear map tr: M --, K such that tr(xy) = tr(yx) for x,y E M. It is faithful if the bilinear form (x,y) H tr(xy) is non-degenerate. If K is given as an extension of the real field R, a trace tr is positive if tr(e) > 0 for any idempotent e in

m M. A trace tr an M = e Mat (K) is completely descfibed by the rn =

i=1 r% (tr(fi))lSiSm where fi is a minimal idempotent in the factor Mat (K).

4 Consider a multi-matrix pair N C M and assume that there exists a faithful trace tr

on M with faithful restriction to N. (This is always possible if K is of characteristic zero.) Then there exiats a unique K-linear map E : M -+ N, called a conditional expectation from M onto N, such that .

E(Y) = Y for all y E N E(y1xy2) = y1E(x)y2 for all x E M and ~ 1 ~ ~ 2 E N

tr(E(x)) = tr(x) for all x E M.

[ One has of course E E L = ~nd&(M), and we will show that L is generated as a vector

,space by the elements xlEx2 for xl+ E M; in short: L = (M,E).

Although any multi-matrix pair N c,M generates a tower by iterating the fundamental constiction, traces and conditional expectations in general do not propagate up the tower. That is why we single out a special class of traces as follows. Given 8 E K* and N c M as ahve, define a Markov pf modulus 8 on N c M to be a faithful trace tr on M with faithful restriction to N for which there exists a (necessarily unique) trace Tr on L = ~ n d & ( ~ ) such that

Tr(x) = tr(x) I for all x E M 0 Tr(xE) = tr(x)

To know whether or not such traces exists on a pair N c M, one has again to consider the inclusion matrix A, and its natural action (here from the right) on vectors with . coordinates in K (after reduction of A modulo the characteristic of K).

,''-s+-r Chapter 2: Towers of multi-inatrix algebras 5 2.1. Introduction

Theorem 2.1.3. Let N c M be a multi-matriz pair with inclusion mat& A and let

P E K* ( I ) ~ h e d e&ts a Markov trace tr of modolw B on N C M if and only if there

ezists a row vector $ E K~ with

I = &, a 1 1 coordinates of k distinct from 0, and a I 1 coordinates of k~ distinct from 0.

If this hokkr, tr is described by a scdar multiple of k. (ii) Let tr be a Markov trace of modulw on N C M, and let Tr be the eztension

of tr io L = ~ n d i ( ~ ) = (M,E) such that @r(xE) = tr(x) for all x E M. Then Tr is

again a Markov trace of mudulzls P on M c L. (iii) I f K is of characteristic 0, the modulus of any Markov trace on a pair N c M is

a totally positive algebraic integer.

A pair N C M is called wnnected if the intersection ZN n ZM of ,the centers of M

and N consists only of scalar multiples of the identity. Using the Perron-Frobedus- theory of matrices with positive entries, one has:

Theorem 2.1.4. Assume that K is given as an eztension of the real field IR. Let N c M

be a connected multi-matriz pair with inclusion matriz A and let /3 K*. There exists a positive Markov trace of modulus P on N c M i f and only i f

Any two positive Markov traces on N c M are proportional.

This implies:

Corollary 2.1.5. The set E = K(Q0 of Chapter 1 (seeq~roposition 1.1.5) is also the set r

of square roots of moduli of positive Markov traces.

We now return to the tower I M0=NCM1=Mc c M ~ c M ~ + ~ C ...

generated by a multi-matrix pair N c M. We assume that there exists a Markov trace tr of same modulus B E K* on N c M. As Markov tracea propagate according to c W m W (ii) of Theorem 2.1.3, one has for ueh k 2 1 ' a Markov trace trk on Mk-l c Mk and a

conditional expectation Ek : Mk + Mk-l, which can also be viewed as an element of

Mk+l. We denote by Atr,k(MocM1) the subalgebra of Mk generated by the unit and

El,. . . ,Ek-1.

Theorem 2.1.6. Let Mo c M1 be a multi-math pair on which there ezists a Markov

trace tr of modulus P. For each k 2 1, let Mk and % be as above. Then

(i) Mk is generated by M1 and El,-

(ii) The idempotents El,. . . ,Ek_i satisfk

W.E.E. = Ei if li-jl = 1, and . 1 J 1

E.E. = E.E. if li-j( 2 2. 1 J J 1

The-proofs of Theorems 2.1.3 to 2.1.6 appear in Section 2.7. In the "generic case" (see below), it is remarkable that (ii) is a complete set of relations for the Eil s. This motivates

the introduction of the Temoerlev-Lieb ,$+, which first appeared in statistid

physics (see [TL] and appendix II.b.), For any P E @ and for any integer k 2 1, the &algebra (with unit) A is defined by the presentation with generators el,. . .,ck-l

P,k and with relations:

Choosing a number q # 0,-1 (in K or in a quadratic exthsion of K) with D = 2 + q + q-l, we say that p is generic if q is not a root of 1 (or if q = 1 when K is

of characteristic zero). First, we describe the structure of A P,k'

Theorem 2.1.7. (i) For any B E k? and fir any ,k 2 1, the algebra APTk hf

, and the natural morphism AP,k + AP,k+l is an injection.

(ii) I f /3 is generic then ABBk is a multi-matriz algebra isomorphic to c.

(K), where $1 = 1 and {f] = M - bFl] for j 2 1. Moreover there

ezbts a faithful normalized trace trk on Aflyk such that @trk(wtj) = trk(w) for w E

and j s k-1.

Chapter 2: Towers of multi-matrix algebras i 8 2.1. Introduction

Next, we describe the algebra Atrlk(MocM1) previously introduced.

Theorem 2.1.8. Let Mo c M1 and let t r and /3 be as in Theorem 2.1.6,

(4 If /3 is generic, the map AAk -4 Mk defined by e. * Ej for j = 1,. +. ,k-1 ,is an J

isomorphism onto Atrlk(M0cM1). Moreover this isomorphism is compatible with the trace

trk on and the Markov trace trk on Mk.

( ig Assume K is given as an eztension of W, and that the Markov trace t r is positive (so that p = [Ml:MO]). Then Atr,k(Mo~M1) is isomorphic to a certain quotient B

Plk AAk which is ezplieitly described in Section 2.9.

The braiding relations, which appear above first in Theorem 2.1.6, suggest a strong connection with Artin's braid groups and with Iwahori' s Hecke algebras. This observation haa constituted a breakthrough in the study of knots in !R3; see [Jo~], [Jo~], [Fre], and

[HKwl. For any q E @ and for any integer k 2 1, define the HBELB

HqYk to be K-algebra with unit presented by generators gl, ,gk4 and by relations

2 gi = (Q-114 + q i = l,...,k-l

gigi+l& = gi+lgi&+l i = 1,. .-,k-2

gigj = gjq if (i-j ( 2 2 i,j= l,...,k-l

gi+l Assuming q # -1, one may also set ei =

q+l and /3 = 2 + q + q-l, and check that

Hqlk has a presentation with generators el,- .,ekql and with relations

e? = e. 1 1

i = 1,. . .,k-1

! , 1

qei+,q - /3- ei = ei+leiei+l - /3-'ei+, i = 1,. . ,k-2

Then HQsk is an algebra of dimension k! over K, and the natural morphism

where IA is a factor for each X E lk; we refer to Section 2.10 and 2.11 for a more precise

description.' That there is a relationship between Hecke algebras and the algebras appearing in towers is clear from the presentation of Ablk with eifs and from that of

Hqlk with eil 8. More precisely:

Theorem 2.1.9. Let q E K* be a number which is not a root of 1 (with q = 1 allowed

when char(K) = 0), and set /3 = 2 + q + q-l. Consider an integer k 2 1. Then Am is

isomorphic to the quotient of the Hecke algebra H by the relation q1k

and one has

where 4 the subset of lk consisting of those partitions with Young diagrams having at

most 2 rows.

The relation (S) was pointed out to us by Steinberg. In terms of the generators el,. . ,ek-l of Hqlk introduced above, it may also be written as

Our exposition is organized as follows. Section 2.2 collects preliminary material on pmmutantt. Section 2.3 shows how to

define the index matrix or inclusion matrix of a multi-matrix pair N c M, and how to encode the relevant information in a Bratteli diagram. Concerning chains of multi-matrix algebras (a natural generalization of pairs), we have added an exposition of the m t h m o a due to Ocneanu and Sunder. Theorem 2.1.1 and Corollary 2.1.2 about the of a pair N C M are proved in Section 2.4.

Hq,k + Hq,k+l is an injection. Section 2.5 concerns in general, Section 2.6 pnditiond mctat iong, and Section 2.7 in particular. Theorem 2.1.3 to 2.1.6 are proved in Section 2.7.

If q is not a root of 1 (with q = 1 allowed if the characteristic of K is zero), then Section 2.8 is about the algebras Ap,k, with emphasis on generic /3, and Section 2.9 Hqlk is a multi-matrix algebra with factors in bijection with the set 'Pk of partitions of

k. We may thus write discusses non-generic /3. Section 2.10 is a leisurely digression introducing Recke m, and the final section ahows how Ahk is a quotient of the appropriate Hecke algebra.

40 T Chapter 2: Towers of multi-matrix algebras

<, , ,-'

If K is our favorite field of p m ~ l e x numb=, multi- matrix (or semi-aimple) algebras "are the same" as &$.& gimensiona Q -. This is of course well-known, and made precise in Appendix II.a. If follows that the present Chapter 2 may be looked at as a study of certain c*-aJgebras which are AF, namely approximately finite; see [Eq. In Appendix II.b we report briefly on the appearance of the algebras lo in statistical

mechanics. Appendix I1.c contains additional material on the algebras Ap,k non-generic p.

2.2. Commutant and bicmmutant.

Let F be an dgebra over a field K. (By an algebra we will always mean a;n algebra with an identity element I.) The ~ommutant CF(S) of a subset S of F is the algebra of

all elements of F which commute with each element of S. One obviously has S c CFCF(S) for any subset S, and the reader can check that CFCF(CF(S)) = CF(S)

for any S. If A is a subalgebra of F, then CFCF(A) is an algebra containing A, a

sort of closure of A in F. "-,

Let W be any K-vector space (not necessarily finite dimensional). We remark that for each d E OI there are canonical dgebra isomorphisms

(For atly algebra A, Matd(A) denotes the algebra of d-by-d matrices over A.) For

1 r i I d let ri : wd -. W be the projection on the ith component, and for 1 I j I d let d e. : W -, W . be the injection such that ri o e. = 41 on W for all i. Given

J J x E ~ n q w d ) d h r matrix [ 3 , j E Matd(En%w)) by 3,j = ri 0 x 0 ei. One can

check that the map x n [xi .] is an algebra homomorphism, and that it ha. an inverse, ,J

which t a k a a matrix [xi,j to x r i o 3,j o 7 E ~ n d l ( ( d ) . This establishes the first

i , j isomorphism. G t {ei,j 1 I i,j r d) denote the standard system of matrix units in

Matd@); the matrix s . has a 1 in the (i,j) position and zeros elsewhere. Then the 1 J

is an isomorphism of Matd(Endl((W)) onto E n q W ) % Matd(K). We will identify

3 2.2? Commutaht i d bicommutant

Matd(En%(w)), and En%(W) e Matd(K) via these iaomorphiams. For any

subalgebra A of EndK(W), the algebra A 4( Matd(K) is thus identified with Matd(A),

aqd A @ 1 with the set of diagonal matrices

a

or with the set of diagonal endomorphisms

The following unassuming lemma is the basis of both the Jacobson density theorem ([BA8], p.39) and of von Neumann's bicommutant theorem ({Talc], p.74), two cornerstones of non-commutative algebra.

~ e & 2.2.1. Let W be a vector space over # and let A be a subalgebra o f

EndK(W) with 1 E A. Write A' for CEnd (A). K(

(i) T ~ C commvtant of A B I in ~nd,,p,vd) ir A' % Matd(K).

(b) The commvtant of A t Matd(K) in ~ n d ~ ( ~ ~ ) is A' @ 1.

' - ~ r m t (a) Since ( a o l ) ( X x i j e eij) = I ? , j @ eij, and ( C x i j @ e i , j ) ( l )

= Z I , j a e eij, it follows that C x i j @ e i j lies in the cornmutant of A @ 1 if and O ~ Y

if x. . E A' for all (ij). 1,J (b) If x = C x i j @ e i j commutes. with A % Matd@), then in plut idar it

1 , J

commutes with the matrix units 1 @ e for all (p,~), A"

It follows that x . = 0 for j # u and x. = 0 for i # p, and x = x Thus ",J "," P,P'

x = Cy @ ei,i = y @ 1 for some y E EnddW). Since in addition x commutes with i

A @ 1, it follows that y E A'. On the other hand any element of the form y @ 1, y E A', COmm~te~ with 4 % Matd(K). #

38 Chapter 2: Towers of multi-matriu algebras 1 3 2.2. Commutant and bicornmutant 1 39

I Consider a f e F g En%(V), where V is a finite dimensional K-vector space. The

basic facts abput the representation theory of F are:

(i) Any finite dimensional F-module is completely reducible, and (ii) any irreducible F-module is equivalent to the standard module V.

. , .

This index is the square of an integer by Lemma 2.2.2. In case M "itsel'f is a subfactor of a factor F, the same leinma implies

(By a module over a K-algebra A, we will always mean a module; that is the which is a preview of Proposition 2.3.5 below.

identity of A acts aa the identity on the module. Thus an A-module is also a K-vector .

space.) If W is any finite dimensional F-module, it follows from (i) and (ii) that Pro~osition 2.2.3, Let M be a multi-matrix subalgebra of a factor F with 1 E M C F. dimK(V) divides %(W), say dimK(W) t dim@) = dl and W is equivalent to the

d F-module V with the diagonal action I (a) CF(M) is a multi-matrix algebra, and has the same minimal central idempotents

We call d the mult i~l ici t~of the F-module W. Proof. Let pl,. .,pm be the minimal central idempotents of M, and identify M

Lemma 2.2.2. Let F be a factor and let M be a subfactor of F with 1 E M c F. Then

(a) CF(M) is a factor, \

(b) CFCF(M) = Mi

(c) M % CF(M) is isomorphic to F.

w. Identify F with En%(V) for some finite dimensional K-vector space, V.

Similarly M is isomorphic to Ends((W) for some W, and any isomorphism

a : En$(W) -4 M c En%(V)

is a representation of E n q W ) on V. It follows that a is equivalent to the diagonal

representation~"of En%(W) on wd for some d; thai is r e can identify F with -

~ n q w d ) and M with E n q W ) @ 1 c ~ n d l ( ( ~ ~ ) . Then by the previous lemma

CF(M) = 1 @ Matd@), which is a factor. Furthermore CFCF(M) = EndK(W) @ 1 = M.

m' with @ piM; each piM is a factor. As the center M n CF(M) of M is contained in

i=l m -

the center CF(CF(M))nCF(M) of CF(M) and a8 X p i = 1 , one has i=l

m CF(M) = @ piCF(M)pi. It is straightforward to check (using pi E M il CF(M)) that

i=l 8

piCF(M)pi = CpiFpi(piM), and this is a factor by lemma 2.2.2. Thus c ~ ( M ) is a

multi-matrix algebra, and its center coidcides with that of M, so that (a) holds. Similary one has

*

.-.

and the lemma implies

I . so that (b) holds: #

Let N be a subfactor of a factor M with 1 E N c M. Recall that the index of N in Remark@.

M is (1) If M is a subalgebra of a factor F and if M is not semi%imple, one may have M C (C (M)). An example is given by the algebin M of matrim of the form :] # F F

[M : N] = (dimK~)(diXilK~)-'. I in the factor F = Mat2(K), for which CF(CF(M)) = F.

Chapter 2: Towers of multi-matrix algebras,p8% ,,"""4

1 3 2.2. Commutant and bicommutant

i 41

:" .d (2) The biccommutant theorem is valid for all finite dimensional s&%imple algebras;

see [Weyl], Theorem 3.5 B. Brauer [Br] and Weyl (op.cit.) stressed the importance of the using CF(CF(M)) = M. Now take cornmutants and apply Lemma 2.2.2 in qFq to obtain theorem for invariant theory. The ultimate form of the theorem is Jacobson's density theorem ([Jacl], [Jac2]). It was noted by Bourbaki ([BA8], p. 39) that Jacobson's the conclusion. #

theorem follows fkom von Neumann's proof of his bicornmutant theorem for operator .

algebras ([vN] or [Tak]). Prowsition 2.2.5 Let M be a multi-math subalgebra of a factor F with 1 E M C F and let q E M U CF(M) be a nonzero idempotent. Then (3) Let V be a vector space such that F = EnUV). The usual notation in

(a) qMq is a multi-matriz algebra. functional analysis for CF(M) is M'. In some books of algebra, V viewed as a (b) CqFq(~Md ' qCF(M)q.

CF(M)-module is called the counter-module of the M-module V.

EIQQf. The notation being as in the proof of Proposition 2.2.3, set % = piq and 2.2.4. Let M be a subfactor of a factor F with 1 E M c F and let

q E M U CF(M) be a nonzero idempotent. Then t observe that q = qi. One obviously has

(a) q ~ q is a factos; i=l E (b) CqFq(~Md = qCF(M)q- I I

EIQQf. Assume first that q E M. (a) If M is identified with EndK(V), then qMq is isomorphic to EnUqV). I m m

(b) Consider first x E qMq c M and y E qCF(M)q. Choose z E CF(M) ' with 4CF(M)q = @ qiCF(M)$ = @ ~ ~ ~ ~ ~ ~ ~ p ~ ~ ) q ~ ' i=l i=l

Y = qzq and compute Claim (b) follows because

I I Cq Fq (qiMqi) = qiCp.Fp.(~iM~i)(li

i i i 1 1 It follows that qCF(M)q C CqFq(qMq). Consider then s = qsq E CqFq(qCF(M)g) and

t E CF(M). As tq = qt one has I by Lemma 2.2.4. We leave details of (a) to the reader. #

st = sqqt = sqtq = qtqs = tqqs = ts.

It follows that CqFq(qCF(M)q) c qCF(CF(M))q, the last term being qMq by Lemma

2.2.2. Taking now commutants and applying Lemma 2.2.2 in qFq, one has

Assume now that q E CF(M).

, (a) The linear map p from M to qMq which sends x to qxq is a morphism of algebras and its image contains q = qlq. As M is simple, p is an isomorphism.

(b) By the first part of the proof one has

Remarks. ( I) , The algebra qMq is called the reduction of M by q. (2) It is easy to find' examples which show that one cannot suppress in Proposition

2.2.5 the hypothesis that q is either in M or in its commutant.

Next is a version of the Skolem-Noether theorem:

Pro~osition 2.2.6. Suppose that M is a factor, N and W are subfactors of M containing the identity element of M, and p : N + W is an isomorphism. Then there is an inner automorphism B of M such that .BIN = p. In paPticular, any automorphism of

M Is inner.

a f . We identify M with EndK(V) for some vector space V over K. Let W be 4

an irreducible N module; then any N module V is equivalent to the direct sum of d

' 42 .-. Chapter Z: Towers of multi-matrix algebras

copies of W, where d = dimK V s dirnl( W. In particular the two N module structures

on V defined by the actions (n,v) H 'si! &d (n,v)G 4 n ) v are equivalent. Hence there is an inuerti&le u E End@ = M such that u(nv) = dn)u(v) for all n E N and v E V.

Thus d n ) = wuU1. #

We end this section with the finite dimensional case of the coupling constant theorem of Murray and von Neumann (see Theorem X in [MvN I], and also Theorem X in [MvN Iq). This is a digreasion motivated by the importance of the theorem for II1-factors (see

Chapter 3).

Consider a factor M = EndK(V), where V is of dimension ,u over K, and a

I representation r of M in a vector space W. Assume that s is of multiplicity d, so that dimK(W) = dp, and view n as aninclusion M c F with F = EndK(W). Any non

zero vector f E W determines a cyclic M-mbmodule M t of W, as well' as a cyclic CF(M)-submodule CF(M)< of W.

ition 2.2.7. With the notations above one hw

di%(CFtM)O/a%(M€l =: d / ~ = [F : Ml/amK(W) 3 jorall ~ E W with t#O.

w. Identify W with vd. Let f = . .,td) d)B W and let st be the dimension

of the subspace U of V generated by fl,. . +,Ed; one has 1 s L s min(#d).

One may choose 4 linear independent 6 's and write the d-st reqaaining ones as j

linear combinations of the chosen ones, so that clearly dim(Mf) = tp. On the other hand CF(M)f consista of vectors of the form

with (ai,j)lii&d € Mat&), 80 that dim(CF(M)f) = a. ~he'conclusion ~ O ~ O W S . #

2.3. hMon matrix and Bratteli diagram for inclusionti of mnlkt r ix algebras.

Let M he a multi-matrix algebra over K, with minimal central idempotents pl, - - .rpm. Each minimal two-sided ideal piM is a factor, isomorphic to Mat (K) for

Y

5 2.3. Inclusion matrix and Bratteli diagram 43

m m some integer $' w e identify M with te piM, and the center ZM of M with @ Kpi,

i=1 is1 and~denote by p or 3 the m-tuple (~,. . . ,pm)t of dimensions. (The superscript t

means "transpose", because we think of ,u as a column vector,) The isomorphism class of M is completely described by the class of $ modulo permutation of its wordidtea.

m Observe in particular that the K-dhe11sion of M, which is x p ; , is the square of the

, i=l

I Euclidean norm of 3. '

Let N be a multi-matrix subalgebra of M with 1 E N (I M.' Denote by ql,. . . ,qn n

the minimal central idempakpts of N; we identify N with @ q.N and ZN with j=1 J

n @ Kq.. If q.N g Mat,(#), then the isomorphism class of .N is described by the vector

j=1 . J J J t v = (ul,...,un) .

Set, moreover,

M. . = p.q.Mp.q. and NiTj = p.q.Np.q. (liism, lsjsn). 1,J 1 l 1 J 1 J 1 J

' As pi is central in M, the product p.q. is an idempotent in piM. Lemma 2.2,4 shows 1 J

that -M. . is either 0 or a factor, depending on whether p.q. is 0 or not. If p.q. # 0, 111 1 J 1 J the map x -4 piApi from Nq. O N. . is a morphism of algebras containing p.q. in its

J 1 J 1 J M image; as q.N is simple, the map is an isomorphism. Recall that the index matrix AN

J for the inclusion N C M is h e m-by-n matrix with entries

A . . = o 'if p.q. = 0, and I J 1 J

Ailj = ([Mi,j : Ni,j) I /2 . if p.q. I J # o

Note that MiYj 1 Ni . @ MatAi(K), using Lemma 2.2.2, and, under this i s o m o ~ , Nij ,J

1 J M is identified with N. . @ 1. We will see that, for multi-matrix algebras, the matrix AN

1J

together with the vector of dimemions 3 determines the inclusion N c M up to isomorphism (Pmpositiom 2.3.3); r e therefore J s o refer to A! as the inclusion matrix for

the pair N c M. We next observe that A! can be described in terma of the representation theory or,

. . m equivalently, the K-theory of the rings N and M. Identify M with. @ EndK(Vi)., where ,

j=1 m

Vi - is a vector s p a e of dimension &, acting on V = te Vi. An i sv-~phism i=l

Chapter 2: Towers of multi-matrix algebras /;U-' 3 2.3. Inclusion matrix &d Bratteli diagram

n r : @ Maty.(K) -, N c M with r(Mat,.(K)) =.Nq. for each j is then a representation of

j=1 J J J i I

n @ Mat,(g) on V. Each Vi is a non-trivial N-submodule and we let ~r~ denote the

j=l J

corresponding subrepresentation. If q.p. # 0 then q.p.V. = q.V. is again a non-trivial J 1 J 1 1 J 1

N-submodule (although an M-submodule), and the corresponding subrepresentation

Iri,j is zero on Mat (K) (l # j), while the restriction of r. to Mat,.(K) is a unital ul 1,j J

isomorphism onto N. . c M. = EndK(qjVi). The multiplicitx of this representation is w 1,j A. so there is a basis of q.V. such that the matrix of qj(xl, , . +,xn) is 1,j3 J 1

diag(x.,. . .,x.), with x. being repeated A. times; cf. the proof of lemma 2.2.2. Hence J J J 1,j

there is a basis of Vi such that the matrix of ri(xl,. . . ,xn) is

', diag(xl,. .,xli3,-. ~ ~ 3 ; . . .;xI1,. e , ~ ) ,

, \ I I

with x. being repeated A. . times. J W

Next we introduce some terminology in order to give the K-theoretic 'interpretation of the inclusion matrix. The KO group of a ring R is an abelian group constructed from the

representation theory of R; the set of equivalence classes of finitely generated pr?j&ive R-modFes forms an abelian semigroup, the operation being direct sum, and KO(R) is the

< 1

'quotient ,goup of this semigroup. For a multi-matrix algebra M g @ piM g @ EndK(Vi),

Ko(M) is the free aabean group generated by the classes of the standard representations 1 ai(x) = xpilV.. This can also be expressed in terms of idempotents. For any idempotent

1

e E M, let [elM denote the class of the left, ideal Me regarded as an M-module. If ei is

any minimal idempotent in piM, then Mei = Mpiei is a minimal left, ideal of M, and as

a left M-module it is equivalent to the standard module {ai, Vi); thus {[?IM} is a basis

of KO(M). Similarly, choose minimal idempotents f. of q.N and denote by P. the J J J

corresponding'irreducible representation of N, for j = 1,. . . ,n. There is a canonical map

KO(N) -+ KO(M) induced by the inclusion N c M, which is the unique morphism of

abelian groups taking [elN to [elM for any idempotent e in N.

Pro~Osltlo . . n 2.3.1. Let N c M be a pair of multi-matrix algebras and let the notation

be as above. (a) A is the matriz of the.map Ko(N) KO(M) induced by the inclusion

N 3 M, with mpect to the basis ([?]M)lrism 01 Ko(Mly Zm and ([fjN)lijln of

M (c) AN is irredundant.

(d) The repmentation IndE(/3.) obtained by induction is equivalent to x4,j4 for J i

j = I, .--,n. (e) The repressntatiqn ailN obtained by restriction is equivalent to x + , j b j for

j i = l,...,m .

(f) A: is indecomposable.if and only if zM n ZN = K.

u f . (a) We have to compute the image . [f ] € KO(M) of the basic element m '

j M m -

[fjN E K ~ ( N ) ; where j E (1,. p}. As f. = x p . f . one bas [fdM = b{lM. But J 1 J i=1 i=l

f.p - f.q.p. is the sum of A minimal idempotents in Mij, as one sees from the proof of J ~ - J J ~ i,j 2.2.2 or the analysis of representations above. Furthermore, a minimal projection in Mi j

m -

remains minimal in Mpi, and is thus equivalent to ei. Thus [fdM = ZAij [eiIM, aa was i=l

to be shown. (b) This follows because $ = [1IM and 8 =

n

(c) Given i E (1,. . . ,h), there exists j with p.q. # 0 because 1 J

x qj = 1, hence I

i=l th the i- row of A: is not zero. Similarly, no column of A: can be zero because

i=l

(d) Observe that Ind;(fli) acts on M % Nf. which is canonically isomorphic to J Mfj; thus indK(4) is left multiplication by elements of M on Mf.. The proof of (a) J

n shows that lnd$(pj) = ,Aijq.

(e) This follows directly from the analysis of representations above, i.e. the diagonalization of the representation r.

(f) If di%(ZM, il ZN) > 1, there exists an idempotent r E ZM n ZN with r # 0 and

r # 1. With an appropriate ind 'n of the pi's and q.'s, one has r J i " g J

I

Chapter 2: Towers of multi-matrix algebras i I

5 2.3. Inc1usion matrix and Bratteli diagram 47

for some m' ,n' with 0 < m' < m and 0 < n' < n. It follows that M. . = 0 and Xi,j = 0 llJ

unless

l i i i m ' and l s j j n ' m l + l < i < m and n 1 + l i j < n

so that A! is decomposable. One checks conversely that, if A: is decomposable, then n,

ZM fl ZN contains a nontrivial idempotent which is a sum of some pi's and also a sum of

some q.' s. # J

Gorollarv 2.3.3 Let L,M,N be multi-matriz algebrcrs with 1 i N c M c L. Then L M A: = AMAN.

EIPPf. By functoriality of KO (or by counting multiplicities). # .

Here is one detail one has to pay attention to. Let N c M be a pair of multi-matrix algebras and let A! be the mnespondi~g index matrix; assume that M [respectively N]

is a direct sum of m [respectively n] factors. As a matrix with rows [respectively columns] indexed by the minimal central idempotents of M [respectively N], the matrix

M AN .is well defined by N c M. But as a matrix with rows [respectively columns] indexed

M by (1,. ; .,m) [respectively {I,. . .,n)], AN depends on orderings pl,' -,pm and

q,,. . .,qn of these idempotents, and thus A: is only defined up to pseudoequivalence by

N c M. Taking for a moment the first point of view, the following makes sense: let N c M and F C 1GI be pairs of multi-matrix algebras; if there exists an isomorphism 8 : M -4 ICP with 4 ~ ) = then A! - with strict equality. This has a converse that we - formulate as follows.

Pro~osltlon 2 . . .3.5 Let N, IT be two multi-mat& subalgebras of a multi-matriz

algebra M, given together with an isomorphism p : N -4 R. If A! = AM then ip m'

&&. We show this for M a factor, and the general case will follow; we may thus identify M with En%(V) for some vector space V over K Set

M A! = A = (Al,. . .,An) E Ml,n(W). R'

Let

.be decompositions into factors, where the indices are chosen so that dq.) J J = C. for

j = 1,. . . p. Each idempotent q. is a sum of (say) v. minimal idempotats in qjN, and J J thus of A.v. minimal idempotents in q.M (these we still minimal in M). The same

J J J holds for G.. J Consequently there d s t s an inner automorphism 8' of M such that - 8' (qj) = 5. 3 for j = 1,. . ,n. It follows that one may assume from the start that q. J = qj

.

for j = 1,. . .,n. For ' j E 11,. . b,n), set V. = q.(V). By the Skolem- Noether theorem, Proposition J J

2.2.6, there exihs g. E GL(Vj) such that d y ) = gjy4' for .U Y E qjN. As J

qj = I , one hw V = @ V. and one may define g = e g. E GL(V). Then 1s j in 1s jsn 1s j in

has the desired property. #

w l a r v 2.3.4. +g multi-matriz pair N c M is isomorphic to N~~~ C MO~'.

w. This follows from 2.i.3, as the two pairs have clearly the same inclusion matrix. n

Here is another way to look at it: Let P be the antiautomorphism of N = @ Mat,(K) j=1 J

which coincides with the transposition on each factor, and view P as an isomorphism N -4 then P may.& extended to an isomorphism cu : M -I M ~ ~ ~ . #

Remarks. (1) Our A: is as in [Jo 21, but is the transpose of A! in [Jo 11. SO our proposition

-, M 2.3.l.d corresponds to 3.2.1 in [Jo 11, which reads & = nAN.

(2) Proposition 2.3.3 shows that a multi-matrix pair N C M is characterized (up to M

isomorphism) by the data 3 (which describes N up to isomorphism) and AN (which

describes the inclusion). The following simple examples show how partial descriptions fail to be complete.

(i) bouii'der the two subalgebras (both of dimension 62):

N = K @ Mat5(K) @ Matg(#)

m = Mat2(K) @ Mat3(K) @ Mat7(K)

of the factor M = Mat12(K), both inclusions being described by 4

I Ll

Then A: = = (1 1 1) though N and A are not isomorphic. : IT

(ii) Consider N = K @ Mat2(K) included in M = MatgQ by (x,y)

M M H = Matg(K) by (x,y) - . Then A N and A are pseudo-equivalent to (2 1) m but M and are not isomorphic.

(iii) Consider finally N = K @ Mat2(#) included in M = Mat5(K) by

. Then the first inclusion matrix (3 1) is not

pseudo-equivalent to the second inclusion matrix (1 2).

The next proposition is a.special case of a statement which appears in [BA 81, 55, 1 exercise 17. .

Pro~osition 2.3.5 Consider two multi-matriz subalgebras M,N of a foctor F with 1 E N c M c F. The inclvsion matriz for CF(M) c CF(N) is the transpose of the inclusion

mcatriz for N c M. . .

EEQPf. The proposition is obvious if M and N are factors (see the Remark following 2.2.2). In general; write

and denote by X. . the entries of the inclusion matrix for , 111

n n CF(N) = e q.C (N) 3 CF(M) = @ piCF(M).

j=1 J i=l

One has by definition

and by Proposition 2.2.5.b,

., A . . = [C (N. .) : C J J qjpiFqjpi 1,J qjpiFqjpi

As N. . and M. . are factors in q p.Fq.pi one has 1,J 1 J 5 1 J

by the particular case observed in the remark following 2.2.2, #

The Bratteli diaeram. It is useful to describe a pair of multi-matrix algebras N c M by its Bratteli diaaarq

B(NcM), which is a bicolored weighted multigraph defined as follows. ("Multigraph" mean's that two points may be joined by more than one line, "weighted" means that each point is given together with a positive integer, an& "bicolored" means that points are given one of two colors, in such a way that any edge in the multigraph connects points of

m n different colors.) If M = @ Mat (K) and N = e MatJK) are as above, then

i=1 Y j=1 J

B(NcM) has m black vertices bl, . . . ,bm with respective weights ply. . . ,pm and n th white verticea wl,. . .,wn with respective weights vl,. . . ,vn; moreover, the i- black

vertex and the $ white vertex are joined by A. . lines. (These diagrams were first 1,J *

introduced in order to study inductive limit systems of finite dimensional C -algebras; see [Bra] and [Effj.)

3v

Exam~le 2.3.6. If N = Matv(C) @ 1 C M = Mat,(C) gMat3(C), then B(NcM) is 111 S

and A! = [3].

Examole 2.3.2 Let G be a finite group and let H be a subgroup of G. As complex group algebras are semi-simple by Marrchke's theorem (example I1.2), C[E[I C C[G] is a multi-matrix algebra pair.

In particular, let e3 be the group of permutations of {1,2,3) and let e2 be that of

{1,2}. Minimal central idempotents of C[e3] correspond to Young frames, and ako to

50 Chapter 2: Towers of multi-matrix algebra 5 2.3. Inclusion matrix and Bratteli diagram

1 3 2 3 1 irreducible representations of e3. We denote them by

p, cofiesponding to the identity representation of e3 1 1 ] 'W. As in the examples, we always draw Bratteli diagrams on two levels, with the upper

arresonding to the 2-dimensional irreducible representation level representing the larger algebra; the coloring of the vertices is actually superfluous,

since the two types of vertices are labelled by their level. The equation A? = has the - following interpretation: For a given black vertex v, consider the set of edges entering v,

Similarly for and for each edge take the weight of the white vertex incident to that edge. The sum of these weights, over all such edges, is the weight of v.

4, corresponding to the identity representation Ir, of e2 Prowsition 2.3.9. (a) Let N c M and c rn be two multi-matriz algebra pairs with

, the same Bratteli diagram. Then there ezists an isomorphism 8 : M - with

~t is easy to check that the representations (b) A bicolored weighted multigraph B (with positive integer weights) is the Bratteli diagram o f a multi-matriz algebra pair i f and only i f the weights and the multiplicities A.

~t , follows that the inclusion matrix and the Bratteli llj respectively a8 rm , irm@ rB , r~

diagram for C[e2] c C[e3] are

w. As (a) is nothing but a restatement of Proposition 2.3.3, we are left with the

Let y, . . . ,lr, be the weights of the black points in B and let vl, ,vn be those of

the white points and suppose = A. .v.. set c. 1,J J j

permutations of {1,2,3,4). The group 6 4 h a irreducible representations m n M = @ Mat (K) N = @ Mat,(K).

j=1 4 j=l J

= l m 3 $ r[ of lines joining the i& black point with the $- th white point in

B. Define a map N - M by associating to (yl,. . . ,yn) E N the element

of respectivb dimensions 1,3,2,3,1, whose restrictions to e3 are res~ec t ivd~ (xl,. . J ~ ) E M with xi the block-diagonal matrix

*a $ IF IF@ I! Xi = diag(yl,, . .'Yli' ' .;Yn,' ",yn)

times. This map identifies N with a subalgebra of M and sel for example, [Serl], Example 5.8. It follows that the inclusion matrix and the Brattdi 14 B(NcM) is the B originally given. # diagram for ([e3] c C[6pl are as follows. (The reader will check that 112 = f )

f-

52 ,- Chapter 2: Towers of multi-matrix algebras - t

'%. ..

Chains of multi-matrix algebras. Now consider an increasing chain (finite or infinite)

of multi-matrix algebras over K,

k k Let pl,... ,pm(,) denote the minimal m t r a l idempotents in Mk, let = k be k the inclusion matrix for Mk c Mk+l, and let g be the vector of dimensions of Mk, so

that p ! ~ ~ g Mat k(K). (Thua $ = A(~- ' )A(~-~) 1(O)p0.) We associate with the pi \

chain of algebras a (finite or infinite) Bratteli diagram B, which is the union of the diagrams B(MkcMkS1), the upper (black) vertex of B(MkcMk+l) drresponding to

p;+l being identified with the lower (white) vertex of B(Mk+1~Mk+2) corresponding to

the same idempotent. For example the diagram for CG1 C CG2 C CG3 C CG4 is

(See examples 2.3.7 and 2.3.8.) We say that the v&ca 3 corresponding to the minimal

central idempotents p! in Mk belong to the kth f l s of the diagram. The vertices of

the kth and k+lSt floors together with the edges joining them - i.e., the image of th B(MkcMk+l) in B - constitute the k story of B. The Bratelli diagram B is thus a

weighted multigraph with the following features: (1) There is a function cp from the set of vertices of B to I = {0,1,2,-• *), which assigns to each vertex the floor which it occupies.

(a) There are only finitely many vertices on each floor; that is p-l(k) is finite k foi ail k. n C'B) t 0, we write ~ - l ( k ) = (4,. . . , v ~ ( ~ ) ) .

(b) The range of cp is either an internal [O,p] in I , if B is finite, or all of I, if B is infinite.

(2) Two vertices v and w are adjacent only if Idv)-y(w)l = 1. There are X k i ,j

,edges joining $ and v:+l. '.

(3) If both the kth and k+lst floors are occupied (i.e., if cp-'(k) # 0 and cp-l(k+l) # 0) then each vertex on the kth floor is adjacent to at least one vertex on the k+lst floor, and each vertex on the k+lst floor is adjacent to at least one vertex

I 3 2.3. Inclusion matrix and Bratteli diagram 53

I I on the kth floor. That is, the m(k)-by-m(k+l) matrix A ( ~ ) = (A! ) is

,J irredundant.

k (4) Each vector v! has a weight pi E {l,2,ee -) called its dimension. The k dimensions {pi) and the "multiplicities" {A! .) satisfy

9J

Conversely, given a weighted multigraph B with properties (1)-(4) above, we can, by iterating the procedure of Proposition 2.3.9.b, construct a chain of multi-matrix algebras with Bratteli diagram B.

Pro~osition 2.3.10. Suppose

i ~ E M ~ c M ~ c ..-, and

~ E A ~ c A ~ c

are two chains of multi-math algebras with the same Bratteli diagram. Then there is an isomorphism gl' of Ma = UMk onto Am = UAk such that #(Mk) = Ak for all k.

k k

We have to produce a sequence of isomorphisms % : Mk -I Ak such that

I '+lIMk = &. Let gl'o : 'MO -' AO be any isomorphism. Suppose gl'o,s . . ,% have been -

defined. Then by Proposition 2.3.9.a, there is an isomorphism %+1 : Mk+l -' Ak+l

such that %+l(Mk) = Ak, and by Proposition 2.3.3 there is an inner automorphism

h+l of extending Q L Y ~ ~ ~ . T ~ U S we uo set q+l = #

2.3.11. A ~ a t h model. Let B be a Bratteli diagram; we use paths on the diagram to construct a natural model for the chain of multi-matrix algebras associated to the diagram. We will suppose that B is infinite; it will be obvious how the construction must be modified for a finite diagram. First we produce an augmented diagram B by adding a (-l)st story corresponding to the inclusion K l c Mo; that is we append one vertex * with

d*) = -1 and dim(*) = 1, and we connect * to v? by edges (1 s j s m(0)). J J , An oriented edge on any graph is an edge together with an ordering of its two vertices;

we will call the first vertex of an oriented edge its start and the second its g&. A path is

a (possibly infinite) sequence (6) of oriented edges such that end(ti) = tart((^+^) for

Chapter 2: Towers of multi-matrix algebras [ $2.3. Inclusion matrix and Bratteli diagram 55

all i. A path (. . ,,tk) has end equal to end(&); a path (G,. . .) has start equal to

start(@. If ( and p are paths such that end(0 = start (p) we define @q to be the

path "fi18t (, then f l A path t on B is monotone increasing if q(end(tk))

= + 1 for d k.

We let n denote the set of infinite monotone increasing paths on B starting at *; n the set of infinite monotone increasing paths starting on the P floor of 8; nrl the

tr set of monotone increasing paths starting at * and ending on the rth floor; and [ r d

the set of monotone increasing paths stutiog on the rth floor and ending on the sth floor (I < a). Given ( = ((o,(l,- a ) E n, set:

\

trl = ( & , " . , t p r ] (0 * r), tt,s] = (kl,.,.,tJ E nb,s] (-1 i < 4 9

and $ '(tr+17...)En[r (-1 s 1)-

Also let be the vertex end($) = tart((^+^). Suniluly if ( = (to,. . ~0 E and

r i s we can define = (lo,. . . ,tJ E 11 , and SO forth.

Let W ) be the K-vector space with basis n. For each r E {0,1,2,. .} we define an algebra ~ ~ $ ~ n d # U 2 ) as follows. Let Rr = {((,r)) E nrl x 11 : end(d(O= = end($}. For

(6,d E Rr define TS,n E En%(KQ) by

TLpw = 6(nr],~r])tr]o~[r (w E "1.

Let ' A ~ be the K-linear span of {TL, : ( 4 ~ ) E l$) in EndK(W)); since

(2.3.11.1) TL$tI,pt = 6(%(')Tt,p., and 1 = ,,;he

i

A, is an algebra. Set

so that nrl = : nil (disjoint union), and = : (n',] x nil). It,follows from the

multiplication law (2.3.11.1) for the T that

so that

Note that the minimal cent14 projections pi in Ar have the form

since each ( E can be extended in Af . ways, by adjunction of an edge A in rl 17J

n[r,r+ll, to a path @A in It follows from this and property (4) of the Bratteli diagram that #(d ) = pi for all r and i (0 i I, 1 i i i m(r)). Thus

11

m(r) Ar : $ Mat r(K).

i=1 4

Finally A, c because for ( 6 , ~ ) E R,,

t . . aa operators on W). If (Lq) E o:] x Q:], a, T E A:, then E, n

Chapter 2: Towersof multi-matrix algebras

and the Brattpli diagram for the chain 1 FA,, C Al C . . . It follows that AA .= (Xi,$, 1

I

is B. As an example of the utility of the path model, let us i t CA!Ar) for 1 < s. k t

and let Arg = spa%{Tfln : (&q) E Rrls}. Then Arls is an algebra, since again

T ~ l $ ~ ' ,v , = 6 ( a t f ) T t ,. (((.n),(tl .nl) E RrlS), and 1 = C { T U : f E n[rl~l).

We have A c As, because if (&q) E R,,,, then r1s

T&n = x { ~ A o [ , X o ~ : '[I] = ([I]= n[r] 1,

as operators on WL. Clmly Arc c CA/Ar).

ElPPfL For x E As define P(x) =

: ( M I ) E Rr}.

One verifies that P is a linear projection of A, onto CA 8 (A,). But for (&q) E Rsr

It is an easy exercise to check that the factors of ArlS are in bijection with pairs of

vertices (v,w), with v in floor r and w in floor s. The factor corresponding to a pair (v,w) is the algebra of endomorphisms of the free vector space over the set of paths from v to w.

Remarks. (1) The path model presented here is due to V.S. Sunder [Sun] and A.

Ocneanu [Ocn]. Compare however [SV], in which a maximal abelian subalgebra of Am = UAk is identified with W1.

k (2) In case K = C, the action of the "path algebrastt A, on Cfl extends to a

representation on the Hilbert space 8(fl) with orthonormal basis fl. It is evident that Ttln ' is then a rank-one partial isometry with adjoint T* (,, = T,,S' So A, is a

c*-subalgebra of ~(F(f l )) .

2.4. The fundamental construction and towaa for multi-matrix algebras.

We consider a pair of multi-matrix algebras 1 E N c M, and the associated tower of

obtained by iterating the fundamental construction, as described in the chapter introduction. It turns out that all the Mk are then multi-matrix algebras:

Pro~osition 2.4.1. Let N c M be a pair of multi-matriz algebras and let M c Endi(M) be the pair obtained by the fundamental construction. Then

(a) End;(M) is a multi-matrix algebra and its minimal central idempotents are of the

form p(q), where q is a minimal central idempotent in N, and p(q) is r&

multiplication by q. M (b) The inclusion matrix for M c end;(^) is the transpose of AN.

u f . Set F = En%(M) and dehne maps Alp : M - F by X(x)(y) = xy and

p(x)(y) = yx for x,y E M. The homomorphism X is the composition of the inclusions M c ~nd&(M) and E n d i ( ~ ) c F; the map p is an algebra isomorphism from MOPP into F. As the pair N c M is isomorphic to the pair NOPP c MOP* by Corollary 2.3.4, it is also isomorphic to p(N) c p(M). But ~ n d ; ( ~ ) = CF(p(N)) and M = X(M) =

CF(p(M)). Consequently (a) follows from 2.2.3.a and (b) from 2.3.5. #

-->

Chapter 2: Towers of multi-matrix algebras

and zi > 0 for all i, and t. > 0 for some j. It follows that J

t k The inclusion b t r i x for MO C M2k is (A A) , and that for MO C M2k+1 is ( A A ~ ) ~ ~ t k + 2 t k Thus dink MU = ll(A A) ull and di% M2k+1 = ll(AA ) A S I ~ ~ . Therefore

Lemma 2.4.3. Let N c M be a pair of finite dimensional algebras over a field K, and

let (Mk)k20 be the associated tower. Then

[M:N] = 1 i m sup {di% Mk} Ilk. k+ m

Proof. As Mo and Mk are finite dimensional K-algebras, one has -

and therefore

[M : N] = 1 i m sup {rk(Mk I M&}'/~ = 1 i m sup {% M ~ } ' / ~ . # k+ m k+m .

Prowsition 2.4.4. Let . I E N c M be a p&r of finite dimensional algebras over a field K, let E be any extension field of K and set

M ~ = M % E and N ~ = N % E .

, Then

(a) ~ n d i ( ~ ) % E Y Endr E ( ~ E ) . N

E E (b) [M:N] = [M : N 1.

&QQ$. (a) This is an example of a theorem on "change of rings in Hom"; see for example [R], p.24. We give a simple proof appropriate to the special case at hand.

Define

5 2.4. The fundamental construction

a : En$(M) % E -I EndE(ME)

by cu(Pa)(*b) = d x ) @ ba (v a En$(M), a,b e E, x E M).

Define also

aa follows. Let {ai} be a basis of E over I( For each @ a EndE(ME) and each i, there

is a unique cpi E En$(M) such that

@(*I) = 1 cpi(x) @ ai (X E M)..

Only finitely many cpi(x) are nonzero for any particular x, and since M is finite

dimensional over K, mly knitely many cpi are non-zero altogether. Then 0 can be

defined by

It is easy to check that a and P are isomorphisms of E-algebras which are inverse to each other.

Next observe that

cu(X(m)@a) = X(m@a), and a(p(n)@a) = p(n@a) (m E M, n a N, a E E).

It follows from this that

(b) Let (Mk)k>O be the tower of extensions generated by N c M and let (Ak&>,, - - : be the tower generated by N~ E ME. We produce a sequence of isomorphisms I $ : Mk % E 4 Ak such that ak+l k

= ak for all k. Take %,al to be the $ I ~ k v

[ identity and 02 to be the isomorphism defined in part(a); we have o2 1 (2.4.4.1). Suppose orl,. . .,% have been defined. Let

I C

62 Chapter 2: Towers of multi-matrix algebras 3 2-4. The fundamental construction I b3

Exam~le 2.4.5 Consider two integas m' ,me L 1 and set m = m' + m'. Let M be

I %+l: Mk+l . E = E ~ ~ ~ ( M ~ ) . E - . E ~ ~ ' ~ (M!) the factor Matm(C), let P be its "parabolic" subalgebra Mk-1 Mk-l

A B = {[o c] E M : A E Matm. (c), B E Matm. ,m,(C), C F at,.(^)}

be the isomorphism defined as in part (a), and let

E and let L be the "Levi" subalgebra {[t ;]} of P. hen L and M are semi-simple, a+1 : End (Mk) ' Ak+l = Ak-1 (Ak)

Mk-1 and [M : L] = 2 as above, but P is of course not semi-simple. We claim that [M : PI = 1. Indeed, from left multiplication

\ one has the inclusion be induced by the pair of i S 0 ~ 0 r p h i ~ ~

E Set ak+l = 7k+1 o $+,; this extends 4 becaw extends the identity on Mk,

and &+l extends at

C n q e t l , we b e d ) = ( M ) = ( A ) for dl k, the

E E equality [M : N] = [M : N ] follows from this and Lemma 2.4.3. #

Proof of Theorem 2.1.1 and Co o l l m 2.1.2 r .. Because of 2.4.4 and the definition of M

AN for arbitruy semi-simple algebras (given in the chapter introduction), it suffices to

consider the case that K is algebraically closed, so M and N are multi-matrix algebras. But then

{M : N] = l im {diml( M ~ } ' ~ = ~ I A E ~ ~ , k

by 2.4.2 and 2.4.3. The corolluy follows fmm Kronecker's Theorem 1.1.1. #

where AA is left-multiplication by A (and pA below is right multiplication). As the

cominutant of P in M is reduced to the center C of M, the cornmutant of XfP) in EndC(M) is isomorphic to M; moreover the natural morphism from M to

CEnd(M)(X(P)) is an isomorphism. Consequently the tower generated by P c M is

P c M c M c . . . and the index is 1.

We also claim that [P : L] = 1. From left multiplication

one has the inclusion

I L End (P)

R e . The norm of a product of two matrim is not, in generd, the product of their Thus CEnd(P)(L) isthe subalgebra norms. It follows that, given a nested sequence 1 E L c P c M of semi-simple algebras, the inequality

[M: L] !, [M: P][P : L] \

e Mat2m, (C) and V E Mat,,(C)

, I is in general stria. ~0-, even this inequality fails to hold for *bras with radicals, as

we now show. Endc(p), isomorphic to (Mat,, (C) @ Mat2([)) @ Matm.(C). As right mdtiplicaion

9 4 [t g] is r e p r ~ t e d in Endt(P) by the matrix

Chapter 2: Towers of mult~-matm itlgt:uryn- Y 4.3. LUG I U U ~ ~ ~ . L U ~ U L ~ L L UJUW,~UCI,~"~ rcX%? 00

the canonical morphism P 4 N is given by

[ P - N = (Mat,, (C)@Mat2(C)) @ Matm.(C)

The argument used to show [M : PI = 1 shows also that the canonical construction applied to P c N gives an algebra isomorphic to N. Finally, the tower generated by L c P is L c P c N c N C . . . andtheindexisalso 1.

2.4.6. A reprise of Proposition 2.4.1. Let 1 E N c M be a pair of multi-matrix

algebras with inclusion matrix A. Write {q j ' . 1 < j < n} and {pi : 1 < i < m) for the

minimal central idempotents of N and M respectively. Let B be the twmtory Bratteli diagram whose 0" dory is B(NcM) and whose lSt story is the reflection of B(NcM); that is A(') = A and A(') = At. Let B be the augmented diagram, as in 2.3.11. For example for CS3 c CG4, B is .

We identify the pair N c M with the pair A. c At, of path algebras d a t e d with 8. (See 2.3.11.) Write {"q j ' 1 i j < n) for the minimal central idempotents of the path

algebra A2. According to 2.4.1 and 2.3.9, there is an isomorphism of ~ n d h ( ~ ) onto A2

which takes X(M) onto M. Our purpose here is to use the path model to provide an explicit isomorphism. Except as noted above, our notation is as in 2.3.11.

An edge on f3 is specified by the data q = (k;i,j,e), where k is the story on which q k lies, v. and v!" are the two vertiegl of q, and the index L distinguishes among the J

k Ai J . edges joining vk j and vf''. Define an involution * of n[o,ll U "1,21 by

Thus * is the reflection through the first floor. (Nevertheless w@ regard the reflection of an upward oriented edge to be upward oriented.)

n Let vj = Knj e lmj and V = @ V.. ~ e f i n e a linear i p U from M to V by

21 01 j=1 J

requiring.

U is a linear isomorphism, its inverse F being determined by

F('e'~) = T(~>( i ) , (C , t i* )

(C E n j and qo E n j ; i s j s n). 21 01

Note that U breaks and unfolds the round trip path toq-', while F folds and joins the pair ( t ' ,%). For example: . .

V carries both a right action of N and a left action of A2, arising from the right action

of N on Kf2 and the left action of A2 on Kn2]: 01

It is easy to check that A2 is in fact the cornmutant of p(N) in EndK(V), and that U

intertwines the right actions of N on M and V. Hence

a : Q - U o p F

is an isomorphism from ~nd;(M) = CEnqM)(p(N)) to CEn (PO)) = A2.

) Let (4.d E % and ( 0 , ~ ) E RI (so TLq E $ and E M). One checks that

66 Chapter 2: Towers of multi-matm rtlgebras

"'(~~,)('u,r) = ~(~~~or;)~(Gytl),(ro,t~). i - ..a

It follows that a-l(x) = X(x) for x E M c AT Also

P(4, = a - l ( C TL$ = p(qj), '4

as required by Proposition 2.4.1. \

Remark. Later we will want to modify the definition of U somewhat. If

c : fl[O,ll 4 @ is any function and we instead d&ne U by

U(TgS = c ( ~ ~ ) ( t ( , . ~ ~ { ) @ 70.

1 then Q H U o p v is another isomorphism of ~ n d & ( ~ ) onto A2.

2.5. Traces.

A K-linear map Q from K-algebra M to a K-vector space V is said to be faithful if

the corresponding bilinear map

(X,Y) d v )

is non-degenerate; that is for each nonzero x E M there is a y E M such that d v ) # 0. This is a o n ~ i d e d notion, but if M is finite dimensional and Q : M 4 K is linear, then Q is faithful on one side if and only if it is faithful on the other. Furthermore, in this case, for each linear 4 : M :K, there is an a E M such that fix) = dxa) for all x E M.

A on M is a linear map tr : M --+ K such that tr(xy) = t r ( p ) for all x,y E M.

On a factor, any nonzero trace is faith$l, and any two traces are proportional. In fact a trace on Matd(%) satisfies tr(e. 1,J .) = 4 tr(el,l), where {e. 1rJ .) are the standard matrix

units. m Let M be a multi-matrix algebra over K, written as before a& M = @ piM, with i=l

piM r Mat (K). We associate to a trace t r on M the Y

c

I

5 2.5. Traces

where ei is a minimal idempotent in Mpi For example, the trace defined by

= 1 and = 0 for k # i mmaponds to the ith vector of the canonical 1 pkM

basis of K ~ . Any row vector E K~ determines a unique trace

with associated vector g. A trace tr on M is faithful if and only if the associated vector 5 has no zero entries.

When the characteristic i f K is zero, we say that tr is positive if si 2 0 for all i. (There

is an ambiguity here; if K is given as an extension of the reds, the meaning of si 2 0 is

clear. Otherwise we take si 2 0 to mean that there is an imbedding of Q(sl, . , ,sm) in C

such that si 2 0 for all i.) A positive trace is faithful if si > 0 for all i.

Pro~osition 2.5.1, Let 1 E N c M be a pair ofmulti-matriz algebras with

M and with inclusion rnatriz AN.

(a) Let u be a trace on M corresponding to g E K~ and let 7 be a trace on N cowuponding to t t K ~ . Then u eztends r if and onb if t = b:.

(b) If char(l() = 0, then there exists a faithful trace on M with faithful restriction to N . If char(K) = p > 0, then a s~ f f i cbn t condition for the ezistence of a faithful trace on M with faithful rmtriction to N is that fir aU j, the sum EXij is not divisible by p.

bf. (a) If fj is a minimal idempotent in q.N, then f.p. is the sum of X i j J J 1

minimal idempotents in piM. Hence the restriction of a to N is described by the vector + t ' with components

1

Chapter 2: Towers of multi-matnx rugeorau, -- OY

not divisible by characteristic, then the weights t. We nOn*rO, and the restricted trace is J

faithful. #

Remarks (1) With the notation of the proposition, one has, when 0 extends 7, (Note that On any Bratteli it is actually superfluous to record the dimemiom

except on the first floor. Similarly on a finite Bratteli diagram it is superfluous to record m the weights of a trace except On the top floor, but on an infinite Bratt& diagram it is not

(5, $1 = En = u(1) = ~ ( 1 ) = ( 6 3. (in general) su~erfluow to record the traces, since the tr- on the higher floors are not i=l determined by those on the floors below.)

(5) s( 3 R, and N c M is a pair of multi-matrix algebra over s( with By Propositions 2.3.1.b and 2.5.l.a, this implies ZN mM = and with inclusion matrix A. ~ e t ( M ~ ) ~ ~ ~ be the tower obtained by

t M + (5, A!?) = (sAN, v),

iterating the fundUWXltal c0IlStrllction. Then it follows from Perron-Frobenius theory that there is a unique positive normalized (tr(]) = 1) trace on M~ = UM

k k' which is, of course, obvious! In fact, let t(O) be the Perron-Frobeniw eigenvector for i \ t ~ , nor&zed by

(2) A faithful trace on M may have zero restriction-to N. Consider for example Xtfo)vj = 1. Define t(2k) = l l ~ 1 1 - ~ ~ @I t(2k-1) = $ W A t (k l). Then

2 I

as r E-ple 2.3.7, and the trace on M associated to the vector (1,-1,1) E 8. Or consider the two elemat field F2 and the pair F2 C Mat2(F2) (with inclusion matrix (21); trace An m ~ m m t similar to one given in the proof of 2.4.2. shows that n ( ~ t ~ ) r ( ~ ; )

rLO on M ~ ~ ~ ( F ~ ) has zero reatdction to the center F$ One may thus say about tram On consists of Perron-Frobeniu~ eigenvectors for A ~ A . Suppose tr is m y positive normalized

multi-trix algebras, that pdikvitY is hereditary, but faithfulness is not. traceon Mm and f(li) is the vector determining t r on MI Then for k r,

(3) The assignment of a vector f E Km to a trace t r : M -+ K has been defined above via the values of t r on (classes of) minimal idanpotents of M. In Chapter 37 we f(2k+2r)(zA)r = :(2k),

consider a new situation, where M is a finite direct Sum of COntihuous (type 111)

N~~~~~~ factors; since no minimal idempotents are present in this situation, we shall whence f(2k) is a Perron-Fmbenius eigenvector for n t ~ . since t(ik)(AtA)k = ~ ( ~ 1 , we have $(2k) = l l ~ l l - ~ ~ f(O).

describe a trace t r by the vector z = ( tr(p1),-' .,tr(pm) ) of values of tr On

projectiom of M. In principle, the description of tr via is possible for

2.6. Conditional expectations. present chapter.

(4) Given a Bratteli diagram representing a sequence of iXlClUSi0ns of multi-matrix We are primarily interested in the following situation:

(1) N C M is a pair of inulti-matrix algebras. (2) has a faithful trace with faithful restriction to N.

corresponding factor,.t,h& is the value of the trace on a minimal idempotent in the

70 Chapter 2: Towers of multi-matrix algebras 3 2.6. Conditional expectations 71

(3) E : M -+ N is the orthogonal projection of M onto N with respect to the inner product determined by the tram.

However, to clarify somewhat the roles played by semi- simplicity, the pair of faithful traces, and the conditional expectation E, we begin in a more general setting.

A conditioa wectation from a K-algebra M onto a subalgebra N is an (N,N)-Iinear map whose restriction to N is the identity. Recall that such a map E is faithful if for each non-zero x E M there is a y E M such that E(xy) # 0. For example, if M is a factor, M = Matp(K), where K has charaxtersitic 0 or ,u is relatively prime to

char(#), then the trace on M normalized by trm(1) = 1 is a faithful conditihnal expectation of M onto K. L

Consider H O ~ ~ ( M , N ) , the set of right N-linear maps from M to N, with its left

N-module structure defined by (xcp)(y) = xdy) (x E N, y E M, cp € H O ~ ~ ( M , N ) ) . We

associate to a conditional expectation E : M 4 N the left N-linear map E~ : M % H O ~ ~ ( M , N ) defined by E ~ ( X ) ( ~ ) = E(xy) for x,y gy. M. Then E is faithful if

and only if E~ is injective. We say that E is very faithful if E~ is an isomorphism.

Lemma 2.6.1. Let N c M be a pair ojfinite dimensional K-algebras. Suppose N has a jaithful K-linear functional. Then any faithful ezpedation E fiom M to N is very faithful.

w. Choose a faithful functional T : N ---1 K and set u = T o E. If x E M is such that u(xxl) = 0 for all x' E M, then u(xyz) = 7(E(xy)z) = 0 for al l y E M and for all z E N, so that E(xy) = 0 by faithfulness of T and x = 0 by that of E. Thus u is

, faithful. It follows that any K-lineax map M 4 K is of the fom x I+ u(ax) for some a E M, since M is finite dimensional. J

Consider a right N-linear map Q : M -+ N. There exists a E M with r(p(x) = u(ax) for all x E M. Define $ : M 4 N by $ = ~ ~ ( a ) ; i.e., $(x) = E(ax). We claim that $ = Q. It is enough to check that A$ = XQ for any K-linea~ X : N -+ K. But as T is faithful, such a X is given by y I+ 7(yb) for some b E N. Now one has for all x E M

Xflx) = 7(E(ax)b) = rE(axb), and

Rema&. (1) If N is a multi-matrix algebra, then N has a faithful K-linear functional.

\

(2) Let V be a K-vector space and define a multiplication on A = KP V by (X,v)(Xf ,vl) = (AX' ,XV' +X1v). The result is a K-algebra for which any subspace of ' 0 P V is an ideal. Suppose dim V 2 2. If (p : A + K is any K-linear functional, then

3 ker((p) n V is a non-zero,ideal in ker(~). So A has no faithful linear functional.

But if V is one-dimensional, spanned by v, then the functional (a,bv) n a + b is faithful on A.

The next proposition concern the existence of faithful conditional expectations.

Pro~osition 2.Q. Let N c M be a pair of K-algebras with N finite dimensional, and let tr : M -I K be a jaithjW trace with faith@ restriction to N. Then there ezists a anique K-linear map E : M -+ N such that

(i) tr(E(x)) = tr(x) x E M (ii) E(Y) = Y Y E N (iii) E(xy) = E(x)y x E M, y E N. !

Moreover E is a faithful conditional expectation from M to N, namely

(iv) E(~x) = yE(x) x E M, y E N r

I i

(v) E(xy) = 0 for all y implies x = 0. If M is finite dimensional, then E is very faithful; that is

(vi) E~ : M --I HO~;(M,N) defined by a I+ (m(E(ax)) is an isomorphism.

w. We coqsider M together with the nondegenerate symmetric K-bilineax form (x,z) +I tr(xz) and with the associated orthogonality relation. As tr and trlN are

faithful one has M = N P N'. We begin by checking uniqueness. Let E : M + N satisfy (i) to (iii). As E is

,- defined on N by (ii), it is enough to check that E = 0 on N". Let t E N'. For any y E N one has by (iii) and (i) >

J

tr(E(t)y) = tr(E(ty)) = tr(ty) = 0 'd

so that E(t) 1 N. But E(t) is also in N, so that E(t) = 0. To prove existence, define E to be the projection of M onto N along N'. It is

obvioufr that (ii) holds. For x E M, one has E(x)-x orthogonal to N and hence to 1, so (i) holds. t, ?, , . . - ,

I .

Note that N' is a right N-module because of the trace prop~rty of tr. Namely if y,y' E N and z E N". Then

so zy 'E N". Now xy - E(xy) and x - E(x) are in N", and hence also xy - E(x)y E N". The difference

2

F (xy-E(xy)) - (xy-E(x)~) = E(x)y-E(x~)

is in N' n N = (0), which proves (iii). One obtains (iv) similarly.

i?

PY"-" Chapter 2: Towers of multi-matrix algebrr? 1 ) 2.6. Conditional expectations *'u%A31.p 73

5 .ar

Since tr * tr o El the faithfulmas of E follows from that of tr. Finally, if M is

finite dimensional, then E is very faithful by Lemma 2.6.1. # %ma& Conditions (i)-(iii) are equivalent to the single condition

tr(E(x)y) = tr(xy) for x E M and y E N, I as the reader may verify.

, The relevance of conditional expectations for the fundamental construction comes from the following fact. .

Pro~osition 2.6.3. Let M, N be K-algebras with 1 E N c M; set L = ~ n d i ( ~ ) and

let X: M 4 L denote the inclusion. Assume moreover that (i) the right N-module M is projective o f f i i t e type, and (ii) there ea5sts a very jaithfil conditional expectation E from M to N . Then L is generated by M and E (viewed as a map from M to M). More

precisely, L is generated as a K-vector space by dements ofthe f o m X(x)EX(y) raith x,y E M. Furthenore, the map x @ y H X(x)EX(y) from M % M to ~ n d i ( ~ ) is an

isomorphism.

mf. Hypothesis (ii) says that E~ : M -+ M* = HO~;(M,N) is an isomorphism. As

projective modules of finite type are flat (see [BAC 11, page 28), the K-linear map

is an isomorphism. Let

be the canonical homomorphism. By (i), it is an isomorphism (see, e.g., [BA 21, page 111). ~onseq'Gentl~, the composition

is an isomorphism. Routine computations show that 1

The proposition follows from the first of these. #

Remarks. (1) It could be that M is projective of finite type as a right N-module but not as a

left N-module, as observed in [BA 81, page 53. ' (2) In the situation of the previous proposition can we conclude'that L is projective

of finite type over M (as a right X(M)-module)?

For pairs of multi-matrix algebras, the situation regarding pairs of faithful traces and conditional expectations is the following:

(1) If char K = 0, then for any pair of multi-inatrix algebras N c M over K, there exist faithful traces on M with faithful restriction to N (2.5.1), hence faithful conditional expectations E : M -+ N (2.6.2).

(2) Whenever E : M 4 N is a faithful conditional expectation, it is very faithful, since N always has a faithful functional (2.6.1).

(3) If char K > 0, M need not have a faithful trace with faithful restriction to N. For example there is no pair of faithful trams for F2 c Mat2(F2). Note that nevertheless

:] I+ a + b + c defines a faithful conditional expectation Mat2(F2) 3 FY

Sorollarv 2.6.4. Consider a pair of multi-matriz algebras

n m 1 E N = .@ q.N c M = @ p.M

~ = 1 J i=l 1

as well as

Szlppose there is a faith& conditional expectati~n E : M -+ N. Then (a) L is generated as a K-vector space by elements X(x)EX(y) for x,y E M;

(b) The K-linear map cp: N -+ ELE defined by ~ ( x ) = X(x)E is an isomorphism of algebras.

(c) If f. is a minimal idempotent in the factor q.N, then X(f.)E is a minimal J J J idempotent in the factor p(q.)L.

J

u f . (a) Condition (i) of Proposition 2.6.3 is fulfilled because any module over a semi-simple algebra is projective, and condition (ii) is fulfilled by Lemma 2.6.1.

To prove (b), first note that Q is a morphism because E is an idempotent which commutes with A(x) for all x E N. If x E N and cp(x) = 0, then also x = cp(x)(l) = 0, so 9 is injective. Finally Q is surjective by part (a).

, -Z bnirpter a: lowers or mgi-matnx a.tgeDraa $2.6.. Conditional expectations 75 '

For j E (1,. . * ,n), the idempotent p(q.)E = X(q.)E is not zero and lies in the factor J J

p(qj)L. The rwulting r e d u d factor is p(qj)ELE = p(qj)X(N)E. As Q is an

isomorphism, i t s restrict ion Q. to q.N is also an isomorphism onto p(qj)ELE. It J J

follows that the idempotent ~.(f.) = X(f )E is minimal in the factor p(q.)ELE. But if e J J j J

E L is an nonzero idempotent in L dominated by X(f.)E, and thus also by J

X(q.)E = p(q.)E, then e = p(q.)E e p(q )E E p(q.)ELE, and therefore e = X(f.)E: In J J J j J 1

other words, X(f.)E is also minimal in L. # J

.&mark: The following instructive proof of 2.6.4.a was given by Wenzl [We&]. First note that the map Q of 2.6.4.b is an injective homomorphism. Now consider the subalgebra A of L generated by X(M) and E, and note that

A = { I(yo) + Z A ( ~ ) E X ( Y ~ ) : 4 Yj E MI, and i EAE = gP(N) g N

If $ is a non-zero element of rad(A), then there &st x, y E M such that E(y*x)) f 0 (using the faithfulness of E). But then EX(y)$X(x)E = X(E(y*x))E = dE(y*x))) is a non-wo element of rad(A) n EAE = rad(EAE), a contradiction since EAE is isomorphic to the semiflimple algebra N. Thus A is semiflimple. Note that A' = X(M)' n {E)' = p(N), so A' = p ( ~ ) ' = L, where primes denote centralizers in Ends((M). Since A is semi-simple, A = A' = L. Finally observe that X(M)EX(M) =

{ ~ . \ ( X ~ ) E A ( ~ ) : 3, yi E M) is art ideal in L, and if $ is a central projection in L i

orthogonal to this ideal, then for all x, y E M,

Bence * = 0 by faithfulness of E, so L = X(M)EX(M). # I

' 2.6.5 Reprise of 2.6.4 using the path mod$. Let N, M, and L be as in 2.6.4.

Suppose tr is a faithful trace on M with faithful restriction to N, and let E : M -, N be the conditonal expectation determined by tr, as in 2.6.2. Let B be the Bratteli diagram for N c M c L, and let f! be the augmented diagram, as in 2.3.11 and 2.4.6. We identify N c M with the pair of path algebras AO c A1, but we distFguish between

L = ~ n d i ( ~ ) and the isomorphic path algebra A2. Let 5 and t be the vectors

determin the trace tr on M and N. We dso regard 5 and t as functions of vertices B 1 on the Ot and lst floors respctively: t ( 4 ) = t. and s(vi) = si. R e d l the *-operation J J

which reflects edges through the first floor. We define the reflection on vertices as well:

We first give a formula for E E ~ n d i ( ~ ) . R e d that E is determined by the

requirement tr(E(z)x) = t r (a ) , for z E M and x E N. If ( t 9 ) E Rl and (o,fi E %, so that T E M and N, then one verifies that t,v I

."

while

Hence

(Remark that 4 = tql] and t[Ol = if E(Tt,J f 0, so the expression is not so

asymmetric as it may first appear.) Let

rod F = u'~, as in 2.4.6. Next we compute e = UoEoF, the image of E in A2. For

( @ '10, art elementary tensor in V. for some j, J

It follows that ..

f [0] [0]

Remark. If K = C, and the trace t r is positive, we prefer to use the inner product

(x,y) = tr(xy*) on M, where * is the natural * operation on the path algebra M, rather than the bilinear form (x,y) w tr(xy). (The orthogonal projection E : M --, N is unaffected by the change.) We give V the inner p r o d for h i Ll "A1 @ nil is an

j orthonormal basis. Then the choice

(2.6.5.3) u(T&R) = &K$ (G,~~,R;) @ vo

makes U into a unitary operator from M onto V. In this case e is given by

Then e is a self-adjoint projection in the c*-algebra A2. Thisformula for e is due to

Sunder [Sun] and Ocneanu [Ocn]. The formulae (2.6.5.3) and (2.6.5.4) are also sensible.if K is any quadratically closed field.

We know from 2.6.4 that any (p E ~ n d k ( M ) has' a decomposition

(p = z h ( x i ) E I(yi) where xi,yi E M, but so far we have not co~~sidered how to compute i

such a decomposition. Since the isomorphism a : (p w U o p F of ~ n d k ( ~ ) onto A2

trlna A(r) to x (x E M), it suffices to decompose z E A2 into a s y z = x x i e y i with i

xi,% E M. For (cu,y) and (6,P) E R1 (so T,,y and T 6 , ~ E M) one computes from

-(2.6.5.2) that

Hence for (a,@) E R2,

where yo is an abitrary edge in with 7[0~ = end(b)* = st&(&). In particular if

we use the convention (2.6.5.3), and formula (2.6.5.4) we get

Another way to write this is '

T4b = t(endtb)*)~(a@ 70) e F(P@ %I*.

As an exercise in using (2.6.5.6) we compute a decomposition for the minimal central idempotent p(qi) in Endi(M). We have

for any yo E nbl. Taking the average over the u. element8 of nil, we mive at 3

In the remainder of this section we discuss, following [Wen31 and [BW], the notion of an extension of an algebra with respect to a conditional expectation. This type of structure appears frequently in Chapter 4.

78 Chapter 2: Towers of multi-matrix algebras

Definition 2.6.6. Let N c M be a pair of algebras over a field K, and E : M + N a

faithfizl conditional expectation. jkc tens i~g pf M is a pair (L,f), where L is an algebra containing M, f E L, and

(i) L is generated aa an algebra by M and f. (ii) f2 = f.

(iii) fyf = E(y)f = fE(y) for all y E M. x H xf is injective.

(iv) The morphism

The model example of an Eextension is the fundamental construction (E~~&(M),E),

when E is very faithful and M is projective of finite type as a right N-module.

Lemma 2.6.7. Let (L,Q be an E-extension of M. n

(i) Ang element of L has the firm yo + zy'.fy:, with yo,yj,yj E M. In J J

j-1

particular MfM is an ideal of L. (ii) There is a unique conditional ezpectation E: L 4 N extending E and

satisbing qx)f = fxf for x E L. Moreover q x ) = qxf) = qfx) for all x E L.

(iii) For x E L there ezist unique bl,b2 E M vrith xf = blf and fx = fb2.

Proof. (I) is immediate from the ddnition 2.6.6.

(ii) ~ e t $ denote the isomorphism x w xf from N to L, whose range is exactly fLf. Then F: x H fl(fxf) has the desired properties.

then bl = yo + C y j ~ ( y j ) satisfies d= blf. ~f b E M

j j and bf = 0, then for all y E M, 0 = fybf = E(yb)f = +E(yb). Since E is faithful and $ injective, b = 0. This proves the existence and uniqueness of bl. Proceed similarly for

Fkmarks 2.6.8. (1) If N ; M, then F is never faithful since f # 1 and

E((1-f)x) = 0 for all x E L. (2) Let x E L. One has qxy) = 0 for all y E M if, and only if, fx = 0. Similarly

&x) = 0 for all y E M if, and only if, xf = 0. Let us check the first assertion. Suppose qxy) = 0 for all y E M. Then for all y,

\

0 = qxy) = afxy) = E7fb2y) = qb2y) = E(b2y).

Since E is faithful, b2 = 0 and fx = fo2 = 0.

3 2.6. Conditional expectations 79.

(3) If N, M and L are *-algebras, E = E*, and f = f*, then F is self adjoint, because $ is a *-morphism.

(4) If N and M are c*-algebras, L is a *-subalgebra of a algebra, E = E* and f = f*, then is positive. Indeed x H M is positive and is positive.

Proposition 2.6.9. Assume that M L projective of jnite tgpe as a right N-module and

E is very faithfil. Let (L,f) be an eeztension of M. Then

defines u (wn-unital) Domorphism of end$(^) onto the ideal MfM of L. Moreover

there is a morphism of algebras p : L -4 end&(^) such that L = MfM @ ker cp (direct

sum of algebras).

Prmf. Identify M with its image in E n d i ( ~ ) . Since by 2.6.3, 1 y: @ y: ct 1 yjEyj - J J is an isomorphism of M % M onto end&(^), the map a is well-defined, and it is an

algebra morphism with image MfM, by definition 2.6,6. We set

{ L -' ER~;(M)

We have to check that cp is well-defined. Let x = yo + x y j f y j and

j a = yo ~ ' f ~ j ~ ~ j with yO,yj,yj E hen for y l y n E M

j

while

E

If x = 0, then Eylay'E = 0 for all yf,y" EM, so MEMaMEM = 0; but

1 end$(^) = MEM by 2.6.3, and since this algebra has a unit, a = 0.

It is clem that is a surjective algebra morphism (indeed Ip(MfM) = ~ n d i ( ~ ) ) , k i and that cpou is the identity. Hence a is injective and L = MfM @ k g cp as vector

I F

spaces. Since both MfM and ker cp are ideals in L, this is actually a direct sum of algebras. #

i

2.7. Markov tr- on pairs of multi-matrix algebras.

Let N c M be a pair of multi-matrix algebras and let X : M + L = ~ n d & ( ~ ) be the

pair obtained by the fundamental construction. If E : M + N is a faithful conditional expectation, we know from Corollary 2.6.4 that L is generated as a vector spke by elements of the form X(x)EX(y) with x,y E M. Any trace TI : L -+ K satisfies

for all x,y E M, and hence TI is determined by its values on elements of the form X(x)E for x E N.

Let tr be a faithful trace on M with faithful restriction to N and let E :M + N be the conditional expectation defined in Proposition 2.6.2. Let P E K Define tr to be a Markov trace of modulus 4 if there exists a trace TI : L -4 K such that

Tr(X(x)) = tr(x) ] for all x E M. PTr(X(x)E) = tr(x)

Observe that this relation implies P # 0, because tr is faithful. If such a TI exists, it is unique in the following strong sense.

Lemma 2.7.1. Let N c M be a pair of multi-matriz algebras and let P E K*. Let tr and E be as above. Then there ezists at most one trace TI on L such that

Tr(X(y)E) = tr(y) for all y E N. If such a Tr ezists, then it is faithfirl and satisfies

/3 Tr(X(x)E) = tr(x) for all x E M .

I f ? is the vector describing TI and t the vector describing tr then ?/l= t. I N'

PI.oof. We use the notation of Corollary 2.64. If such a trace TI exists, then for

j, ,

p rj = flr(X(fj)E) (by 2.6.4.c)

3 2.7. Marlrov traces z6%,

01

,

so that ?P = t. Uniqueness and faithfulness of TI follow. Finally

Pro~osition 2.7.2. Let e K*, let N C M be a multi-matrix algebra pair with inclusion matriz A and let X : M + L be the pair obtained by the firndamental constwtion. Let the decompositiom into factors be

q.N 2 MatJK) piM p Mat (K) o(qj)L 2 Mat,.(K), J J '5 ' J

Let tr be a faithhl trace on M with faithfirl restriction to N and associated conditional ezpectation E : M 4 N. Let f E Km and E K~ be the c o ~ p & ~ o n d i n ~ vectors, so that in particular = !?A. Finally, let P E K*.

Then the following are equivalent. (i) tr is a Markov trace of modulus P. (ii) f ( h k ) = /3 f and t ( kA) = p t. In particular, i f char(K) = 0 and if P i s the modulus of some Markov trace on M,

then /3 is a totally positive algebraic number; that is, P > 0 for any imbedding of Q(P) in c.

M. (i) * (ii) Let Tr be as in the definition of a Markov trace, and let ? E K' be the corresponding vector. Then = ?Ath because Tr extends tr, and t = f i by the previous lemma, so that p t = h t A . One has also f = ?At, sp that

(ii) 4 (i) Set ? = rlt and let TI : L + K be the corresponding trace. Then Tr exten& tr because

i i l t = p - l t ~ t = / r l . f ~ ~ t = f.

82 Chapter 2: Towers of multi-matrix algebras

Consider the linear map "7 N 4 K defined by TY) = /3 Tr(X(y)E); it is a trace, because E is N-linear and idempotent. If f. denotea some minimal idemotent in q.N, one has

J J i

y(f.) = Dr(X(f.)E) = @ - t., j = I,,. . .,n J J j - J

by CoroVary 2.6.4.c and the definition of t, so that 'i = t r J w Thus TI satisfies the

Markov condition Dr(X(x)E) = tr(x) for all x E M by the previous lemma. Finally matrices of the form AtA have totally positive eigenvaluea,' when

char()() = 0. #

Remarks. (1) Take A = [i i] and k = (3,1), so that t = (44). Then tAtA = 4t, but

t s4At is not a scalar multiple o f t . This shows that one cannot delete the first equality in condition (ii). Although tAtA = t p follows from tAAt = gp (because t = $A), we prefer to state (i) in a symmetric form.

(2) We stress that ,D> 0 holds without any positivity assumption on tr, in case char()() = 0.

Theorem 2.7.5 Let K be a Peld eztension of IR. Let N C M be a pair of multi-matriz algebras over K with inclusion matriz A, and with ZM n ZN = K. Let /3 E K*.

A necessary and'sufficient condition for the ezistence of a positive Markov trace of

modulus on M is that /3 = [M:N] = llA)12. Any two positive Markqv traces on M are proportional.

Proof. Since ZM n ZN = K, it follqws that A is indecomposable and AAt is - irreducible (2.3.lf and 1.3.2b). Recall that [M:N] = llA112 by Theorem 2.1.1.

If tr is a positive Markov trace of modulus P on M, then P = llhAt[l = [M:N] by the previous proposition and Pmon-Frobenius theory.

Conversely, set P = [M:N]. Let k be a Perron-Frobenius vector such that t s AAt = fi. Let t = $A; it follows as in remark (1) above that tAtA = t. Hence if tr is the (positive) trace corresponding to the vector 'ZT, then tr is a Markov trace of modulus @ by 2.7.2.

The final statement follows from the uniqueness of the Perron-Frobenius eigenvector for Aht. #

--....

A crucial property of a Markov trace tr on a pair N c M is that the trace TI on L = ~ n d i ( ~ ) entering the definition of the Markov property is again a Markov trace on

M c L. More precisely:

Pro~osition 2.7.4. Let tr be o Markov trace of modulus /3 on a multi-matriz pair N C M , set L = ~ n d i ( ~ ) as yual, let TI : L -t K be the extension of tr to a trace on L as

in Lemma 2.7.1, and let D : L -t X(M) be the conditional expectation defined by TI and tr. Then

(a) TI is a Markov trace ofmodulus /3 (with respect to X : M -t L);

(b) P D(E) = 1; (c) P DX(E)D = D, where I ( . ) means left multiplication on L;

(dl f l o w ( E ) = V ) .

EEPPf. (a) Let k and t be the vectors defining the trace tr on M and N respectively. Aa t r 'is a Markov trace of modulus P, one has

+ S A A ~ = ~ ~ , ~ A ~ A = P ~

by Proposition 2.7.2. From t,he proof of 2.7.1, we know that TI is described by ? = flit. Consequently

c t A A ~ A A ~ = P $

and (a) now follows from 2.7.2. . (b) The b i i e a r form (u,v) H Tr(uv). is nondegenerate on L and its restriction to

X(M) is nondegenerate; thus L = X(M) e x(M)', where orthogonality is meant with respect to this biinear form. For all x E M one has

so that PE-1 E x(M)'. Aa D is theorthogonal projection of L onto X(M), this implies D(@) = 1.

(c) By M-linearity of D one has DX(E)D = X(D(E))D, so (c) follows from (b). (d) Choose x,y E M and set u = X(x)EX(y) E L. The maps from M to M,

are equal by (N,N)-linearity of E. By (M,M)-linearity of D one has

Consequently, using (a),

Chapter 2: Towers ot mut1-ma,tm ageuraa,,-

which proves (d). #

This completes the proof of Theorem 2.1.3 and2.1.4.

We now analyze the role of Markov traces for towers. Changing our notation slightly, we consider a multi-matrix pair Mo C M1, the tower (Mk)kyO it generates, and a trace

tr = trl on M1, which is a Markov trace of modulus P on the pair Mo C M1. We denote

by tr2 the extension of the trace to M2 denoted previously by Tr, and by

E1=E:M1+Mo, E1EM2

E2=D:M2+M1, E2€M3

the associated conditional expectations. According to Proposition 2.7.4, the process of extending a Markov trace on Mk to MkS1 iterates; namely, if

%: Mk+ Mk-l is the conditional expectation associated to trk and trk,l, and

trk+l : Mk+l -' K is the unique extension of trk satisfying

P trk+1(xEk) = trk(x) for all x E Mk (see 2.7.1),

then trk+l is also a Markov trace, aqd the process can continue. Note that Mk+l is the

algebra generated by Mk and Ek, for short Mk+l = (Mk,Ek). Denote by Moo the

inductive limit (union) of the nested sequence

MoCMIC -.. CMkCMk+lC....

This is a K-algebra with unit which is the union of its finite dimensional semi-~imple subalgebras, and which has a finite dimensional center isomorphic to ZM tl ZN. The union

of the trkl s constitutes a trace t r : M + K which is nondegenerate (namely, tr(xy) = 0 (9

for all y E Moo implies x = 0). If K 3 IR and tr = trl is positive, then t r is also

positive in the sense that tr(r) > 0 for any non zero idempotent r in Moo. If this holds,

and if moreover ZM n ZN g K, then tr is the unique positive trace on Moo, up to

normalization; see Remark (5) at the end of Section 2.5. \

9 2.7. Markov traces / x-'T 00

Pro~osition 2.7.5 Let Mo c M1 be a pair o f multi-mate algebras and let

tr : M1 -' K be a Markov trace of mod& P. With the notation above one h u

(a) P E.E.E. = Ei for i j 2 1 with li-jl = 1; l J 1

(b) E.E - E E for i j 2 1 with li-j ( y 2; 1 j - 2 . t (c) P tr(wEk) = tr(w) for aU w E Mk. In particular, if tr is normalized by

tr(1) = 1, then tr(%) = for all k 2 1.

U f . Statements (a) and (c) follow from (a), (c) and (d) of Proposition 2.7.4. If j 2 i+2, then Ei E Mkl, and (b) follows because E. is Mkl-linear. #

J

Observe that this Proposition contains Theorem 2.1.6. .

2.7.6. The path model for M and the idempotents Ei. Let Mo c M1 be a pair of

multi-matrix algebras and let

be the tower generated by iterating the fundamental construction. Let B be the augmented Bratteli diagram of the tower and

A0cAl C * - ' c A ~ c A ~ + ~ C - * '

the chain of path algebras associated to B as in 2.3.11. Having identified Mo c M1 with

AO C A1, we can obtain an explicit sequence of isomorphisms ak : Mk -4 Ak with

= for all k, by iterating the procedure of 2.4.6.

. If tr is a Markov trace of modulus P on MI, then tr extends uniquely to a trace

on Moo which is faithful on each Mk and which has the Markov property: if

Ek : Mk -' Mk-l is the conditional expectation determined by the trace, then

Btr(%x) = tr(x) for dl x E Mr If t p ) denotes the weights of the trace on the kth

floor of B, then tIk) = ,K1tjk-2) for all k and j. We also write tr for the J J

correeponding trace on Am = UAk. I k

Assuming (just for the sake of having definite formulae) that # .is quadratically closed, we can choose the isomorphisms {ali) so that $ = %(%) =

Chapter 2: Towers. of multi-matrix algebras 5 2.8. The algebras for generic b .87

[k-1 ] =q [k-l]

where Sk denotes reflection of an edge through the kth floor of B. In fact we h o w that

this choice determines (4 completely because of the decomposition 2.6.4.(a). Then

{%} is a sequence of idempotents (self-adjoint projections on ?(n) in cas? '-K = C and

tr is positive) satisfying (a)-(c) of 2.7.5. Iterating the decomposition (2.6.5.6), we can write any matrix unit T in Ak as a

a,P monomial in the matrix units of A1 and the idempotents el,- e,ek-l. For example for

(a,@) E R3 (Ta,p E A3) one finds

where denotes the edge in B(MocM1) directly below the edge ai, and $,$ are

arbitrary edges in 0 with the appropriate endpoints. The constant C(@) can be 01

evaluated by computing tr(T .T ), using the fact that y e k = %(x)ek = ek%(x) a;P B,a (x e Ak) and the Markov property of tr.

Let Atrlk(M0cM1) be the subalgebra of Mk generated by l,E1, . ,Ekl. Our next

goal is to understand the structure of these algebras. We shall see that, when the modulus ' p of the Markov trace tr lies in a certain generic set, these algebras depend only on P

and k, and not on any other data pertaining to the inclusion Mo c M1 or the trace tr.

For p in this generic set, Atrlk(M0cM1) is isomorphic to an abstractly defined algebra

whose structure we describe in detail in the next section. For non-generic P, new

phenomena can occur, and our knowledge is much less satisfactory in this case; see Section 2.9. The following two sections borrow heavily from [Jo 11.

2.8 - The algebras Afl$ for generic f l \

For any integer k 2 1 and for any number /3 f 0 in the basic field K, let AP,k be

the algebra abstractly defined (as an associative algebra over K) by

the generators cl,c2,. . and the unit I 2 the relations ri = ri

,%.c.e. = ri if li-jl = 1 1 J 1

(Observe k indexes the algebra generated by idempotents up to k-1; this agrees with the usual convention for Artin's braid groups, but is not as in [Jol] or [Jo~].)

A monomial in AP,k is a product c. c. . , .ti where each ci is one of el,. . , rk-l; '1 '2 q j

the unit 1 of Aplk is a monomial (the empty product).

Proposition 2.8.1. Any monomial w E A may be written in one reduced form P,k

where r E M is an appropriate integer and where

Moreover dur4( AP,k s ["I .

Proof. Consider an integer m with 0 < m < k-1; we prove the first part of the lemma - by induction on m for a monomial w in {el,-. .,cm}. Asthis is obvious for monomials

with m 5 1, we may assume that m 2 2 and that the claim holds for m-1. Suppose w is a monomial in which cm appears at least twice. Then w has one of

the forms W = W E ae w l m m 2

or w = ~ ~ ~ ~ a ~ ~ - ~ b ~ ~ w ~ ,

where a,b aremonomials in { E ~ , - - . , E ~ - ~ ) . As cm commutes vhth these, w equals

Chapter 2: Towers of multi-matrix algebras +

either Wlrmaw2

1 or . wlaF fmbw2,

and the number of em's has been reduced. Consequently we may assume that w

involves exactly one rm.

Let w = wlrmw2 with wl and w2 monomials in {el,. .,rm-l). Using first the

induction hypothesis on w2 and then the commutation r r. = e.e for j < m-2, we caa m J J m

reduce to the case that w = wlcmrm-l... cn, with wl a reduced monomial finishing,

say, with el. If 1 n one has

Consequently we may assume that 1 < n, so that w is of the form

with all desired relations for the i' s aqd the j's. This ends the induction argument.

We now count the number of reduced monomials, following Chapter 111 in [Fell. By a path in the lattice z2, we mean here an oriented connected polygonal line with vertim at integral points and with edges being either horizontal and directed to the right or vertical and directed upwards. A path starting at (a,b) and ending at (c,d) has c-a + d-b unit edges, c-a horizontal ones and d-b vertical ones. The number of these paths is consequently the binomial coefficient

c-a+d-b N[$] = [ c-a 1.

Assume first a > b and c > d. To each of these paths touching the main diagonal, associate the following "reflected" path: if ( j j ) is the diagonal point on the path with smallest j, replace the subpath from (a,b) to (j,j) by the reflected path (with Iespect to the diagonal) from (b,a) to (j,j) and leave the subpath from (jj) to (c,d) qchanged. This defines a bijection between the set of paths from (a,b) to (c,d) which touch the diagonal and the set of paths from (b,a) to (c,d). Thus the number & paths from (a,b) to (c,d) which do not touch the main diagonal is N t i ] - N[:$].

Assume now a = b and c = d = a+n for some n > 0. Consider the paths from (a,a) to (a+n,a+n) whose vertices are on or below the main diagonal. These are in bijection

y LhU. - 5 S W r P r rr P,k "' 6--'" I- .* >

- -

with paths from (a+l,a) to (a+n+l,a+n) which do not touch the main diagonal, and their number is

Consider finally a sequence (il,jl,, , i ' ) corresponding to a reduced monomial in , P"P We may associate to this sequence the following path from (0,O) to (k,k), and any

path from (0,O) to (k,k) which remains on or below the diagonal can be obtained in this way.

it follows that the number of reduced monoinials is . # ["4 Remark. The Catalan numbers may be defined by

0 ! With this notation, dim lo I See e.g. n 2.7.3 (page 111) of [GJ]. i E : c We shall also need the following computation. We agree that a binomial coefficient 1 b] is zero if the integers a,b satisfy b < 0 or b > a. k. $

I S

90 Chapter 2: Towers of multi-matrix algebras

I

Lemma 2.8.2. Let k ;! 1 be an integer and set m = [k/2], the greatest integer less than

or epudl to k/2. Then

proof. By comparison of the coefficients of tC on both sides of - ( l + t ~ ~ ( l + t ) ~ = (l+t)a+b, one has

for ahy integers a,b,c ;! 0. (See for example Section 11.12 in [Fell.) Assume first that k is even: k = 2m. Setting a = b = c = k in (*), one obtains

- '["I + '[k12, and -.Z k Z m

ik1l2 =;KI -:[;12. j =O

Setting a = b = k and c = k + 1 in (*), one obtains

k k k

2 i k] El] = C [j] [i-l] = EL] j = O j = O

For k odd (k = 2m+l), one obtains similarly \ \

5 2.8. The algebras Ai,k for generic /3

so the conclusion follows for k odd as well. #

Define now a sequence (Pk)k20 of polynomiitls in Z[A] by

so that in particular

(Observe Pk here is as in [Wenl], but as Pk-l in [Jol].)

Pm~osition 2.8.3. Consider an integer k ;! 0 and set m = b] . Then

(i) The polynomial Pk is of degree m. Its leading coefficient Is (-l)m if k = 2m

is even and (-l)m(m+l) i f k = 2m+l b odd. (ii) Pk has m' distinct' roots which are given by for j = 1,2,..;,m.

. - (iii) Assume k 2 1. Let A be a real number with

1 A < -q

Then P1(A)> 0, P2(A)> 0,. . . ,Pk(A)> 0, Pk+l(A)t 0.

(iv) Set Qk(A) = P~(x(x+I)-~). Then

/CS'* 92 Chapter 2: Towers of multi-matrix algebras ~7 9 'JJ. Ine lLlgeDrw generic P a - k -

Y J

Proof. Claims (i) and (iv) are easily checked by induction. - For (ii), we compute in the ring Q[A,-] and proceed as in the proof of 1.2.2. The

difference equation for the Pkl s has an indicial equation j? - p + A = 0 with roots

k so that Pk = Cpl + D&. By adjustment of the constants C,D to fit PO,P1 k e find -1 k+l k+l Pk = (pl-p2) (pi -p2 ) for each k 2 0. Consider now a real number 0 with --

1 i 0 -i 0 e e O < 8 < 7r/2 andset A=-, so that pl =- and h=pco~f Then

4cos 0

pk(A) = *, .2 cos ( 0) s m 0

which vanishes when 0 = & with j = 1,2,. . ,m.

Claim (iii) is obvious for k = 1, and we may assume k 2 2. For 6 (2,. . .,k}, the smallest root of Pl is and PAX) > 0 for

>A, one has PdA) > 0. The two smallest roots of Pk+l are

and Pk+l < 0 on ]A1,A2[. As < A2 one has in particular Pk+l(A) < 0. #

Since the polynomials Pk have coefficients in H, it makes sense to evaluate them at

any number in our referende field K. Given an integer k 2 1, we define ,b E K* to be k-genkc if

Say that P is generic if it is k-generic for all k. '.

For example, any /3 E K* is l-generic, and /3 is 2-generic if and only if ,d # 1. If K is not algebraic over its prime field, transcendental numbers are obviously

generic. If K contains the reals, Proposition 2.8.3 (ii) shows also that any P outside the interval ]0,4[ is generic.

For ,b E K*, let q be a number distinct from 0 and -1;in K or possibly in some quadratic extension of K, such that ,8 = q-'(q+l)'. Claim (iv) of Proposition 2.8.3 shows

m . that ,b is not generic if and only if z q l = 0 for some integer m 2 2. In particular, if K

j=O is a finite field, no ,b is generic.

For generic /3 € K*, we shall define inductively a neated sequence (BAk)k21 of

associative K-algebras with unit, and a normalized trace on each of these. Set Bp,l = K and denote by trl the tautological trace on BPll. Set

BA2 = Kel @ K(1-el) where el is an idempotent, not zero. Define tr2 : BPY2 -J K by

1 tr2(el) = 8 and tr2(1-el) = 1 - 8'. Identify B,b,l with the multiples of the identity

in B The Bratteli diagram of the pair Bp,l c Bp,2 is P,2'

(see the end of Section 2.5 for the notation). In the next lemma, we set

Lemma 2.8.4. Consider an integer n 2 2, and assume P E K* is n-generic. Suppose

there is given a nested sequence (B P,k ) of K-algebras, together with traces

. trk : BPjk -' K eztending one another, such that the following hold for k E (2,- . .,n}:

(i) BP,k is generated by its unit, by elements el,. + . ,ek-2 (all in BAk-l) and by

ek-l. Denote by B' the two-sided ideal in Bp,k generated by el,. , . ,ekv1. P,k , (ii) The generators satisfi the relations

: e: = ei,

k Peieje,=e, if l i - j l=l ,

g e.e. = e.e. if 1 i-j 1 > 2 1 J J 1

{ f o r d i , j~{l , . . .k-1) .. L

k (ii) BPYk is a direct sum of b] + 1 fuctom Q:, with Q. isomorphic to the algebra J

k of matrices of order {;), for j = 0,1,- . -,El. One has Bh,k = .@ Q j. Denote by dk J) 0

94 Chapter 2: Towers of multi-matrix algebras 9 2.8. 'I'he agebras A b,k tor genenc p YO

k the (unique) nonzero idempotent in QO. - Proof (see $5.1 in [Jol]). During the proof, we write Bk for Bp,k.

(iv) The inclusion BWk-l c Bak is desc~bed by the Bratteli diagram: Both trn and it restriction trn-l to Bn-l are nondegenerate by (vii), since P is . I

tional expectation associated to tr, a,q in

Then Enen-lEn = ?En. Indeed, for all y E Bn and all z E BW1, one has, first by

2.6.2(i) and then by (vi)

trn-l({*n(en-lEn(~)) - 7En(y)}z) = trn(en&,(yz)) - drn-l(En(yz)) = 0.

Thus E,(~,-~E,(Y)) = TE,(~) because trn-l is non-degenerate, and in particular

Next we claim that

en-lEn(en-l~) = e p l x (*I

The j of the subfactors increases from right to left, So the white [ ~ M P . black] k Obviously (*) holds for x = 1 because En(en-l) = 7 1 by the previous claim.

vertez on the eztnme right reprrcsnts 9i-l [ rap . Qd. 1 Next we check that (*) holds if x = yen-l for some y E BPI, First, if y = ylen-2y2 -

with y1,y2 E Bn-2, then en-lx = ry 1 e n-lY2 = 7en-1Y1Y27 and

2 en-lEn(en-lx) = enn1En(en-1)~1~2 = 7 en-lY1Y2 =

by Bn-2-.linearit~ of En. If y E Bn-2, then en-lx = ePly, and again

(viii) trk is faithful. en-lEn(en-l~) = e n - l X '

Thus (*) holds when x = yen-l, for any y € Bn-2 + Bn-2en-2Bn-2, namely for all

Now using the Bn-l-.linearity of En, we see finally that (*) holds for all Suppose in oddition that K = c, that each BOyk (k 1 3 has a c*-&ebw stmdure

x E Bn-1 + Bn-len-lBn-l, namely for all x E B,. mating the idempotents ei self-adjoint projectiom, that 0 > 0, and that p k ( r l ) > 0 for

Define BhS1 to be the algebra obtained from the pair Bn-l c Bn by the I 1 k 5 n+l. Then Bp,n+l aka has a c*-algebra structure making en a w-adjoint . fundamental construction, and set

projection, and the trace trn+l is faithhl and positive.

Bn+l = Bh+l @ %+l

Chapter 2: Towers of multi-matrix algebraf '--'

where g+l is a central idempotent. By Corollary 2.6.4, the two-sided'ideal BA+l is

generpted by Bn and En. From now on, we write en (an element in BnS1) rather

than En (a mapping from Bn onto Bnel). Then Bn+l is a multi-matrix algebra by

2.4.1 in which el,-. .,en generate BA+l, so that ],el,. . .,en generate Bn+l. We have

checked (i) and (ii). Define a map J : Bn -+ BA+l by J(x) = (x,O) if x E BA and

J(dn) = (dn,dn+l). (This is of course an abuse of notation: the first component of J(B) is the element of Bk+l = End (B,) which is left (or right) multiplication by dn!)

Bn-l

Then J is obviously an injective morphism, so that we may (and we shall) identify Bn

with a subalgebra of Bn+l. NOW the shape of the diagram in (iv) follows from the

induction hypothesis and Proposition 2.4.lb, and the dimensions from the relations

and (all n and jj.

This shows (iv), and consequently also (iii). Now (v) follows from L e y 2.8.2. Define the trace trn+l : BnS1 7 K by assigning the weight ~JP,+~-~.(T) to the

factor Qn", as desired for (vii). Let fk denote a minima( idempotent in Q\. When n j J

is even and j = n/2 we have

n+' - P I ~ P ~ ( ~ ) = p12, trn+1(fi/2) = 2) - while

trn(fit2) = P / ~ P ~ ( T ) = PI2.

In all other we8 we have

\ by the three term recursion for the P's. Consequently trn+l extends tr,, and in

particular t ~ ~ + ~ ( l ) = 1. (This point shows precisely why the factor Q;+' = had

to be introduced in Bn+l!) Incidentally, this gives the relation

$2.8. The algebras Ap,k for generic /3

which could also be checked directly. We next verify the relation

Ptm+l(wen) = trn+l(w) (**I for all w E B,.

,

We check this first for w E Bn-l. We may then as well assume that w is some

minimal idempotent f?-I of Q?-', where j is an integer with 0 I j i p+]. But then J 3

we know from Corollary 2.6.4~ that f?-'en is a minimal idempotent in Q ~ + ~ J j +l'

Comequently

'+1 trn,+l(q41en) = 7J Pn+l-2(j+l)(r)

= = fi~n-~($-l)

and (**) follows because trn+l extends t rn l .

We now set w = xen-ly for some x,y E Bnq1. Then enwen = xenen-leny = rxe,y

by (ii) and, using the case of (**) already checked

On the other hand, by the induction hypothesis

Tlim (**) holds for w = xen-ly.

Consequently (**) holds for all w in Bn-l + B,len-lBn-l, namely for all w E B,.

This proves (vi) and (vii). If P is (n+l)-generic, then (viii) follows from (vii). Finally, if K = C, and the Bk are c*-algebras for k 5 n, then B6+1 also may be

given a ~ * - s t ~ c t u r e making the idempotent en self-adjoint; see the discussion in

Appendix IIa, or the remark under 2.6.5. Clearly Bn+l also has a c*-structure.

Moreover the weights of the trace on Bn+l are strictly positive by (vii). #

98 Chapter 2: Towers of multi-matrix algebras 5 2.8. The algebras APjk for generic P 99

Theorem 2.8.5. Consider an integer k 2 1 and a number P E K* such that

P ) # 0 r j < - 1 w e e (P ) are the polynomials ofProposition 2.8.9. j j>l 1 2k ( a ) A , is a multi-ma* algebra of dimension + [ k], isomorphic to

m

&"il(K), where m = j =O

M and 1) = M - El].

(b) There ezists a unique normalized trace trk : 4 K sach that

whenever 1 < j < k-1 and w is in the subalgebra generated B$I l,tl,. ~ , e ~ - ~ . Moreover

tr, is faithful i f pk(/T1) # 0.

(c) The natural map A P,k-l -I is injective and trk extends trk-l.

(d) If Bp,k is as in Lemma 28.4, the assignment 6. I+ e. (1 j s k-1) eztends to an J J

isomorphism from ApIk onto Bp,k.

(e) The trace trk on also satisfies

whenever 1 5 j< k-2 and w is a word in {ejSl, . . . , c ~ - ~ ) . More generaUy we have

whenever u is a word in {cl,. . .,4.) and w is a word in { c ~ + ~ , . . . , c ~ - ~ ) . J

( f ) The map ej I-+ ek-j extends to a trace preserving automorphism ok of 2 Furthermore q is inner in case K contains a solution q of the equation q"(q+l) = 8.

Claims (i) and (ii) of the previous lemma show that the map of (d) is a morphism onto. Claim (v) of the lemma and Proposition 2.8.1 show that this morphism is injective. Consequently, assertions (a) and (c) and the existence of trk in (b) follow from

the lemma. But the relation in (b) together with the normalization trk(l) = 1 and the

trace property trk(q) = trk(yx) suffice to compute the trace on p y word in the

generators {ei) of A so the trace is unique. I

P,k2 We prove by induction on m ( j + l l m < k-1) that the formula of (e) holds for

u 6 alg {l,el,. . 6.) and w E alg {I ,C~+~, ,em}. The case m = j+l is clear from (b). ' J

Suppose that m > j+l and that the result is verified for elements of dg { l , ~ ~ + ~ , . . ,eml). It suffices then to deal with a reduced word w = xtmy where x

and y are words in { c ~ + ~ , . . . ,em-l}. Then trk(w) = /T1trk(xy), and trk(uw$ =

trk(ywem) = ~ l t r ~ ( U x y ) = trk(u) ~ l t r ~ ( q ) , where the last step follows from the

induction hypothesis. Let q be an element of K, or of a quadratic extension of K, satisfying q-1(l+q)2 =

p. Define elements [ 7 = + 1 ~ ~ - 1 and

i ci = (r1r2.. .rhl). ..(7172)~1

in AS+ % K(q) for 1 < i 5 k-1. Thae are invertible, with $ = (q-I + l)ri - I, and one

verifiea by induction that' C .~ .C ;~ = yhl and c.e.c;' = e for i 5 j-1. In particular, J 1 J J 1 J hl

ak: x ++ cpc i l is the automorphism of part (0, This automorphism is trace presewing,

because the trace trk extends uniquely to Ap,k % K(q). #

Corollary 2.8.6. Consider an integer k 2 1 and an grbitraa number @ E K*. Let cp

be Qe homomorphism -' Ap,k+l which, for j 5 k-1, maps 6. viewed as a generator J

of Ap,k to e (sic) viewed as a generator of A j P,k+l

(a) is of dimension

(b) cp is an injection and any element x E A P,k+l can be written as x =

du) + Zdui)%q(wi), whae u, vi, and wi are elements of A P,k' (c) There is a sequence of traces tre : 4 K (1 < l ( k) such that

tre (1) = 1, and trl+l(p(u) + xq(ui)ee "(i)) = trl (u) + ~ ' Z t r ~ (viwi) for all U,

EEnefL I t is enough to prove the corollary for any extension of the field K, so that we may assume K to contain infinitely many generic numbers.

Assume first that is generic. Then AAk has a basis over K made of the ' ["I reduced monomials (see 2.8.1 and 2.8.5a), say ( c ~ ) ~ ~ . The structure constants are

Proposition 2.8.1 shows that, for any given pair ((r,~), aKbut one if the cF vanish and U,T

100 d w h h Chapter 2: 'lowers 01 muiwmauu argeur- f i 3 A . 0 rllt: ~ g t : u ~ w A P , ~ Ior generic p

1 the one non-zem c$,, is a power of r depending on u and r. In particular there are

monomi* ~ $ , ~ ( t ) E Kit] such that c$,, as above is just cp 0,r (8') for any u,r,p E S.

Define now the "generic" algebra A @n,k over the polynomial ring K[t] as the free

K[t]-module over S, with canonical basis denoted again by (c,),,~~, and with

multiplication defined by

The relations which express that this multiplication is associative are polynomial, and they hold when t is specialized at r1 for any generic P EK*, by Theorem 2.8.5. Hence they hold identically, and Agen,k is a well-d&ned associative algebra. Indeed, it is the algebra

with unit over K[t] abstractly deGned by generators el,-. . , E ~ - ~ and relations

Consider finally an arbitrary P E K*. Then Ap,k is isomorphic to A ' where K is made a K[t]-module by c(t)X = c ( ~ l ) X for c(t) E K[t] and X E K. This shows claim (a): That cp is an injection follows similarly. As observed in the proof of (a), there exist bases of AS+ and AP,k+l consisting of the reduced monomials of 2.8.1, and

claim (b) follows from this. We leave the details of part (c) to the reader; compare, however, 2.9.6. #

&mark: In general the traces tr! of claim (c) are not faithful; see Theorem 2.9.6.d.

Consider now the situation at the end of Section 2.7: One has a multi-matrix pair M,, c MI and a Markov trace t r : M1 3 K of modulus B; these ge&rate a tower, and the

conditional expectations E. : M. -I Mkl for j = 1,. . .,k-l generate (together with 1) a J J subalgebra Atryk(M0cM1) of Mk'

promition 2.8.7. Suppose that /3 E $ satisfies pj(F1) # 0 for i <\j 5 k.

(a) Suppose that x : A P,k -+ C is a surjective homomorphism of K-algebras anathat

C has a trace t r satishing b tr(wx(6.)) J = tr(w) for i < j 5 k-1 and w E x(A fl, j ). Then

x is an isomorphism and t r is nowdegenerate.

(b) In particular, with the notation above, the map ,y : E. I+ E. eztends to an J J isomolphbm of onto Atr,$MOcM1), and the restriction to Atr,k(Mo~M1) of the

Markov trace t r : Mk ---I K is nowdegenerate. I

b&& (a) It follows from 2.8.5(b) that t r 0 x = trk' Hence if x E ker(x), then for

all y E one has trk(xy) = tr(x(x)x(y)) = 0, so that x = 0, by the non-degeneracy

of trk. Thus x is an isomorphism and t r is non-degenerate.

(b) By 2.7.5, the map x extends to an homomorphism of AD,k onto Atr,k(MO~M1),

and 2.7.5 together with 2.8.5(b) imply that t r o x = trk. Thus (b) foHows from (a). #

Suppose ,8 E s(* is generic. The following picture sums up the structure of the traced 1 algebras introduced in this.section (with r = F ).

102 Chapter 2: Towers of multi-matrix algebras

2.9. An appmach to the mn-generic case.

If ,13 E is non-generic, then (1) The algebra AP,k defined by generators and relations as in Section 2.8 need not be

semiflimple. (2) Given a multi-matrix pair MO c Ml and a Markov trace t r of modulus on

MI, the restriction of tr to Atr,k(MocM1), the algebra generated by {],El,. . ,Ek_l)

in Mk, need not be faithful.

(3) Given a second such pair M0 c and a Markov trace & of modulus /3 on

M,, the algebras Atr , k(MO~M1) and A ~ , ~ ( M ~ c M ~ ) need not be isomorphic.

All this contrasts with the generic case described in 2.8.5 and 2.8.6. The modulus /3 = 1 illustrates these phenomena.

Example 2.9.1. The algebra +,3 is not semi-simple. (This is a particular case of

Theorem IL10 in Appendix Kc.)

Proof. Let T = - { [i i]) be the algebra of 2-by-2 upper triangular matrim over K.

. As T is not semi-simple, it suffices to show that T is a quotient of But the

assignment

extends to a homomorphism from .$3 onto T. #

Example 2.9.2. Consider the pair Mo = C @ C imbedded in M1 = Mat3(C) @ Mat3(C)

with inclusion matrix A = [i 11, together with the trace t r on M1 with weight vector

(1,-1). Then tr is a Markov trace of modulus 1 on M1. Consider also a pair MO =

with any faithful trace ?r on then ?r is evidently a Markov trace of modulus 1 on

M1. We have

(2) The restriction of t r to Atr,l(MO~M1) is not faithful.

(3) A t r , k ( M o ~ ~ l ) and A ~ , ~ ( M ~ c M ~ ) are non-isomorphic for all k 2 2.

(: t] brs eigenvectors (1,-1) a+ (1,l) with &&. The matrix = A =

eigenvalues 1 and 9 respectively. The Perron-Frobenius eigenvalue 9 is also the index [M1:MO]. But the other eigenvector (1,-1) also defines a Markov trace t r on M1 with

modulus = 1.

3 2.9. An approach to the non-generic case 103

Let Mo C M1 C M2... be the tower generated by Mo t M1. Since Mk is generated

as an algebra by M1 and Atrjk(M0cM1), if for some no the algebras A and tr,no

AtrTE0+, were equal, then M4 = Mno+l as well, and therefore Mk = M for all

k k no

k 2 no. But dimc Mk increases as [Mi:Mo] = 9 , by Proposition 2.4.2, Hen&

for all k. On the other hand h;ll = Mn for all k and A ~ , ~ ( M ~ c M ~ ~ 2 C for all k.

This proves (3). The algebra Atr,2(MO~M1) is spanned by 1 and El, and is of dimension 2, since

AtrS ; Atr,l 1 C. The trace tr on M2 restricted to Atr,2 is given by

tr(a+bE1) = a + b (a,b E C).

It is not faithful because

for all a,b E C. #

We do not intend to make a detailed study of the algebras Atr,k(Mo~M1) when P is

not generic. But we want to describe the structure of the unique quotient of on

which the usual rules tr(1) = 1 and p tr(wei) = tr(w) for w E alg {&el,. . ~ , e ~ - ~ )

defines a faithful normalized trace. (Here ei denotes the image of ti in the quotient.)

The algebras BP,k, For the rest of this section we fix a P E s(* which is n-generic

but not (n+l)-g~eric for some n 1 1. That is pk(/3-l) # 0 for k < n, but 1 Pn+l ( r ) = 0. We again define a nested sequence (B ) of multi-matrix algebras P,k k>l

over K, and a consistent fami ly of normalized faithful traces trk on these algebras. ' For k i n, define Bp,k and trk exactly as in Lemma 2.8.4; since P is n-generic there

is no problem i n doing so. For k 2 n define Bp,k+l to be the algebra obtained by

: applying the fundamental construction to the pair BSk-l C BPyF Observe that B P,n+l c isthesameas B'

P,n+l in 2.8.4. For k < n+l, define trk as in Lemma 2.8.4; then

trn+l is also faithful because Pn+l does not appear in the computation of the weights of

I.

I V Y ,4-, V " * p Y Y L ... &"I."*" "1 -.'A". YI".". lo""-- ,.">* ZVO

\

the trace on e Q:+'. A h since Pn+l(~) = 0, the trace on B h + l =

Bp,n+l = extends that on Bp,,; it thus follows from 2.8.4(vi) (with k = n+1)

that trn is a Markov trace of modulus @ on Bp,n-l c BPp For k 2 n+l, we define

trk as in Proposition 2.7.4. Thus trk is a Markov trace on Bp,k-l C Bp,k for k 2 n,

but not for k < n. Note that B is a multi-matrix algebra generated by the identity P,k

and idempotents {el,- . . ,ek-l) satisfying the relations 2.8.4(ii); in fact these relations

hold for {el,- .,en) by 2.8.4 and for {en,en+l,. . .) by 2.7.5. For k 2 n+1 the identity

is contained in the algebra generated by {el,- .,%-l), in contrast to the case of generic

fl this follows from 2.6.4. Note that if K = C and P = 4ms2(d(n+2)), then the algebras Bp,k can be given a

structure such that the generators {ei) are self-adjoint projections, and the trace is

faithful and positive. This is shown in 2.8.4 for k 5 n. The assertion for k 2 n+l follows, because the tower construction for a pair of finite dimensional c*-algebras with a positive Markov trace produces a chain of c*-aIgebras with a positive trace, and self-adjoint projections ei; see the discussion in Appendix IIa.

Example 2.9.3. Let P = 1, so that n = 1. The definitions above (cum grano saris)

give Bp,k = Bp,l = K for all k 2 1.

Example 2.9.4. Assume that the characteristic of K is not 2 and let P = 2, so that

n = 2. The structure of the algebras Bp,k and of the traces trk is shown in figure 2.9.4

below.

Example 2.9.5. Assume that K contains Q(z/S) and choose /3 E

2 2 (4 cos (lr/5);4 cos (2lr/5)}, so that n = 3. The picture (with T = 8' satisfying = r2 - 37 + 1 = 0) is given below in figure 2.9.5.

B2,l 1 , l

Figure 2.9.4.

Figure 2.9.5.

In general, the picture for the B 's is obtained fmm that of the .4P,k's at the end P,k

of Section 2.8 by deleting the factor Q:" (represented as the extreme right point in the

(nt1)st row) as well as all factors above and to the right.

106 vnapter z: lowers or I I I U I G I - L ~ ~ I ~ itlgaula t 5 2.9. An approach to the non-generic case

Theorem 2.8.5 gives a complete description of A when P is (k-1)-generic. The P,k

following theorem indicates how part of the picture changes when P is not generic. Recall that we may (and do) always identify with a subalgebra of (see Corollary

2.8.6) and that Bp,k is also a subalgebra of Bp,k+l.

Theorem 2.9.6. Comider an integer n 2 2. Let /3 E I(. be such that p j ( r l ) # 0 for

j 3 n and P ~ + ~ ( B ' ) = 0, where (Pj)j21 are the polynomiak of Proposition 2.8.3. Then

one has for all k 2 1, (a) BP,k is a md tha t r i z algebra, and there ezists a homomorphism rk of APYk

onto B mapping each generator c. onto e. (1 < j < k-1). P,k J J

(b) There ezists a normalized trace trk : Bak 4 K such that, for any j E 11,. . .,k-1)

ptrk(wej) = trk(w)

whenever w is in the subalgebra B p,j B ~ , k . M O T ~ O U ~ ~ trk is faithfir and the

restriction of trk to B is tr. for j s k. P,j J

(c) For k > 2 the following diagram commutes.

(d) There is a unique family of normalized traces trk : -+ K S U C ~ that

If IPYk denotes the two sided ideal in AP,k consisting of those x such that trk(q)) = 0

for all y E Ap k, then IP = ker(rk), so that Bp dAp k/IB k. 9 > 1 , 1

(e) Suppose (Ck)k21 is an increasing sequence of K-algebras and & : 4 Ck

are surjective homomorphisms such that & = for all k. ,Suppose further that

1

each Ck has a faith@-nomlized trace trk : Ck 4 K satbbing

- trk 1 - trkI, and Ck-l

for w E Ck-l. Then Ck Bp,k g AP,k/Ip,k.

(f) The trace trk on also satispes

P trk(Ejw) = tr(w)

whenever 1 5 j 5 k-2 and w is an element of alg {l,ej+l,. . ~ , e ~ - ~ ) . More generally, we

have trk(uw) = trk(u) trk(w)

whenever u E alg {&el,. . .,e.) and w E alg {P,ejSl,. . . ,ek-l). J

(g) The map 4 I+ % extends to a trace preserving automorphkrn I?k of and

e. - ek-j eztends to a trace preserving automorphism Fk of Bp,k. These automorphisms J

2 are inner in case K contains an element q satishing q1(q+l) = P.

Proof. Claims (a) to (c) follow from the construction of the Bp,k above. The traces - tr o rk on APIk satisfy (*). The uniqueness statements in (b) and (d) are proved

as in 2.8.5(b). We have trk(xy) = trk(rk(x)rk(y)), so that if x E ker(rk), then x E 1 P,k'

Conversely if x E I then rk(x) = 0 by faithfulness of trk on B This proves (d), P,k' P,k' and (e) follows similarly. Statement (f) ia proved as 2.8.5(e), and statement (g) as 2.8.5(f). #

Corollary 2.9.7. Suppose that K 3 R, that Mo c M1 is a pair of multi-mat& algebras

over K, and that tr is a positive Markov trace on M1 of modulus P = [M1:MO]. Then

, Atr,k(Mo~M1) is isomorphic to Bp,k for all k 2 1.

I r Proof. This follows from 2.8.5 and 2.8.7 when P is generic, so we suppose that P is B -

non-generic. Let (Mk)k21 be the tower of algebras generated by Mo c MI, and tr the

extension of the trace to UMk, as described in Section 2.7. Both B = 1 1 A ~ ~ 1 1 2 and the 1 k 1 weights of the trace are real and positive; see 2.7.3. Using the path model (2.4.6 and 2.6.5),

/ we aee that it is possible to choose a system of matrix units T for the algebra Mk so

I

a"" F- vuapm A, r u w c ~ u ur L l r u i u l u m e i r A olgcuroa

$

that the idempotents Ei (1 5 i 5 k-1) are positive linear combinations of oertain minimal B idempotents T see especially 2.6.5.2 and 2.6.5.4. Let Mk be the B-linear span of the 6, 6

mat* units geneiating Mr Thus 4 is a mul t i -mat e b a o r I, and

Mk = M! % K Let A! be the Csubalgebra of 4 generated by {&El,. . .,Ek-l}. B The trace t r restricts to a positive Rvalued trace on 4. Note that Ak is closed under

I the Mnear involution r of Mk defined by ~ f , ~ = TlYf. Positivity of the trace implies I that tr(x*x) > 0 for all non-zero x E Mk, and as this holds in particular for x E Ak, we

conclude that t r is faithful. It follows by linear algebra that t r is also faithful on 1 A! W Atr,k(MO~Ml) = Ak % K, and therefore 2.9.6(e) implies the conclusion. #

The proof of Theorem 2.1.8 is now complete.

Theorem 2.9.8, ([Jo~]). Let n 2 2 be an integer and suppose that /3 E K* is n-generic but not (n+l)-generic. Then the generating function fn(x) for (dimK(Bp,k+l)k20 b

where the P: are the polynomials of Proposition 2.8.8. J

m. Set An = ABpvn and = di+BAk). Also let ((n(n.k) be the vector of Bp,n-l

dimensions of the multi-matrix algebra Bp,k.

Note that the Bratteli diagram for

Bp,n-l c BP,, is the Coxeter graph An+1, with a particular bicoloration and labelling of

the vertices. (See 2.8.4(iv) for the picture, substituting n for k.) Thus for n odd An is

the -by- Jordan block

while for n even An is the ( i + 1) -by- i matrix

$is&'-,, An approach to the non-generic case

In order to accomodate vectors and matrices of different sizes, we adopt the convention that Id imbeds in Id+' via

' I

With this convention we have for n &l

t(n,k) = I (A&:) (")I2( for k odd

A:(A~A:)~/~-~( for IL even,

where t = (O,O,-. .0,1)~. Hence

(2.9.8.1) t$ = 11((n3k)l12 = ((A~A:)~-~(I 0 (n odd).

The corresponding formulae for n even are

(hk) = I (A;A,) (k-1)/2( for k odd

A ~ ( A ~ A ~ ) ~ / ~ - ~ ~ for k even,

' Hence

- (2.9.8.2) bf: = ( ( A ; A ~ ) ~ - ~ ~ I (n even).

One can visualize these results quite easily by adding to the Bratteli diagram of the

: chain (Bp,k)k2 some "phantom" vertices with zero dimension. The picture for n = 5,

for example, is >

Chapter 2: 'l'owers ot muln-matnx algebras

Recall also that our labelling of the vertices on each floor increases from right to left, Since An< = < (n odd) and A:C = 4 (n even), (2.9.8.1) and (2.9.8.2) give

(2.9.8.3) b i = ( ( A : A ~ ) ~ O ~ ~ t ) (n ~ d d ) , and

(2.9.8.4) b: = ((A,A:)~( I 4) (n even).

Finally one verifies that

(2.9.8.5) A:+~A,+, - = E (n odd), and

(2.9.8.6) A ~ + ~ " + ~ -A&= E (n even),

where E is the ortihogonal projection onto IRS, in the Euclidean space of the appropriate dimension.

We claim that the functions ( f , ( ~ ) ) ~ ~ ~ satisfy the first order difference equation

fn+l(z) - [Zf,(z)+ll = zfn+l("[zfn(z)+ll.

Erst consider the case that n is odd. Then zfn(z) + 1 = .

3 2.9. An approach to the non-generic case

t using 2.9.8.3. Setting B = A A we have n n .

Similarly using 2.9.8.2 and setting A = A:+~A,+~, we have

The difference fn+l(z) - [zfn(z)+l] is computed using 2.9.8.5, and the resolvent identity:

The case n even is entirely Siinilar. Pn-l(z)

Next we observe that the functions sn(z) = GPJ satisfy the same difference

equation. First note that

by the second*rder difference equation for the P.. Hence J

using the defining relation for the P. again. But this last expression is ~ s ~ + ~ [ m ~ + l ] . J

Since (fn)n21 and (snlnLl satisfy the same first order difference equation, it suffices m

k 1 now to check that f2 = s2. But bE+l = 2 for all k, so f2(z) = 1 2kzk =

P1(z) 1 while s2(z) = = #

2.10. A digression on Hecke algebras.

As a general reference for this section, we use [BL%e], especially exercises 2 22 in 8IV.2. See also [CR], 811D. For the origin of the term "Hecke algebra", see p. in [Lus].

2.10.a - The com~lex Hecke al~ebra defined bv GL- (a) and its Bore1 suberou~.

If G is a finite group and Go is a subgroup, the complex Hecke algebra H(G,G,,) of

the pair Go c G is the cornmutant of the natural representation of G on the complex

vector space C[G/Gd of functions from GIGO to C.

We denote by C[G] the algebra of complex functions on G, with the convolution product. We identify CIG/GO] with the subspace of this algebra consisting of functions p

with dgh) = cp(g) for g E G and h E Go, and we denote by CIGO\G/GO] the

subalgebra of C[G] of Go-bi-invariant functions.

Proposition 2.10.1. The algebras H(G,GO) and CIGO\G/GO] are isomorphic.

Proof. More generally, consider first an associative algebra A with unit, an - idempotent e E A, and the left A-module Ae. It is easy to check that the map x I-+ p(x) = right multiplication by x is an anti-isomorphism from eAe to EndA(Ae).

Now let A = C[G]; for each g E G, denote by 6 the characteristic function of {g}b g

Set e = & %. Then Ae = CIG/GO] and eAe = CIGo\G/Go], so that H(G,Go)l h€G0 @

and CIGO\G/GO] are antiismorphic. But C[G] has a monical anti-isomorphism p I+ $, defined by h g ) = dg-l), which restricts to CIGO\G/Gd, so the proposition follows. #

Corollary 2.10.2. Let e be the central idempotent in C[Gd corresponding to the

trivial representation GO 4 GLl(C), and denote by pl,-. -,pm the minimal central

idempotents in C[G]. Then

H(G,GO) @ epiCIGlpie

where the direct sum is over the i ' s with epi # 0. The Bratteli diagram for the pair

C c H(G,Go) i s &at part ofthe Bratteli diagram /or the pair CIGO] c C[G] which lies above

the wertez corresponding to e.

Proof. This follows from Section 2.3. (See Corollrtry 11.26 of [CR] for a - generalization.) #

g i As a first example, consider the permutation groups G2 c e3; the diagram for

Then C C H(G3,G2) is described by

1 . 1

0

1

In particular H(e3,e2) 8 C @ C. It is easy to check that there are two double cosets in

62\G3&2' One shows similarly that H(Gk+l,6k) 8 C @ C for any integer k 2 1.

But the case of main interest here is when q is a prime power, G = GLn(q) .for some

n 2 2, and Go is the (Borel) subgroup B of upper triangular matrices. (The letter q

will no longer denote an idempotent below.) Identifying the double w e t s is a special case

114 Chapter 2: Towers.of multi-matrix algebras 9. z.ru. . necKe algebras 115

where W is the I1Weyl group1', namely here the symmetric group Gn embedded in

GLn(q) as permutation matrices (see 5IV.2 in [BLie]). Thus to each permutation w E 6,

there is associated an element aw of the Hecke algebra H(G,GO), which is the

characteristic function of BwB divided by the order of B. For i = 1,2,. . .,n-1, let si

be the element of W given by the matrix

1. 1

'1 0 1 1 0

1,

where the first diagonal 0 is the (i,i)th entry, and set 4 = asi.

Proposition 2.10.3. With the notation above, one has

( 4 S: = (~-1)gi + q i=l, . . . ,n-l

(b) gigi+lgi = gi+lgigi+l i = 1,...,n-2

(c) gigj = g j ~ if 1 i-j 1 1 2 i,j = 1,. . -,n-1

Furthermore the elements 4 (1 6 i < n-1) generate the Hecke algebra H(GLn(q),B).

PEeefL (see [BLie] as well as Propositions 11.30 and 11.34 in [CR]). For each permutation w E 6, set C(w) = BwB. Let a, E C[B\G/B] be the quotient by I B I of

the characteristic function of C(w); then is a C-basis of the Hecke algebra.

For w1w1,w' E 6, and for g E C(we), one has

If C(w) n gC(wl)-I is not empty, there &a bl,. ..,b4 E B with blwb2 = gbjwf-1b41

so that g E C(w)C(wl).

For s in the set S = {sll*. .,s,-~) of generators of en, we need to compute

I C(s) I. Observe more generally that, for any h E G, the map

I B/(B n h ~ h - l ) -, (B~B) /B

c l a s s of b class ofbh

is well defined (if b,bl are in the same class modulo B n hBh-l, there exists b' E B with b' = bhb'h-l, and blhB = bhB) and bijective. Then the number of left classes

- modulo B in BhB is the index [B : B n hBh-'1. It follows that

Let us compute (aw*aw,)(g) when w = w' = s. As C(s)C(s) = B u C(s) this is

zero unless g E B U C(s). For g . ~ B one has by (w)

As al is a convolution unit in C[B\G/B], this implies

as * as = Xas + q

, for some X E C. Introduce the restriction ,u to C[B\G/B] of the augmentation homomorphism C[G] -+ C, mapping p to dg). Then Z

gEG

and consequently X = q - 1. This shows (a). Introduce the length function l : 6, -, {0,1,2,. . . } with respect to the generators S.

Then

Indeed, if 4sw) > 4w)> then C(s)C(w) = C(sw) by no IV.2.4 in [BLie], so that as * a, is a scalar multiple of as, by (*). Let g,h E C(s) and u,v E C(w) with gu = hv; then

vu-I = h-'g E C(s)C(s) = B u C(s); but vu-I E C(s) would imply v E C(S)C(W) =

r

, .; . Y

:(sw), which. is incompatible with v E C(IV); ' hence g E hB, ' and thus any element in :(sw) can be written in exactly ( B I ways as a ~roduct of one element in C(s) by one in' :(w). This shows that as * a, = a,,. It follows in particular that {as) generates

[(GLn(q),B). 3 Consider finally s,t E S with (st) = 1. Then 4s) = 1,qst) = 2,4sts) = 3 and thus

* at * as = asts by (**). Similarly at * as * at = at,, and (b) holds. Claim (c) . .

>llows in the saine way. # . . . . . . .

Now remember that 'the symmetric group in n letters has a presentatioh with enerators the transpositions .si = (i,i+l) for 1 s i s n-1 and relations

2 i+lsi = si+lsisi+l s.s - S.B. if, Ji-jl L 2. si = 1 . s s 1 j . - J I

'here is an easy proof of this which shows at the same time that the abstract algebra enerated by n-1 generators subjected to the relations of 2.10.3 is of dimension at most !. (See the beginning of $4 in [HKW].) For q a prime power, it follows then that the gations of 2.10.3 give a presentation of the Hecke algebra H(GLn(q),B). But we shall see

lat it is important to consider a more general family of algebras, defined for all q f 0.

l0.b -The Hecke -bras Qq,, . Let K again be an arbitrary field. Consider an integer n L 1 and a parameter q E K.

Te define H to be the associative K-algebra. with-unit presented by q,n

generators: gl,g2, . . ,gn-1 relations: as in 2.10.3.

Proposition 2.10.4. One has di- H = n! for all q E K and for aU n L 1. q,n

Proof. We take for granted the presentation of 6, in terms of the transpositions - 3i). Each of the n! dements r of Gn can be written uniquely as a reduced word w in

le isi) with ,

(i) minimum length among all words representing Ir,

(ii) the largest si in w appearing only once, and moved as far to the right as

possible, and (iii) all subwords of w reduced according to criteria (i) and (ii)..

1

he corresponding n! words in the generators {gi) of H span H linearly, because q1n q,n

le Hecke algebra relations 2.10.3(a)-(c) can be used

8 fP*w newt: ageoras 117 ,-----,

(i) to reduce the length of a word in the {gi) (i.e., to write it as a linear combination

of shorter words), and (ii) to reduce the number of occurences of the largest gi in a word, and to move it to

the right, whenever the corresponding operation can be performed on the corresponding word in the {si). It follows that d i ~ & H is at most n!. On the other hand, we will exhibit below a

q,n sufficient family of inequivalent irreducible representation of H to obtain the other

q,n - inequality. See [HKW,$4] for a more explicit proof. #

For convenience we take 9( = C in the following discussion. For q a prime power, Hq,+ is the same as H(GLn(q),B) in 2.10.a, and is in particular semi-simple. But we

have no reason a pn'ori to believe that there is any relationship between these algebras for different values of q. Also, the decomposition of any H as a direct sum of matrix

q,n algebras is not obvious, each summand corresponding to some irreducible representation of

GLn(q).

Ohewe however that, if we put q = 1, we recognize H as the algebra C[Gn] of l,n

the symmetric group, so H is semi-simple. A necessary and sufficient condition for l,n

semi-simplicity of H is the non-degeneracy of the Killing trace x - tr(X(x)), where q,n

tr denotes the trace on EndC(H ). (FQI a finite dimensional C-algebra A, the radical q,n rad(A) coincides with A' := {x E A: tr(X(xy)) = 0 for all y E A). In fact, both rad(A) and A' are ideals which contain every nil ideal, and to show equality one shows that each

, is a nil ideal.) From the proof of Proposition 2.10.4 one obtains a basis {g, : a E 6,) of

H ~ , + and polynomia~ structure constants PP (q) such that g$T=Cp$,T(q) gII ~t ('97

- P follows that degeneracy is determined by a polynomial equation in q, so for all but a finite set of q E C (n fixed), H is semi4mple of dimension n!. Also Hq,n-l embeds in

q1n Hq,n via the obvious identification of the generators 4 for 1 s i i n-2.

We now argue intuitively, though Mremely plausibly. For the values of q for which and Hqp are semi- simple, the inclusion H q,n-l c Hq,n is completely deecribed

by a vector of integers (for the dimensions of the factors in H ) and an integer valued q,n

matrix (the inclusion matrix). As these should vary continuously with q, they should be independent of q for these values. In particular they can be determined by examining the case q = 1. But then they are determined entirely by the dimensions of the different representations of C5n-l and Gn and the restriction rule from en to FOI this

reason we shall now describe this structure. In 2.10d we will identify a certain singular set

110 bnapcer a: lowers or mulr1-maFnx ageoras 9 2.11~. neme ageoras

St c K and construct for q $ St a complete family of irreducible representations of H . q,n1

this will demonstrate that H z #[Gn] for all n and for q $0. q,n -

2.10.c - Com~lex representations of the svmmetric mou~. The conjugacy classes of the group Gn are naturally indexed by partitions of n, two

permutations being conjugate if and only if they have the same cycle structure. Thus there are as many irreducible representations of Gn (over C) as there are partitions. Although

one cannot expect a natural correspondence between representations and partitions on the above grounds, it has long been known how to construct an irreducible representation from a partition. It is convenient to represent partitions by "Young diagrams", as amply illustrated by the following example.

Example 2.10.5. To the partition X = [X1,X2,XpX4,X5] = [5,3,2,2,1] of 13, one

associates the Young diagram

The most important rule is the restriction rule: if one restricts the representation of 6, corresponding to a Young diagram Y to it is isomorphic to the direct sum of

all representations corresponding to all Young diagrams Y' obtained by removing one box from Y, all occuring with multipIicity one.

Thus the irreducible representations of Gn (and hence the Bratteli diagram for

CG1 c CG2 C CG3 c . . .) arewnveniently pictured by the following important diagram:

figure 2.10.6

The dimensions of the corresponding representations are given by the number of ascending paths on 2.10.6 beginning with o and ending at the Young diagram in question. The above facts will actually follow from the construction of irreducible representations for the Hecke algebras Hq,n, to which we now return.

2.lO.d - Irreducible representations of H for q $ St. q1n The material that follows is taken fiom Wewl's thesis [Wen2]. The K-algebra H

q,n is that defined at the beginning of 2.10.b; in particular, the field K is arbitrary.

By our intuitive argument, we expect that figure 2.10.6 should also represent the structure of H for all but a countable number of values of q. While this could be q,n proved in an elegant manner due to Tits (see exercise 26 in 3IV.2 of [BLie], or Lemma 85 in [Ste2]), two important pieces of information would be missing: there would be no indication of which values of q are."badl' (and that would be particularly frustrating for K countable!), and there would be no construction of concrete representations of H

q,n' We shall now show how to cons- an irreducible representation of HaSn for each

partition of n, provided q is not in the set St defined below. It is first convenient to dispose of another presentation of Hqln than that of 2.10.b.

Proposition 2.10.7. Colnsider a number q E K* and an integer n 2 1. Assume q # -1

and set

g i + l (a) e,=m i=l,...,n-1

These generate H and constitute with the relations q9"

(b) e; = ei i = 1,. . .,n-1

-2 -2 (c) eiei+lei - q(q+l) ei = ei+leiei+l - q(q+l) ei+l i = 1, . . ,n-2

(d) e.e . - e.e. 1 j - 3 1

when li-j1 2 2 i,j = l,...,n-l

a presentation of H 4,n'

Proof. A straightforward computation.' # -

Naturally this demands comparison with the definition of Ap,, in Section 2.8.

However we shall postpone further comments on this until the next section. Define the subset 0 of K to be the union of (0) with those q for which there exists

n

an integer n 2 1 with x q j = 0. Thus 0\ (0) is the set of non-trivial roots of 1 in j=O

characteristic 0 and the set of all roots of 1 in finite characteristic. (As already noticed in section 2.8, if q f! 0 then /3 = q-l(q+l12 is generic.) For each d E a \ (0) and q € K \ S 1 define

l + q + . ..+q d

(Remark: When d > 0, then ad(q) = Note also that - (l+q)( l + q + . . . +qd-l)'

9 ad(q) = Qk(q) where Qk is as in Proposition 2.8.3.iv.) Suppose given a partition 1s $k of n, say X = [A1,. .,Ak], where we allow some of the last X.'s to be zero. We think of J X as a Young diagram. Let VA be the free K-vector space on the set of ascending paths

p fcom to X on figure 2.10.6, and denote by {v ) its canonical basis. We define now P endomorphisms f l, . ,fn-l of V

Let i E (1, ,n-1). For each ascending path \

1 p = ( X =u,X2,-..,Xn=X)

we have to define fivp. The partition Xi+' is obtained from in one of three ways

(a) By adding two boxes to the same column of Xi-'. In this u s e fivp = vp.

(b) By adding two boxes to the same row of Xi-'. In this u s e fivn = 0. r

(c) By adding boxes in differeht rows and columns of Xi-'; mort precisely there is pair of integem (1,s) with r # s and A:-' :'li A:-' such that A; = A:-' + 1 and

A:+' = Xi-' + 1. In this case there is precisely one asceading path from Q to A which

differs from p in its ith vertex only; we call this path p' . For example:

Set

d = (A?'-r) - (Xi'l-s) = (8-1 + (A:+~-A;+~)

and 0 b S e ~ e that d # 0. Define d' in the same way for the path p' and note that d' = 4. Finally, define

fivp = ad(,d(4)vp + (l-ad(~))vp'.

Observe that fi leaves invariant the subspace Kv @ Kv of VX as well as its canonical : P P' , complement; on Kv @ Kv , ,, it is described by the matrix I P P

i We have taken advantage of the equaIity ad(q) + ad,(q) = 1, which follows from the

definition of ad and from d + d' = 0.

Chapter 2: Towers of multi-matrix algebras 3 2.11. AD,, and Hecke algebras

The veiification that fl,. s-,fn-l satisfy the relations (b) and (d) of Proposition

2.10.7 is trivial. They also satisfy (c), but this is more tedious to check and we refer to [Wen2]. We conclude that, for each partition A of n there is a representation rX of

Hq,n in VA defined by rA(ei) = fi' A remarkably easy inductive argument shows that

the rA' s aie irreducible and mutualljr inequivalent when X runs over the set .'P, of all

partitions of n (for q E K \ R). Indeed,, these representations are absolutely irreducible, because the same argument applies to any extension of K. By theorem of Burnside and Frobeni~ls-Schur, this implies that Hq,, has a quotient isomorphic to the multi-matrix

algebra e En%(VX), of dimension n!. But we have already reported that the A€?- u

dimension of H is no more than n!. (See the end of 2.10.a above, and $4 in [HKW.) q,n

Consequently the dimension is precisely n!, we have a complete set of irreducible representations of H for q 6 K \ R, and Hq,, ie isomorphic to e EndK(VX). (In

q,n A€%

particular, setting q = 1, this gives for K = OJ the usual complete set of irreducible representations of the symmetric group an.)

Another trivial consequence of the construction is that the restriction of a representation rA of Hq,n to Hq,+ is a direct sum te XX,, where A' runs over all

partitions of n-1 obtained from the partition of X of n by removing one box from the Young diagram. We reformulate this as follows.

Theorem 2.10.9. Let K be a field and let R c K be the union of {0), ofthe non trivial

roots o f 1, and of 1 in case char(K) # 0. Consider q E K \ R, an integer n 2 1, and the Hecke algebra H generated by gl,. q,gn-l with Me relations of 2.10.b, or equivalently

q,n by el,. . *,en-l and the relations ofProposition 2.10.7. Then

(a) Hqp is of dimension n!.

(b) Hq,, is a multi-matrix algebra.

(c) The natural mapping Hq,, -' Hq,n+l is an imbedding.

(d) The structure of the chain Kq,l c Hq,2 c . . . c Hq,., c q . is gyen by figure

2.10.6.

We make one further comment on Wenzl's paper, and for this we assume K = C. His exposition does not involve the matrix 2.10.8, but rather the related one

with ad written for ad(q). It follows that Hqp has a c*-algebra structure for q E IR

and q > 0, for which ei is an orthogonal projection: In fact, the main interest of [Weh2]

is in the values q = e*2d/n E R, for which Wenzl has constructed c*+dgebras which are quotients of the corresponding Hecke algebras.

2.11. The relationship between Am and Hecke algebras.

It was first pointed out to V. Jones by R. Steinberg that the defining relations of AD,,

(see section 2.8) actually imply the Hecke relations. This is obvious from the definition of Ap,, and Proposition 2.10.7, but we would rather state this in terms of the generators gi

again.

B Proposition 2.11.1. Consider q E K \ {-1,O) and ,O = 2 + q + q-I E #*, an integer

2 1, and the algebra Ab,n. Set

7

7i+1 7i = (q+l)fi - 1 so that ci = -7

q+ i = 1,. .,n-1.

and constitute with

a presentation of A P,n'

Proof. A straightforward computation. # ' -

Corollary 2.11.2. There D a surjective algebra morphism

a : Hq,n -' AD,,

defined by Iln(gi) = 7i for i = 1,. - -,n-1. If n = 1 or n = 2 it is an isomorphism. If

n 2 3 ikr kernel In is the two-sided idea1 o j H generated by q,n

Chapter 2: Towas of multi-matrix algebras

.g1gg1+ glg2 + g2g1+ g1+ g2 + 1.

Moreover the diagrdm

commutes.

Proof. The existence of $n follows from the definition of H Qnd Proposition - q,n 2.11.1. It is clear that 4 and % are isomorphisms, and that for n 2 3 the kernel of

% is generated by

for i = 1,. .,n-2. As q # 0, each 4 is invertible with inverse q-l(&+l-q). By

relations (b) and (c) in Profisition 2.10.3 on has

and consequently

-1 1 (g162.- 'h-1)9(~!1" 'g2 g i ) = i = 1 , s . + , n d

Thus In is generated by xl. The last claim is obvious. #

Thus, if q is a value for which H is semi-simple, it must be possible to identify q,n

AD,, with a certain ideal of Hq,n, given by some subset of the set of partitions of n. We

shall show that this subset is precisely the set of a3l partitions with at most two rows; this was also explained to us by R. Steinberg.

We recall that R has been defined in 2.10.d, and that the relations which constitute with gl,-. -,gngnl a presentation of H are written in Proposition 2.10.3 (see.also

q>n 2.10.b).

p. A@,, and Hecke algebras '1

f Lemma 2.11.3. Let q E K \ 0 , let. n L 3 be an integer, k t A be Young diagram with

n boxes and with at most two rows, and let

r . H 4 End(VA) A . 9,n

be as in 2.10.d. If xi = gigi+lgi + gigi+l + gi+lgi + gi + 1, then rX(xi)'= 0

for i = 1,...,n-2 .

1 . Pmof. We set first n = 3. As,there are twa partitions of 3 with at most two rows, we

split the proof in two steps. First X = [3], pictured as an. Here, according to the definition of

xA(gl) = rX(g2) = -1 and rX(xl) = 0.

Second A = [2,1], pictured as Here, instead of using rA, we may argue with any

2-dimensional irreducible representation of H for example that defined by q,3'

(which is irreducible if q is neither 0, nor -1, nor a nontrivial cube root of 1). A routine calculation shows that Ir(xl) = 0.

Assume now n 2 4, and that the lemma holds for n-1. By the proof of 2.11.2, it is enough to check that n;\(xl) = 0.

We recall from 2.10.d that

where A' has one less box than A.. In particular A' has at most two rows, and 5, (x1) = 0 by the induction hypothesis. Consequently rA(x1) = 0. #

Consider q E K \ 0 and an integer n 1 3. Let 'P, be the set of partitions of n. We

know from section 2.10.d that H is a direct sum @ IA of simple two sided ideals, q P A E . ~ ,

the notationbeing such that, for each AO E Tn, the representation r~ of Hqp restricts 0

to an isomorphism IA -+ End(VA ) and maps IA to (0) when A # A,,. We denote by 0 0

f the subset of 7, of partitions with at most two rows.

126 Chapter 2: Towers of multi-matrix algebras 5 2.11. ApIn and Hecke algebras

Observe that the proof above shows again the equality

d ~ n A ~ , , = for p generic In'= KerMn : -' AAn) = (glg2gl+glg2+g2gl+gl+g2+1)

of Theorem 2.8.5. be as in Corollary 2.11.2. Then To sum up, we have shown that, for generic p and corresponding q, the algebra A

P,n is isomorphic to the quotient of

Hq,n - by the two-sided ideal generated by

gl%gl + glg2 + g2g1 + gl + g2 + 1. This ideal corresponds precisely to the direct

Proof. By the previous lemma for A E f , one has ( 1 = 0 so that mnmands IA of H given by all Young diagrams A E 'P, having at least 3 rows. - (4,''

IA n In = (0). Consequently In c e IA. A n n \ f

Let A = [r,s] E f . From the definition of VA (see 2.10.d) one has

dim V[r,s~ = dim V[r,s-l~ if r = 8,

dim V[r,sl = dim V[r,s-l] + dim V[r-l ,s] if r > s > 1,

dim V[r,Ol = 1.

By induction on r and s, ones deduces from this

Thus 2 1 2n I: (dim = rn[

AEf

by Lemma 2.8.2.

n a contradiction. For A E f U {Ao}, the representation rA of Hqp defines a

representation 5 of A@,, in VA. As the 9 ' s are pairwise inequivalent, so are the

?rA18, a d

But this contradicts Proposition 2.8.1. #

CHAPTER 3 ' ~ i n i t e von Neumann Algebras with Finite D i i o n a l Centers

In this chapter we study pairs of finite von Neumann algebras with finite dimensional centers, and the index of such pairs.

Sections 2 to 4 are purely expository, and may be taken as an encouragement to the reader having essentially no previous experience with von Neurnann algebras. Sections 5 to 7 present a generalization to the present setting of some of the ideas of [Jol] for pairs of factors. Though this chapter c a ~ o t be so self-contained as the previous ones, we have tried to minimize the technical background in operator algebras assumed from the reader.

Let us first describe Sections 2 to 4. Let M be a von Neumann algebra which is a f&@ of type II1. (The definition is given in Section 3.2.) We denote by t r : M -+ C the

normalized trace on M, For every Hilbert space H on which M acts, Mmay and von Neumann have defined a positive number (possibly a) called the constant between M and its commutant; we denote this number by dimM(H). Two

representations of M by operators on two separable Hilbert spaces H and H' are equivalent if and only if dimM(H) = dimM(H1). Section 3.2 is as exposition of the

definition and the basic properties of these coupling constants. Except for the presentation, all this material comes from the original papers by Murray and von Neumann. I

In Section 3.3, we present some geometric ezamples of coupling constants arising in the theory of discrete series representations of Lie groups; they are borrowed from Atiyah-Schmid [AS]. In particular, we show:

Theorem 3.1.1. Let G be a connected real semi-simple, non-compact Lie group without center. Let r be a lattice in G, and let M be the uon Neumann algebra of the discrete group I'. Then M is a 111 factor. If n : G ---, U(H) is an irreducible discrete

series representation of G, then T &ends to a representation of M on H, and 1 r

dimM@) = C O V O ~ ( ~ ) d,,

where d, is the formal dimension of 7.

In Section 3.4, we consider a pair N c M of finite factors and we recall some aspects of the original work [Jol] on this subject. First the of N in M is now defined to be

128

where L ~ ( M ) is the Hilbert space obtained by completion of M for the scalar product (xi y) = t r ( ~ * ~ ) . It was shown in [Jo4] that this definition of index a g r w with the purely ring-theoretic definition of Chapter 2.

If [M:N] < W, the pair N c M generates a tower of IIl-factors

I E M ~ = N C M ~ = M C ... c M ~ - ~ c M ~ c . - .

by a fundamental constructi~g which is defined as follows. The natural conditional

expectation from Mk onto Mk-l can be seen as an orthogonal projection

% : L'(M~) + L ~ ( M ~ - ~ ) , and Mk+l is the von Neumam algebra of oprnntors on 2 L (Mk) generated by Mk and ek. This MkS1 is again a 111 factor. It is aparticular

case of Proposition 3.1.4 below that this way to define the fundamental construction agrees with that of Chapter 2. Moreover the Markov relation holds:

I

[M:N]trk+l(xek) = trk(x) for all x E Mk,

where trk and trk+l denote the normalized traces on Mk and Mk+l respectively.

The sequence (ek)k21 of projections in U Mk satisfy the relations k2O

IM:N] e.e.e. = ei if 1 i-j 1 = 1 1 J 1

e.e. = e.e. 1 J J 1

if (i-jl 2. 2

and provide consequently a representation of the algebras with P = [M:N]. (SW

Section 2.8 and Theorem II.16,) From this follows

2 Theorem 3.1.2. If N c M is a pair of 111-factors, either [M:N] = 4 cos (r/q) for

some integer q 2 3 or [M:N] E [4,m].

There is substantial overlap between Sections 3.2 to 3.5 and Sections I to I11 of Comes' report [Con].

b

F g Let us now describe Sections 5 to 7, where we consider a pair N C M of finite von

! Neumann algebras with pnite dimensional centers. There are projections pl,' .,p, which li.

are centrd in M and projections ql,. . . ,qn which are central in N such that

unapter 3: Finite von Neurnann algebras I 3.1. Introduction 131

plM,- + ., pmM, qlN,. ., qnN are finitefactors, and Then there exists a pair N c M as above with A = AN M and T = T!. Moreover, M and

n m N .=. @ q.N C M = @ piM. , N may be chosen hyperfinite.

j=1 J i=l The Skolem-Noether theorem does not hold for 111-factors and Proposition 2.3.3 does

If dimC(M) < m, this is the situation of Chapter 2. At this stage, let us assume that each not w r y over to the present setting: the matrices A! and T: do not characterim N

)f the factors piM, qjN is of type Ill (see the comment after 3.5.4). as a subalgebra of M. M As in Section 2.3, we define an index & AN = (Ai,j) E Matm,,(R+ U {a}) by Once a faithful trace is given on M, <the fundamental construction gives a new algebra

(M,eN), just as described above in the case of factors.

- [ O i fp iq j=0

1,j - Proposition 3.1.4. Let N be offinite indez in M.

[ M ~ ~ : N ~ , j]112 ifnot (a) The algebras (M,eN) and ~ n d i ( ~ ) are isomorphic,

(b) The algebra (M,eN) is again a finite sum of 111 factors. There is a natural vhere N. . = p.q.N is a subfactor of the factor M. . = p.q.Mp.q.. We say that N is of

1 J 1 J IJ 1 3 1 J een the minimal central idempotents of N and those of (M,eN), inite index in M if A: doe8 not have any infinite entry.

For an analysis,of traces on M and N (see Section 2.4 when dimc(M) is finite), we A convenient isomorphism is described in Corollary 3.6.5, and the bijection of (b) M Lefine also the &race matrix T N = (ci,$ E Matm,,(R+) by appears in Proposition 3.6.l.i~.

The partid description of N c M by A! and T! is useful because, if N is of

ci,j = trPiM(piqj), finite index in M and if L = (M,eN), one may compute AM L and TM L h r n AN M and

here t r denotes the normalized trace on the factor piM. 4 trace on M is described piM L M t

y the vector E ( R + ) ~ with si = tr(pi), and its restriction to N by the vector A,=(A,) T ~ = F ~ T ~ N N

T! E ( R + ) ~ . Traca are always assumed to be positive in this chapter, so that si 2 0 for E Mat,,,@+) is defined by

= 1,. .,m. M If dimE(M) < the matrices AN and T! are simply related by c i j = ~ ~ , ~ u ~ & ~ ,

itb py = dimc(piM) and u? = dimC(qjN). This relation has no analogue when the J

,M's and the q.N1s are factors of type 111: J

and where F! is a diagonal matrix e n s u i q that (TM)j,i)jd=l L for j=l , . .+,n. Proposition 3.1.3. Consider two irredundant matrices 1s i i m

.l See Propositions 3.6.6 and 3.6.8 for the details.

A =(A. .) E Matmp({2 C O S ( % / ~ ) } ~ ~ ~ U [2,m]) and T = (e. E Matqn(lR+) As in Chapter 2, a trace on M is said to be a Markov t r a of p o d u l ~ B for the pair I J N c M if it extends to a trace tr on (M,eN) for which

tisfging: = 0 ~ c . . = 0 and c i j = 1 for i E (l,...,m) . P tr(xeN) = tr(x) x E M.

1 ,J li j ~ n

There exists at most one such extension. As traces are positive in this chapter, ,8 has to be a positive number. The analogues of Theorem 2.1.3 and 2.1.4 hold as follows. Recall that a pair N c M is s@g&$& if the intersection Z(M) n Z(N) of the centers is reduced to C1.

m n Theorem 3.1.5; Let M = @ piM and N = @ q.N be finite direct sums of 111

i=l j=1 J M M pctors, let N be a subalgebra of M offinite indez, and write T, T for TN, TN.

(a) Let t r : M + C be a trace, let 5 E R? be defined by q = tr(pi), and let @ ER;.

Then t r is a. Markov tl'ace of moddw P for the pair N c M if and only if

(b) Ifthe conditions of (a) hold, then the Markov dension (My%) + C of t r is a

Markov trace of modulus P for the pair M c (M,eN). \ -

(c) If N c M is connected, there ezists a unique nonalized Markov trace on N c MI and its modulw P is the spectral radiw of TT.

Comparing Theorems 2.1.4 and 3.1.5, we may define the index of N in M as

[M:N] = p ( ~ T ) where p denotes spectral radius.

I Corollary 3.1.6. Theorem 3.1.2 holds for finite direct s u m of 111 factors.

We note that the definition of [M:N] given above is not the same as that of Chapter 2. However, P. Jollissaint has shown, in unpublished work, that the two definitions of index coincide.

If N C M is a connected pair of finite dimensional multi-matrix algebras with [M:N] s 4, we have shown in Theorems 2.1.1 and 1.1.2 that the corresponding graph is a Coxeter graph of one of the types A,D,E. The chief result of Section 3.7 is that connected pairs N c M of'finite direct sums of 111-factors with [M:N] 5 4 give rise to possible

Coxeter graphs associated with finite and affine groups.

Theorem 3.1.7. Let N c M be a connected pair of finite direct sums of 111-factors. M Assume Uiolt N is of jinite index in M and let A = AN be the inclusion matrix..

(a) If [M:N] < 4, then A is the matriz associated (in Theorem 1.1.3) to a bicoloration of one of the following Cozeter graphs:

Moreover [M:N] = 1 1 ~ 1 1 ~ = 4 cos2(lr/h), where h is the Cozeter number. (See tables 1.4.5 and 1.4.7.)

(b) If [MN = 4, then A: corresponds to one of the graphs

so that [M:N] = 1 1 ~ 1 1 ~ . (See tables 1.4.6 and'l.4.7.).

The index range described by Theorem 3.1.2 appears also in the remarkable family of

W -, which are discrete subgroups of PSL(2,R) generated by two parabolic transformations. We have included an Appendix I11 on these groups. Its purpose is to expose the spectacular comparison with Theorem 3.1.2 as well as to illustrate Section 3.3.

3.2. The coupling constant: definition.

, Let H be a (complex) Hilbert space. We denote by B(H) the *-algebra of all bounded operators on H, with x* the adjoint of the operator x E B(H). :Besides the topology associated to the norm

the algebra has also the ultraweak $@x&gy or w-topology which is defined by the semi-norms

Whenever necessary , we assume H to be separable. A von Neumanu algebra acting on H, or a yon Neumann of . B(H), is a

w-closed *-subalgebra of B(H) which contains the identity. If Mj is a von Neumann

subalgebra of B(Hj) for j = 1,2 and if q : M1 -, M2 is a *-isomorphism, it is known

that p is continuous with respect to the w-topology on both M1 and M2 (corollary

5.13 in [SZ] or section 1.4.3 in [DvN]). A ~ Q Q Neumann &g&& is a *-algebra M which is *-isomorphic to a von Neumann subalgebra of B(H) for some B; by the result just recalled, such an algebra has a well-defined w-topology.

A f a is a von Neumann algebra M with center ZM reduced to the scalar

multiples of the identity. Von Ne- algebras axe known to be principal in the sense that any w-closed two-sided ideal is generated by a central projection (see section 1.3.4 in

[DvN]). Thus a von Neumann algebra M is a factor if and only if m y two-sided ideal J # 0 in M is w-dense. There is not any continuity problem for representations of a factor M in the following sense: any *-homomorphism M -+ B(H) is w-continuous. (See theorem V.5.1 in [Tak]; the separability of H is crucial here.)

A HI &g&g is an infinite dimensional factor M which admits a normalized finite

trace t r : M 4 C such that (i) tr(1) = 1 (ii) tr(xy)=tr(yx) x , y ~ M (iii) tr(x*x) 2 0 x E M.

It is known that, on a 111-factor, such a trace is unique in two senses. First, in the usual

sense for operator algebras: tr is the unique linear form satisfying (i), (ii) and (iii); see [DvN], nos 1.6.4 and III.17; moreoever one has tr(x*x) , 0 for x # 0. But also secondly, in the naive sense: t r is the unique linear form satisfying (i) and (ii), by [FH]. The existence of II1-factors which may act on separable Hilbert spaces is one of the basic

discoveries in the first paper by Murray and von ~eumann [MvN I]. A finite factor is a von Neumann algebra which is either II1-factor, or isomorphic to

B(H) for some H of finite dimension. Such a factor is simple as a cbmplex algebra by [DvN], 111.5.2. Here is a characterization of finite factors; for more on this, see [KvN].

m o s i t i o n 3.2.1. Let M be a c*-algebra with unit and with center reduced to the scalar multiplies of 1. Let tr : M -, C be a faithful normalized trace (namely b linear form satisbing (i), (ii), (iii) above and tr(x*x) > 0 for x # 0). Assume that the unit ball of M is complete with respect to the metric d(x,y) = IIx-~(~, where = (tr(x*x))lJ2. Then

M is a finite factor.

&g& Let H = L 2 ( ~ , t r ) be the Hilbert space obtained by completion of M with respect to the scalar product defined by cxly> = t r ( ~ * ~ ) for x,y E M. Let r : M -4 B(H) be the *-representation of M on H, with r(x) being the extension to H of the left multiplication by x on M. Then r is injective because t r is faithful,. Let 4M)' denote the double commutant of M in B(H), which is, by von Neumann's bicommutant theorem, the w-closure of r(M) in B(H).

To show that M is a von Neumann algebra, it is enough to show that the inclusion of M in n(M)' is surjective. Let a € 4M)' with JlalJ = 1. By Kaplansky's density

9 3.2. Coupling mistant: definition 135

theorem, there is a net (x$ in M with Ilx,ll s 1 for all cu such that Ir(xcu) converges

strongly to a; that is, n(xcu)[ converges to a< for all 4 E H. Taking < = 1, this means

that (r(x&) is a Cauchy net for the )I.))2-distance, so converges with respect to this

distance to some element r(xo) by the assumed completeness of the ball of M. One can

check that the strong topology and the 11. 112-topology coincide on the unit ball of Ir(M)',

so a = r(xO) E r(M). #

Let M be a finite factor acting on some Hilbert space H. We are going to define the Coupling constant dimM@) which is a measure of the size of H aa an M-module, the

definition b n g made so that the standard M-module L ~ ( M ) = ~ ~ ( ~ , t r ) has size 1. Before comparing other M-modules to that one, we recall the following facts.

Lemma 3.2.2. (a) Let J : L 2 ( ~ ) -+ L 2 ( ~ ) be the conjugate linear isomety which

d e n + r3 M,. Then JMJ is the commutant E ~ ~ ~ ( L ' ( M ) ) of M in B(L~(M)). x w x

(b) Let K be a Hilbert space and let M act on L ~ ( M ) e K bg the diagonal action ~ ( $ 4 = (xrj) @ 0. Then JMJ @ B(K) is the commutant of M *I B(L~(M) @ K).

(c) Assume that the space K of (b) is infinite dimensional. For any M -module H, there exists an isomety

~ : H - , L ~ ( M ) @ K

which is M-linear, namely which intertwines the actions of M.

EUMf, (a) Let x,y,z E M. By definitionof J

JxJy = (xy*)* = yx* = yJx.

Applying this twice we get .

JxJyz = yzJx = yJxJz, and setting z = 1,

(JxJ)y = y(JxJ).

Thus JMJ c MI where M' = E ~ ~ ~ ( L ~ ( M ) ) .

Let moreover a E M' . By definition of the adjoint

(Y*x* l a) = (x* lya) =(x* lay) = {a*x*l y) = (x*a*l y) = (a*lxY).

- Now one has (Jq) 8) = (71 JR) for all q,R E L2(M), and consequently

so that Ja = a*. Thus the first computation shows also that JM'J C M' and, taking adjoints, M' c JMRJ.

By von Neurnann' s bicommutant theorem, one has M' = JMJ. (b) Let x E B(L~(M) @ K). Choose an orthonormal basis (q)iQ of K, and

repreah x by a matrix (3 .) over B(L~(M)). If x commutes to the aWm of M, ,I i,jEI

this matrix must have entries in EndM(L2(M)), and thus x E EndM(L2(M)) @ B(K).

Conversely any bounded matrix ("ij) with entries in E ~ ~ ~ ( L ~ ( M ) ) commutes with the

diagonal action of M. (c) Consider H @ ( L ~ ( M ) e K) as an M-module for the diagonal action

X(C @ ( q QD 8)) = x( @ (X q8 0). Then 0 @ 1 is an infinite projection in the commutant of M. By the Murray- von Neumann comparison theory for projections, there exists a partial

2 isometry in the commutant EndM(H@(L (M)@K)) from 1 @ 0 to a subprojection of

0 @ 1. One ~y view as an isometry

~ : H - + L ~ ( M ) ~ K

which intertwines the actions. #

As there will be many traces with various normalizatiob in the sequel, we introduce the following convention. If M is a finite factor, trM will denote its normalized t r m .

So if TI is any other trace on M, then TI = Tr(l)trM, a formula which we will use

often. Occasionally, we will have to consider a trace TI on a factor P which is not finite (for example B(H) or M@B(H), with H of infinite dimension). Let P+ denote the

positive cone of P, consisting of those element of the form z*z with z E P. Then a trace TI is a map P+ --, [O,m] such that

(i) Tr(x+y) = Tr(x) + Tr(y) x,y E P+

(ii) Tr(Ax) = XTr(x) X E R+, x E P+ (with 0.m = 0)

(iii) TI(-*) = Tr(x) x E P+, u a unitary in P.

Given a finite factor M acting in a Hilbert space H as in Lemma 3.2.2, we define now the paturd t r a o TIM' on its commutant. It is crucial for what follows that TIM,

is not necessarily normalized. 2 First, if H = L (M) as in (a), we define TrM, (JxJ) = trM(x) for all x E M; in this

ease, TIM, is notmalied. Secondly, if H = L ~ ( M ) @ K as in (b), consider an

i

orthonormal basis of K; then any element x , in the commutant

EndM(12(~) @ K) is represented by a matrix (J? .J)i,jEI; when x is moreover positive, 31

then the diagonal elements are also positive, and we define

. For example,

TrMt (JxJ@p) = trM(x) dimC(pK)

if x E M+ and if p E B(K) is a projection.

Let 3(K) denote the finite-rank operators on K. If x E JMJe?(K) c EndM(L2(M) @ K), that is if all but finitely many of the matrix entries x. are zero, but

1,j x is not necessarily positive, then TIM, (x) is well-defined by the same formula.

Furthermore, x I--+ TrM, (x) is a positive trace on the *-aigebra JMJ @ 3(K).

. Third, for H arbitrary and fm u as in (c) of Lemma 3.2.2, we define

for x E EndM(H)+, and thus -* E EndM(L2(~) @ K)+. If ul,% are two possible

ChoiC€!~ for U, then u;ul = ugUz = idH and urd% = for X M; as TIM,

is a trace,

and TIM, (x) does not depend on the choice of u.

The word "natural" is justified by the following property (which again shows the independence just observed).

Lemma 3.2.3. Let HI,H2 be two M-modules; let a : HI --I H2 and b : H2 -, H1 be

two M-linear bounded operators. Denote bg T. l e natural trace defined on EndM(Hj) J

as above, for j = 1,2. Then

Proof. Let u . H --, L ~ ( M ) @ K be an M-linear isometry. Then - j ' j

Definition. Let M be a finite factor and let H be a M-module. The ~ o u ~ l i n g

constant dimM@) is defined to be TrM,(idH), where the natural trace

TI : EndM(H)+ + [O,W] is defined as above. If u is as in 3.2.2.~~ one has also

dimM(H) = TrM, (uu*) by 3.2.3.

Proposition 3.2.4. Let M be a finite factor and let H,Hf ,H1,H2,. . . be M-modules

which are separable as Hilbert spaces. Then (a) dimM(H) = dimM(H1) if and only if H and H' are isomorphic as M-modules,

x.,

(b) dimM(?Hi) = x d i m M ( ~ i ) , 1 i 2 (c) dimM(L (M)) = 1,

(d) dimM(H) < m if and only ifthe factor EndM(H) is finite.

Proof. Claim (a) follows from the comparison theorem for projections in the factor. - E ~ ~ ~ ( L ~ ( M ) @ K), claim (b) fmm the o-additivity of the trace T r M on the same

I - factor, and (c) is obvious. 1

In all cases, EndM(H) is a semi-finite factor, and thus admits a non-zero trace which

is unique up to a multiplicative constant. Claim (d) holds because EndM(H) is finite if

and only if it has a finite trace. #

In the next proposition, we continue with properties of dimM. The deep result is ( f ) .

We now describe the main step, the proof of which is in [MvN I] and [MvN N] (see Theorem X in both papers). Again, let M be a finite factor and let H be a M-module; let t r be the normalized trace on M and let TI' be the natural trace on EndM(H).

Choose ( E H with t # 0. Denote by e the orthogonal projection of H onto the C closure of the cyclic module EndM(H)f, and by e' that onto observe that e € M t C and e' E EndM(H). The basic (and difficult) fact is that the ratio f \

tj 3.2. Coupling constant: definition 139

is independent of 5. (When M and H are finite dimensional, this basic fact reduces to Proposition 2.2.7.) Murray and von Neumann define the coupling constant of M and

EndM@) to be

cM = tr(e )/trl (el) - (Tr' ( l ) ) E M E R: C t -

if EndM@) is finite, with tr' the normalized trace on EndM(H) and TI' the natural

trace. In case EndM(H) is infinite, they define cM = +w.

The M-module K gives rise to othm modules as follows. Let e E B(H) be a projection (e # 0), with range denoted by eH. If e E EndM(H) then eH is naturally a

M-module (a submodule of H); if moreover EndM(H) is finite, the value D(e) of the

normalized trace of EndM(H) on e is called the dimension of e. On the other hand, if

e € M, then eH is a eMe-module; the algebra eMe is a finite factor (because it is simple, a fact easy to check) which is called the reduction of M by e. Following common practice, we also write Me for eMe.

Proposition 3.2.5. Let M be a finite factor and let H be a M-module. Assume that

the factor EndM@) is jnite (namely that dimM(H) < w). Then

(e) dimM(eH) = D(e) dimM(H) for any non-zero projection e E EndM@).

( f ) dimM(H) = cM, the coupling constant ofMurray and von Neumann.

1 (h) dimeMe(eH) = dimM(H) for any non-zero projection e € M, where

D(e) = tr(e). (i) I j L is a finite dimensional Hilbert space, then dimM@ @ L)

= &mM(H) d i q ( L ) .

Proof. For (e), one may view e as an M-linear isometry from eH to H. Then if - 2 u : H --, L (M) @ K is an M-linear isometry, we have by definition of dimM(.) and by

Lemma 3.2.3

dimM(eH) = TI (ueu*) E~~,(L~(M)sK)

I-U r-' VIldrpbGl aJ; r U U b G V U U IVGUl l ld l lU iLlgGUldii

where each TI, denotes a natural trace.

Next we show how ( f ) reduces to the result of Murray and von Neumann recalled above. Replacing H by an isomophic submodule of L2(M) @ K, we can assume H c L2(M) @ K. Let p E EndM(L2(M) e K) denote the orthogonal projection from

L2(M) @ K onto H. Then by definition c.

Let C E H with ( # 0 and let q E L2(M) @ K with q # 0. As earlier, denote by e E M C

and e' E EndM(H) the projectiom of H onto Endlyl(H)i and m. Likewise denote by C f R E M and f' E EndM(L2(M) @ K) the projections of L2(M) @ K onto

11

EndM(L2(M)@K)q and fi. With respect to the orthogonal decomposition L2(M) @ K =

H 4 HI, the algebra M acts by operators of the form 6 9, the algebra

EndM(L2(M) e K) is of the f o m :] q the space E ~ ~ ~ ( L ~ ( M ) e K)t is

\

of the form [TI. It follows that pft = eCp, or in matrix f o m that

fC = [:( : 1, so that it is the same element in M which acts as f on L2(M) @ K and C

as e on H. Consequently C

(3.2.5.2) trM(ft) = trM(et).

1

Observe also that, more simply

because c H. To compute E2 = trM(frl) + Tr (f;), we may choose q = 1 @ x with

E ~ ~ ~ ( L ~ ( M ) @ K )

1 E M c L2(M) and x # 0 in K. Then f is the identity on L2(M) @ K and f' is the 'I 5'

p"\a.x voup~lng consram: aen~llrlon k

2 projection onto L (M) @ Cn. Consequently

and .E2 = 1. But E2 can also be computed using ( E H, so one has

The coupling constant of Murray and von Neurnann for M and EndM(H) is

= trM(et)/trEndM(H)(ei)*

since we are assuming that EndM@) is finite. By uniqueness of the normalized trace on

EndM(H), one has

(3.2.5.5) trEnd H (PXP) = Tr d M( ) E . ~ ~ ( L ~ ( M ) ~ X ) E ~ ~ ~ ( L ~ ( M ) @ K )

(x) + Tr (PI

for any x E E ~ ~ ~ ( L ~ ( M ) @ K ) . Putting together (3.2.5.1) to (3.2.5.5) one obtains

= {tr (e ) 6 pEndM(L2(M)@Kfi)' TrEndM(L2fM)sK)

(PI

and claim ( f ) is proved. Claim (g) now follows trivially from (f). As for (h), using (e) and (g) as well as

EndeMe(eH) = e(EndM(B))e, we have

{dimeMe(e~)}-l = dim hdeMe(BA)(eH) = D(e) dimEnd H (HI M( )

= ~ ( e ) { d i m ~ ( ~ ) } - l .

Point (i) follows easily from the definition of dimM(.).

This ends the proof of Proposition 3.2.5. #

142 Chapter 3: Finite von Neumaq algebras I 9 4.4. c;oupnng constant: examples 14Y

If M = Mat (c) for some integer p 2 1, then dimM@) = 3 dimc(H) is of the I'

d form - with d, a integer as in Proposition 2.2.7. This follows for example from claims P

(b) and (c) of Proposition 3.2.4. The objectsf the next section is to describe examples involving factors of type III.

3.3. The coupling constant: examples.

The situation for which the coupling constant is computed in this section is of the following kind: G is a non-comp&t semi-simple connected real Lie group which has the same rank as its maximal compact subgroups, r : G --, U(H) is an. irreducible

representation of G in the discrete series, and M = w*(I') is the von Neumann algebra of an appropriate discrete subgroup I' of G. Then H is naturally an M-module. Theorem 3.3.2 below is a computation of dimM(H), due to AtiyWchmidt [AS,(3.3)].

First we discuss some background; the knowledgeable reader should jump to Theorem 3.3.2.

3.3.a. Discrete series,

Let G be a locally compact group. We assume that G is unimodular,.we choose a Haar measure dg on G, and we denote by lG : G u ( L ~ ( G , ~ ~ ) ) the left regular

representation of G. \ For an irreducible unitary representation r : G 4 U(H) of G, the following

properties are equivalent:

(i) r is a subrepresentation of XG; more precisely, there exists a projection p in

the commutant of XG(G) such that the restriction of XG to the range of p is equivalent

to Ir;

(ii) There exist b q E H - {O) such that g I+ <li(g)gl q> is in L ~ ( G , ~ ~ ) ;

(iii) For all S,q E H the function g I+ <n(g)( ( q> is in L ~ ( G , ~ ~ ) .

If these hold, r is said to belong to the (unitary) discrete seria. On may then attach to r a real number d r > 0, called its formal dimension, such that Schur's orthogonality

relations formally hold. In particular, for any r : G + U(H) in the discrete series

I The f o n d dimension dr depends on r and on the choice of the Haar measure for

I G; if d'g = kdg for some constant k > 0, the two corresponding formal dimensions of r are related by d; = k-ldr. If G is compact and if dg = 1, then d, is the dimension

of H in the naive sense. For a l l this, see section 16 in [Rbt] or Chapter 14 in [DC*]. Given an arbitrary (unimodular) group G, its discrete series may be empty. This

happens for G infinite abelian, or infinite discrete, or G = SL(2,C), or G = SL(n,lR) with n 1 3, to quote but a few examples. When G is a semi-simple connected real Lie group with maximal compact subgroup K, then G has discrete series representations if and only if G and K have the same rank. In particular SL(2,lR) has a discrete series, as well as ~0(n , l )O for n even.

3.3.b. Factors defined bv icc vrouns.

On a discrete group I', we consider always the counting measure; the space of square summable functions from I' to E is denoted by ?(I?). The von Neumann algebra w*(I') of I' is the (u1tra)weak closure of the linear span of +(I?) in B ( ~ ~ ( I ' ) ) ; by von

Neumann's theorem, it is also the bicommutant of Xr(r) in B ( ~ ~ ( I ' ) ) , and w*(I') is

thus also denoted by A#)".

Let 6, c.t2(r) be the function which takes the value 1 at the identity e of I' and 0

elsewhere. It is easy to check that x x(b,) is a linear injection of w*(I') in ?(I?),

and that the map tr(x) = ~ x ( 6 ~ ) (6,> is a normalized finite faithful trace on w*(T); see

the end of 4.2 in [Sak . It follows that the von Neumann algebra W*(I') is finite, and that 1 the Hilbert s p w L (W*(r),tr) defined before Lemma 3.2.2 is canonically isomorphic to

8 e2g). Moreover w*(I') is a factor (and thus a factor of type 111) if and only if I' is an

infinite coniueacy group, or for short an icc m o u ~ (Lemma 4.2.18 in [Sq). The following lemma exhibits a rich class of icc groups. Before this, we recall that the quotient G/I' of a unimodular locally compact group G by a discrete subgroup I' has always a Ginvariant measure, which is unique up to a scalar factor; by definition, I' is a &t&g in

I, G if the meaure of G/I' is finite.

Lemma 3.3.1. A lattice I' in a connected semi-simple r e d Lie group G vithout

center and without a compact factor is an icc group, and W*(I') is consequently a 111 - factor.

IYY ,y' ~ - Y cnaprer 5: r lnlre von lveumann ageoras r

Proof. The main point is Borel's density theorem, which we quote without proof (see - [Bar] or [Zim]): I' is Zariski-dense in G.

Consider h E I' and its conjugacy class Ch in I'. The m&p extends by 7- 7h7 rch

continuity to the Zariski closure If Ch is finite, then i7;;= Ch and

{g E Glgh = hg} is a closed subgroup of finite index in G. But the algebraic group corresponding to G is Zariski- connected, and it follows that {g E Glgh = hg) = G. Thus h is central in G, so that h = e. This shows that I' is an icc group. #

A final remark about this: let rl c and r2 c G2 be two examples of the

situation in the previous lemma. Assume moreover that G1 and G2 have real rank at

least two. It is a conjecture, due to A. Connes and "beyond Mostow and Margulis", that w*(I'~) is isomorphic to w*(r2) if andonly if rl and r2 are isomorphic.

3.3.c. w*(I')-modules associated to subre~resentations of A .

Let G be a unimohlar Lie group with Haar measure dg and let I' be a discrete subgroup of G. In the present context, it is convenient to define a fundamental hmaiq for r in G to be a subset D of G which is measurable and satisfies

7 , ~ n r 2 ~ has nu^ measure for 71.'y2 E r vfth 71 # 72 and

G \ U .ID has null measure. %I'

Such a D always exists. Indeed, as G -+ r \ G is a topological covering, it has a Borel section, and the image of such a Borel section is a convenient D. The measure of D d m not depend on D itself and is called the covolume of I'. (If dg is defined via a . differential form 0 of maximal degree on is a unique form w on I'\G which pulls back to 0, and the covolume of I' is

Of course, cbvol(I') does depend on the choice of the Haar measure on G. If d' g = kdg for some constant k > 0, the two corresponding covolumes of I' are related by COVO~' (r) = k COVO~(~) .

Given r c G and D as above, there is an isomorphism from L ~ ( G , ~ ~ ) onto t2(I') @ L2(Il,dg) which maps s to 61 @ q7, where 67 E t2(r) is the chvacKdstic

function of (7) in r , and where p (g) = 447g) for 7 E I', g E D. It follows from the 7

definitions of XG and Xr that the restriction A G l r to I' of the left regular

representation of G is the tensor product of Xr with the trivial representation of I' on

/+*.a. coup~ing constant: examples ,JAE,

L'(D,W ~ e n c e the von ~ ~ u m m n algebra XG(I')' is isomorphic to w*(r) o c g

w*(r). More generdy, let p E B ( L ~ ( G , ~ ~ ) ) be a projection which commutes with AG(I').

Denote by H the range of p, by r . I' -+ U(H ) the corresponding subrepresentation P P ' P of XGlr , and by r (I')' the von Neumann algebra generated by r (I?) in B(H ).

P P P

Then the ~*-mor~h.ism (b(I')' -r px r~( r ) ' ia obviously surjective. 1f I' is moreover

an icc group, then XG(I')' g w*(r) is a factor of type I I ~ and is in particular a simple

ring, so that the map XG(I')' t r (r)* is an isomorphism. P

We shall particularize below to the case in which the projection p commutes with all of XG(G), and definw an irreducible representation of G in the discrete series.

3.3.d The formula dimM(H) = covol(I')g,

Now the relevant background has been established, and we demonstrate the main result of this section.

Theorem 3.3.2. Let G be a connected semi-simple real Lie group with Haar measure 4

dg, let r be a discrete subgroup in I', k t M denote w*(I') and let r : G -+ U(H) be an irreducible representation in the discrete series. Assume that is an icc grou$. Then dimM(H) = covol(I')dr.

Observations. (1) Lemma 3.3.1 says that. I? is automatically an icc group in case it is

a lattice in a connected simple noncompact Lie group without center. (2) Both covol(I') and d, depend on dg, but these dependences cancel out in the

product.

Proof. From the discussion in 3.3.12, we may assume that H is included as an - 2 M-module in L (G,dg). This inclusion, say u, satisfies u*u = idH and uu* = p, where

p is the orthogonal projection from L2(~,dg) onto H. Also, L2(~,dg) may be identified with L 2 ( ~ ) @ K, where L 2 ( ~ ) is the monical M-module, and where K is the trivial M-module L2(~,dg) associated to some fundamental domain D of I' in G. Thus we have

dimM@) = TIM, (p);

in this proof, M' denotes the cornmutant of M in L2(~,dg) or in L 2 ( ~ ) @ K, and TrMt is the natural trace on M' .

146 ' Chapter 3: Finite von Neubim-algebras

By Lemma 3.2.2.b, 'this cornmutant M' is generated by finite sums of the form

X = C p 7 @ a r For each 7 E I?, the symbol p 7 stands for JXl'(dJ E E ~ ~ ~ ( L ~ ( M ) ) YE^ :

and a is a finite rank operator in B(K). Let (en)na be an orthonormal basis of K. 7

Let @ 5, denote the operator [ I+ (eml ()en on K. One may write

a = a7,m,n <@ en, where the a are complex numbers. By definition of 7 7,m,n

m,nEiN TrMt one has

where trM is the normalized trace on M and where TK is the trace on B(K)

normalized by T K G @ f m ) = 1 for all m E I. With x as above, one has consequently

2 Let q : L (G,dg) --, K be the orthogonal projection given by restricting functions from G to D, and let T denote the trace on B(L~(G,&)) taking value 1 on projections of rank one. Then TK(y) = T(qyq) for y E B(K)+ or y E ?(K). In particular, for x of

the form x = p @ a we have C 7 7 7 a

\

Finally any x E M I is the strong limit of an increasing net of operators of the form

x p 7 @ a+ .B the t r a m are normal, the formula (3.3.2.1) holds for all x E M;, and in YEr

The r i g h t a d term is explicitly gven by

Recall that 6 denotes the characteristic function of (7) in l', and that en, which 7

is a function on D, is also naturally a function on G (vanishing outside D). Thus the

3 3.3. Coupling cbnstant: ,examples 147

orthonormal basis (6 7 @e n } F r l n a of t2(l') @ K is more conveniently viewed as the

basis {AG(7) J%r,na of L ~ ( G , ~ ~ ) . Let rl be a unit vector in L2(G,dg); assume that

E H, namely that pq = q. For any g E G one has (writing X instead of XG)

Consequently, as p commutes with X(G):

By Schur' s relations

and the proof is complete. #

Corollary 3.3.3. In the situation of the p~ev iow theorem, I? is a lattice if and only i f i

the tommutant of M in B(H) is a finite factor.

Proof. The last condition holds if and only if covol(l') is finite. # -

We now particularize G to the group PSL(2,R). For each integer k 2 2, let Hk be

the .space of holomorphic functions on the Poincarb half-plane 'P which are s q w ~ u m m a b l e for the measure yk-2dxdy. (The open unit disc A in the complex plane with the corresponding measure is equally good). As G acts on 'P by fractional linear transformations, there is a natural unitary representation rk of G in Hk. It is a

standard result that Hk is an infinite dimensional Hilbert space and that rk is an

irreducible discrete series representation. (These rk consitute the holomorphic discrete I

I series, and the full discrete series contains a second "half", the anti-holomorphic part.)

I Define the Haar measure dg on G as follows: let T = S0(2)/{*1) be the maximal

g compact subgroup of G, such that {! induces a diffeomorphim G/T g 1 then

1 !

* *" "YY,,"". V. L * I Y Y W .VY *."UYIUYY U ~ W V I I V r- ,"*\

ddz) = ~ - ~ d x d y is a G-invariant measure on 'P, if cp is a continuous function G --t C with compact support, set f(z) = (cz+drP f [a], z E 7, [E 11 E SL(2,Z).

' JGdg)dg = @(z)Jydldgt) Z = b(i) The second one is a growth condition: observe that f(z) = f(z+l), so that f can be defined on the punctured unit disc A* by ~ ( e ~ ~ ~ ) = f(z); the second defining condition is that the Laurent expansion of f in A* is of the form ?(w) = x a n w n for w E A*.

where dt is the Haar measure on T of total measure 1.

Then the virtual dimension of rk is known to be given by dk = $$ see theorem ni 1 It is a result of Hecke that a cusp form f of weight p satisfies If(x+iy) 1 d B ~ ~ / ~ for all x + iy E 7 and for some constant B; see page 1.24 in [Ogg].

17.8 in [Rbt]. (Warning: under the Cayley transform Let M = w*(I'). Consider an integer k 1 2 and the M-module Hk of example

3.3.4. Given a cusp form f of weight p, the growth condition implies that f induces a multiplication operator Af : Hk 4 HkSp, defined by (Afcp)(z) = f(z)dz), which is

chosen in the present section is that which is defined by the Riemannian structure for bounded (in fact IIAfll 5 B with B as above). The invariance condition implies that Af

which 'P has constant curvature -1; the computation may be found, for example, in is M-linear. Consequently, given two cusp fonns f,g of weight p, the operator Section 5.10 of [Car].) t EndM(Hk). Let Tk denote the natural trace on

Now consider an integer q 2 3, set X = 2cos(lr/q) and let rX be the Hecke subgroup EndM(Hk). Then the space of cusp forms of weight p has a natural hermitian form

of PSL(2,lR) generated by the classes modulo *l of the matrices

[: : ] and [-: :I. (f 1 dk = T ~ ( ~ ; ~ ~ ) .

A computation in the same spirit as that presented in the proof of Theorem 3.3.2 shows Then cov01(l'~) = 41 -:) by the Gauss-Bonet formula, because has a triangular.

fundamental domain with angles 0,z,z (see Appendix 111). 9 q (f l dl, = +kmg(z ,~p2dxd~

Altogether, we have shown:

k 1 . with D a fundamental domain for I' in 'P. Up to a constant factor -;i-, this is known Example 3.3.4. Given integers q > 3 and k 2 2, consider the 111-factor

as the Peterson scalar product for cusp forms. M = w*(rX) defined by the Hecke group rA with X = 2cos(lr/q) and the Hilbert space

Hk of the holornorphic discrete series of PSL(2,lR). Then Hk is a M-module of coupling This suggests a natural project, which could be interesting for the study of cusp forms: evaluate ~ L n o r m s defined by constant

d 2 l/q. > Ilflkq = { ~ ~ l ( ~ ; ~ ~ )

The equaUty T~(A;A ) = Tk+ (A A*) should be useful. g P g f

We particularize further, and set q = 3 in example 3.3.4. That is, we consider 3.4. Index for subfactors of 111 factors. r = PSL(2,Z) as a discrete subgroup of PSL(2,lR).

Given an integer p 2 1, recall that a cusp form of weight p is (in this situation) a There were two main motivations for the introduction in [Jol] of the concept of index

holomorphic function f : 'P 4 C on the Poincar6 half-plane satisfying two conditions. The for subfactors. The first was that, if rl < I'2 are two icc discrete groups, the 111 factor first one is an invariance:

IDU uapte r s: rlIllte von NeUnUmU algebras .

N = A(rl)' acts in an obvious way on t2(r2) and dimN(t2(r2)) = [r2:r1].

Furthermore t2(r2) is the same as L ~ ( M ) where M is A(r2)'. This suggested the

following definition:

Definition 3.4.1, The index pf pubfactor N ef a fulite f w &$ is

This was the original definition of index; it was ahown in [J&] that this definition agrees with the ring-theoretic one which we have given in Chapter 2, when M and N are finite factors. The index C a n also be computed as IM:N] = dimN(H)/dimM(H), where H is

any M-module of finite dimension over M; see Proposition3.4.6. The second motivation was a result of M. Goldman ~Gol], who showed that, if N c M

are 111 factors (always with the same identity R) then, if %(L2(~) ) = 2, there is a

crossed product decomposition M = N r 2/22. Consequently if one defines [Ma] as above, Goldman's result is seen to be a beautiful analogue of the fact that a subgroup of index 2 of a group is normal.

It would also have been nice to have been able to call a subfactor N c M, normal when its (unitary) normalizer generates M. But unfortunately standard terminology reserves "normal" for subfactors N such that (N'nM)' n M = N, and the term a is . used for subfactors with the normalizer property described above. We take this opportunity to introduce some more terminology.

Definition 3.4.2. If N c M are factors we say that N is breducibl~ k N' fl M = C

It is not hard to see that a regular irreducible subfactor has integer index (or w which we shall treat as an integer) -see [Jo~]. A more refined analysis based on [,To61 shows that all regular subfactors have integer index. On the other hand dimM@) can be any

positive real number so the question naturally arose:

(a) What are the possible values of [M:N]? {b) What are the possible values of [MN] for an irreducible pair N c M ?

Question (a) was settled completely in [Jol] for M = R, the hyperfinite 111 factor.

Question (b) remains open even for M = R, and question (a) is open for arbitrary 111

factors M. We summarize the mast important known results as follows: . .,

3 3.4. Index for subfactors

Theorem 3.4.3. Let N be a subfactor of a 111 factor M.

(i) Either [M:N] = 4cos2r/q for some integ~r q 2 3, or [M:N] 2 4. (ii) If [M:NJ < 4, then N is automatically irreducible in M . (iii) There ezist subfactors of the hyperjinite 111 factor R with any of the indec values

allowed by (i). (iv) There are ezamples,of factors M for which the set of all possible values [M.N] is

countable.

Remarks: Statements ji) to (iii) are from [Jol]. We prove (i) below. A generalization to finite direct sums of 111 factors is shown in Corollary 3.7.6. A second proof of (i) occm

in Corollary 4.6.6, as a byproduct of the analysis of "derived towers". Statement (ii) is proved as Corollary 3.6.2(c). We will verify Oii) by giving 'several constructions of subfactors of R. The first

construction, in this section, works for all allowed index values. Another construction, valid for the index values 4cos2r/q is given in Theorem 4.4.2. A third construction, in

2 Section 4.5, produces irreducible pairs; the index values 4cos r/q are obtained once more, as well as sporadic values greater than 4. In Section 4.7.d, we give examples of hon-eonjugate irreducible subfactors of R of index 4. We would also like to mention the work of Wenzl [Wen2], in which a family of irreducible subfactors of R of index greater

r than 4 is produced by a construction involving the Hecke algebras Ha(q) for q .a

primitive root of unity. Statement (iv) is from [PP2], md will not be proved here.

2 For arbitrary 111 factors, the question of existence of subfactors of index 4 eos rJq

remains open, more precisely we know of no example of a full 111 factor M having a 2

,J subfactor of index 4 cos a/q, q # 3,4,6. (A 111 factor is called "full" if the group of inner

automorphisms is closed in the topology of pointaise strong convergence in the whole automorphism group - an example of such a factor is X(PSLf2,2))'.)

Proof of 3.43 {i). As for finitk dimensional algebras (2.6.2) there is always a (unique) faithful tracepreserving conditional expectation from M onto N, which, viewed as an

2 operator on L (M) is the orthogonal projection % onto L 2 ( ~ ) . The fundamental

construction again yields a 111-factor

end;(~~gn,tr)) = E~~;{MI = (M,eN).

(See Theorem 3.4.6 below for the first equality,) We claim that the norxdized t r w of <M, eN> has the Markov property

[M:wtr(eNx) = tr(x) for all x E M.

Indeed, the linear fprm defined on N by x w tr(eNx) is a trace (3.6.l.iii). As 1 = 2

tr(eN) [M:N] by Proposition 3.2.5.e applied to the N-module L (M), the property is

valid for x E N, by uniqueness of the normalized trace on N. But then for x E M, we have [M:N]tr(eNx) = [M:N]tr(eNxeN) = [M:N]tr(eNEN(x)) = tr(EN(x)) = tr(x), wing

3.6.1.i . Now the tower construction of Chapter 2 works and yields an increasing sequence of

111-factors

M o = N c M 1 = M c . - - c M k ~ M k + l C ; . . ,

and a sequence of self-adjoint projections (ei)i2 satisfying

(3.4.3.1) P 9eiklei = ei

e.e - e.e. if 1 i-j 1 2 2, I j - J 1

with ,fj = [M:N]. Claim (i) now follows from Theorem 11-16. An alternative proof wing the trace goes as follows: The trace tr on UMk has the

k

Markov property

(3.4.3.2) p tr(we.) = tr(w) for j > 1 and w E alg {l,el,, . .,ej-l). \ J

2 where f l = [M:N]. Now suppase that P < 4 but /.? $ (4 cos r /q : q 1 3). Using 2.8.5 and 2.8.7 (note that the number P is generic) as well as the relations 3.4.3.1 and 3.4.3.2, we obtain for each k 2 1 a trace preserving isomorphism of the algebra Bp,k of Section 2.8

onto the algebra Ck = {l,el,-. .,ek-l)'. By 2.8.4(vii), for each k the trace of the k minimal central projection Qo (necessarily a self-adjoint projection in Ck) is, ~ ~ ( 0 - l ) .

But by 2.8.3(iii), if 4 cos2(r/k) < /3 < 4 cos2(r/k+l), then pk(P-l) < 0, contradicting 2

the positivity of the trace. It follows that if P c 4, then P E (4 cos r/q : q z 3). #

2 Proof of 3.4.3(a), Fix ,!3 E I with P = 4cos r/q for some integer q > 3, or P ) 4.

Consider a sequence of self-adjoint projections (ei)i21 on a Hilbert space, together with a

faithful normal tracial state tr on R = {I,el,e2,. . .)' satisfying the relations 3.4.1.1 as

well as the Markov property 3.4.3.2. First we must recall how such a sequence of projections and such a trace can be

2 constructed. In 2.8.4 (in case P > 4) and in Section 2.9 (in case P = 4cos Ir/q for some

q) we have constructed an increasing sequence of finite dimensional c*-algebras (BP,k)k21, with BPSk generated by its identity and self-adjoint projections el,+ + *ek-l

satisfying the relations 3.4.3.1, and a positive faithful normalized trace t r on B P,k

satisfying the relation 3.4.3.2 for 1 5 j 5 k. Since tr is faithful, the trace representation rtr is faithful as well, and we can take R to be u ~ ~ ( u B ~ , ~ ) ' .

A simpler procedure is available when P is the square of the norm of a non-negative integer valued matrix (i.e. @ E ,&(IN)). In this case there is a connected pair of finite dimens& c*-algebras I3 c A with [A:B] = P, and the tower construction for this pair yields a sequence of projections (ei)i,l satisfying 3.4.3.1, and a positive faithful trace on

alg {&el,. , .) satisfying 3.4.3.2. Cf. 2.7.5 and the discussion at the end of Appendix IIa.

Lemma 3.4.4. [Jol] With the notatioa above, R i s the hyperfinite 111 factar.

llEgPfr It is clear that R is a finite, hyperfinite von Neumann algebra. We claim that if z is in the center of R, then

t tr(uc) = tr(z)tr(x) for all x x~ R .

r It will follow from this and the faithfulness of tr that z = tr(z)l, so R is a factor. For each k, let Ck = alg {&el,. . ~ e ~ - ~ ) . By 2.9.6(e) (is case 0 < 4) or by 2.8.7(a)

and 2.8.5(b) (in case P 2 4), the map c w e. (on the generators { e . ) of A ) induces a J J J Ak trace preserving isomorphism of B onto Ck. It then follows from 2.9.6(g) (for /3 < 4)

B,k or from 2.8.50 (for P > 4) that e. w ekVj extends to an inner automorphism of Ck, and

3 :' hence to an inner automorphism ak of R

Note that tr has the multiplicative property tr(y1y2) = tr(yl)tr(y2) whenever

y1 E Cs and y2 E alg {l,es,. . .eS+,]. (One can veriQ this directly or use the

isomorphism Cm 2 Bp,, together with 2.8.5(e) or 2.9.60.)

It will suffice to verify the relation tr(zx) = tr(z)tr(x) when x E Ck for some k. Let

c > 0, and choose y E CL for some L, such that 11~-y11~ z a Then tr(yak+4x)) =

tr(~)tr(x), since %+e (x) E alg {P,ee+l,. -ee+k-l). Co~~equently,

I tr(zx) - tr(z)tr(x) 1 = I tr(zak+e (XI) - tr(z)tr(x) I (since ak+e is inner)

5 ltr((z - ~ ) a ~ + ~ (XI) I + l t r ( ~ 4 + ~ ( 4 ) - tr(z)tr(x) l

Chapter 3: Finite von Neumanu algebras

= ltr((z - Y)%+! (XI) I + I (tr(y) - tr(z))tr(x) l 5 2 6 llxl12.

Since c is arbitrari, this finishes the proof. #

Lemma 3.4.5, [Jol] Set R - {P,e2,e3,. . . )' . Then [R:RP] = P. P-

We know (by 2.8.5 and 2.8.7 or by 2.9.6) that for each k > 2 , the

relation P tr(elx) = tr(x) holds when x E alg {P,e2,, , .ek), and, taking limits, we have

the same relation also for x E Rg Therefore ER (el) = 8'1. P

Similarly EN(eZ) = B11, where N = {l,e3,e4,-. :I*. For k ? 3, any

x E alg {l,e2,. . .ek} is of the form x = a + bigci, with a,bilci E dg { I q , . .ek}. i

Consequently, EN(x) = s + f l C b i c i and elxel = EN(x)el. Taking limits again, we i

have

(*) elxel = EN(x)el for all x E RP.

One next' verifies that xel = fi ERJxel)el, for all x E R, by first checking this for P

x E alg {l,el,- .q) (that is, for x of the form x = r + xbielci , \with a,bi,ci E

i

alg {keg,. . .ek)) and then by taking limits. Consequently , ,bl = RP el, and RelR

=RP elRP. Observe also that R = RelR, because finite factors are algebraically simple

([DvN], Cor. 111.5.3). 2 2 Let e be the orthogonal projection of L (R) onto L (Rd . One has exe = ER (x)e

D for all x E R, by 3.6.1.i. below, so that in particular, eele = T1e. We claim that also

eleel= riel. Sine! R = R e R it suffica to check this equality on vectors xelyfi, P 1 P ' where x,y E R and fl is the trace vector for R But P

eleel(xel~n) = eleelEN(x)~n (by (*)I = e1ER$elEN(x)y)fi (by definitions of e and E

= e1ER$4)EN(x)y" (by R p e a r i t y of ER. ) P

= 8' elEN(x)yfi

(by (*I).

$3.4. Index for subfactors 155

It follows from the relations eele = T1e and eleel= /Tiel that e and el are

equivale t projections in (R,e). Since e is finite in (R,e) by 3.6.l(v), the projection el b is finite in (Ke). But 1 is the sum of finitely many projections each equivalent in R to a subpr~jection of el, so (R,e) is finite. Hence [RRP] = tr(e)" = tr(e1)" = 8.

This completes the proof of the lemma, and also of 3.4.3(iii). #

It is tempting to guess that the pair R 3 R is irreducible, also for P > 4, since on a P purely algebraic level it is easy to see that there is no element of the algebra generated by e l } which commutes with {e2,e3,. ' v } . V. Jones confesses to spending

considerable &fort to prove this, but it turned out that RP has non-trivial relative

commutant in R when p, 4. A laborious proof of this non-obvious fact was given in ' [Jol] and a simpler proof in [PPl]; we will give a proof due to Popa in 4.7.5. The

difficulty is that m e cannot write down an explicit form for an element in RP' tl R

without invoking a beautiful representation of {el,e2,. '1' discovered by Pimsner and

Popa: We have seen that one way to obtain a sequence of projections (ei)iL1 satisfying the

' relations 3.4.3.1 is to form the tower from an indecomposable pair B c A of finite dimensional c*-algebras. Then, as we have observed in Chapter 2, the restrictions on index are related to restrictions on the type of inclusions B c A which yield a modulus p < 4. This is where the Coxeter graphs of types A, D, and E enter the picture. But to meate the sequence (ei)i21 one can also use a pair N c M of finite direct sums of

111-factors. In the following sections we will see how, if one allows this extra freedom, the

remaining Coxeter graphs appear!

We finish this section by recording one useful fact on indev of subfactors from [Jol].

Pronosition 3.4.6. Let N c M be and let H be aay M-module such that

dimM(H) is finite. Then [M:N] = (,)i (In paPtimlar, dimN(H) 1 dimM@).) m~

EreQfc If HI and Ha are any two M-modules such that dimM(Hi) is finite for

i = 1,2, then there is a finite dimensional Hilbert space K and an M-invariant : projection q such that H1gq(H2@K) as M-modules. Then

dimN(H1) < dimN(H2 @ K) = dimN(H2) dim@), by 3.2.5(i), so dirnN(H1) is finite if

and only if dimN(H2) is. In particular, [M:N] is finite if and only if dim@) is.

1

156 p~ 'p ' 'p ' vnapcer 3: rmce von ne- iugeulaa R--% 3 J.O. UCIWIOIIS 01 nmte von lveumann agebras --(57 t I

Assuming that [M:N] is finite and choosing an M-module isomorphism H 2 q ( L 2 ( ~ ) @ K), as above, we have

dimN(H) = d i m N ( q ( ~ 2 ( ~ ) @ K))

= trNJ (q) d i m N ( ~ 2 ( ~ ) @ K) (by 3.2.5(e))

= tr,. (q) d i q ( K ) dimN(L2(M)) (by 3.2.5(i)),

while dimM(H) = trM, (q) d i ~ ( K ) . #

c..

3.5. Inclusi0118 of finite von Neumann algebras with finite dimensional centers.

We saw in Chapter 2 that a unital inclusion B c A of finite dimensional c*-algebras can be specified by the inclusion matrix A E Matfin@) and a vector d E ~i~ for some n,

specifying the algebra B up to isomorphism. It is impossible to specify an inclusion so precisely in the 111-case since, for example, it is possible to find infinitely many

nonanjugate subfactors of index 4 in R, even irreducible ones, as we shall see in Chapter 4. What we will do is specify enough information to be able to calculate all the needed coupling constants, which will enable us to find the Markov traces as in Section 2.7.

The situation will -differ in two ways from the finite dimensional case. The first is that there are no minimal projections around, so integers do not appear inThis way. The second is that the subfactors can have indices different from squares of integers. This extra freedom allows the appearance of new Coxeter graphs.

m First some notation. Let M = @ Mi be a direct sum of finite factors with

i=l corresponding minimal central projections pl,, . . ,pm. Since the trace on a finite factor iB

unique up to a scalar multiple, a trace on M is completely specified by a row vector !- s = (slle . . ,sm), with si = tr(pi), ( Warning: This is not the same vector which was used

in Chapter 2 to specify a trace on a direct sum of finite dimensional factors; there we used th the vector whose i component is the trace of a minimal projection in Mi.) A trace is

positive (i.e., trace (a*a) 2 0) if and only g has non-negative components. We adopt t& convention that ('trace" means 'Ipositive trace". A trace is faithful (i.e., trace (a*a) = 0

rn

implies a = 0) if none of the components of 5 are zero, and normalized .if si = 1. A 1= T.

trace is automatically n o d , i.e., if {fi} is a family of mutually orthogonal projections, m

m then trace ( V fi) = ztrace(fi).

i=l i=l

Recall that if P is a finite factor, trp denotes its unique normalized trace, and if TI

is any other trace on P, then Tr = Tr(l)trp. n

Let N = @ N. be another direct sum of finite factors, contained in M and having j=1 J

the same identity. Let ql19 . . ,qn be the minimal central projections of N.

M Definition 3.5.1. If N c M are as above, we d&ne the m-by-n matrix T N = (cij)

by

4~ = trPiM(piqj).

Proposition 3.5.2.

(i) The nzatriz T: k row-stochastic; i.e., c. . 2 0 and 1,J

J

(ii) If k apecipes a trace on MI then 5 T! spwijes its ndriction to N.

(iii) If N c M C L are fnite direct sums of fnite factors, then Tfi = T&T!.

(ii) A s ~ ~ ~ = l , i

L (iii) Let {rk} denote the minimal central projections of L, so that TN is the

matrix whose (kj) entry is tr, L(rkqj). Since q - p.q., one has 1 k i-C i 1 1

But in the finite factor A = rkL, if e s f are two projections, then

trA(e) = trA(f)trfAf(e). Thus

158 . Chapter 3: Finite von Neumann algebras 3 3.5. Lnclusions of finite von Neumam algebras 159

If rkpi # 0, then. x w trr p.G. p.(rkx) is." trace on piM whose value at pi is 1, so in k l k l

fact tr Lr p.(rkx) = trpiM(x). Hence 'kpi k I

. . . L

as desired. #

A second piece of data needed is the matrix of indices of the "partial embeddingso'. Note that Nij = Np.q. = {p.q.x : x E N) is a fidite factor, a subfactor of

1 1 1 J M. . = p.q.Mpiqj.

l,J 1 J

M Definition 3.5.3. (i) With notation as above, d d n e an m-by-n matrix AN with

entries A i j = [M. . : N~,]'/~.

llJ

(We note that this expression is the same as in the finite dimensional case. Observe that in M M the finite dimensional case AN determines T N , namely

where pi M g Mat (C), and q.N g Mat,(C).) '5 J J

' (ii) The inclusion N c M is called connected if Z(M) n Z(N) = €1. This is true if and only if A! is indemmpossible.

(iii) A representation . lr of M on a Hilbert space H is called a &i&

re~resentation ef & Q& N c M if 4N)' is a finite von Neumam algebra. (iv) We say that N is of finite index in M if N c M admits a finite faithful

representation. (Note that parts (ii), (iii), and (iv) make sense for vbitrary pairs of finite von

Neumann algebras -not necessarily with finite dimensional centers.)

Lemma 3.5.4. Suppose N c M are finite direct s u m of finite factors. The following . are equivalent:

(i) N is of finite index in M.

(ii) The matriz A! has only finite entries.

(iii) For any faithfil trace tr on M , the regular representation of M on L 2 ( ~ , t r ) is b finite representation of the pair N c M.

(iv) For any faithjkl representation {lr, 3) of M such that 4M)' is finite, the algebra r(N)' is also finite.

Proof. (iv) E=, (iii) * (i) is evident. - (i) 4 (ii). If T is a faithful finite representation of the pair N c M on H, then the

oommutant of lr(N. .) on x(p.q.)H is ir(p.q.)lr(N)'~(p.q.), which is finite. It follows I rJ 1 J 1 J 1 J

that dimN. .(lr(p.q.)H) < m (Proposition 3.2.4.d), and 1,J

1 J

(by 3.4.6.), which is finite. (ii) + (iv). Consider a faithful M-module H for 'which M' is finite. Since

I = z p i q j , to show that N' is finite, it suffices to show that each p.q. is a finite 1 J

i , ~ projection in N' (because a sum of finite projections is finite.) If pq. # 0, then

1 J p.q.N1p.q. is the commutant of N. on p.q.H. By 3.4.6 and 3.2.5.h,

1~ 1~ 1,j 1~

Since M' is finite on H, so is piM'=(piM)' on piH, so by 3.2.4.d,

dim (pH) < m. Hence also dimN. .(3qjH) < m, and by 3.2.4.d again, (N. .)' is piM 1

1 J 1,J

I finite. # I

1 Observe that the analogue for A of Proposition 3.5.3.iii does not hold. For example, 1 let R be the hyperfinite 111 factor, let p be a non-trivial projection in R, let rp be an Y k isomorphism from R to R and set 1 P 1-P'

!

N = { ~ ' E R : y = x + rp(x)forsomex€R ),and P

M = R @ RIep. P

Then

1 . I

A ~ A $ = (1 I)[:] = 2,

and

!

by Corollary 2.2.5 of [Jol] or 4.7.2. These are not equal, unless tr(p) = 112. Of course, if N c M c L is a triple of finite m, then [L:N] = [L:M] [MN] by

Proposition 3.4.6.

If N and M are as in 3.5.4, and the inclusion N c M is connected, then all factors of N and M are of type 111, or d i v ( M ) < CO. It is also known that all factors of N and M

share (or do not share) the property of being hyperfinite (Lemma 2.1.8 in [Jol]) or the property T (see [Ana] and [PP2]).

If r is a finite faithful representation of the pair N c M on H, then the centers of

r(M) ' r(;)'],are the same as those of M and N respectively, and the rows and M

columns of A, , are naturally indexed by the columns and rows of AN. The

generalization of Proposition 2.3.5 to this setting is the following.

Lemma 3.5.5. Let N C M be a pair of finite direct sums of finite factors, as above, as

suppose r is a faith'fil finite representation of the pair. Then

Proof. If M and N factors, the equality holds because [x(N)?n(M)'] = [MN] - by Propositions 3.4.6 and 3.2.5.g. To extend the equality to the general case, one proceeds exactly as in the finite dimensional case (Proposition 2.3.5), with Proposition 2.2.5b being replaced by [DvN], Proposition 1 of $1.2, which says: if Q is a von Neumann algebra on H and p is a projection in Q or in Q' , then Endpgp(pH)) equals pEnd Q (H)p. #

Also note that r(M)' is of finite index in 4N) ' by Lemmas 3.5.4. and 3.5.5.

Proposition 3.5.6. Given an irredundant m-by-n m a t h A over

(0) U (2 cos r/q : q 2 3) U [~,co], and an m-by-n row stochastic mat& T having the same pattern of zero entries as A, there ezists a pair N C M (both hyperfnite) with A! = A and T! = T.

Proof. Take M to be the direct.sum of m copies of R, the unique hypednite 111

factor, denoted %. In each I$, choose a partition of unity {g. . : l 5 j 5 n} with I J

t ~ ( q ~ , ~ ) = (T)i,i If (T)i,j is nonzero choose a 111 'subfa~tor PiJ of R,,j = qi,jTqij

,+-,a%, 9 a.0. l n e ~ u n a a m e n t ~ construction +-=-yl

with [qlj : P. .]'I2 = (A). . (possible by [Jol], Theorem 4.3.2). For each i and j such 1 J 1,J

that (T). . # 0, choose an isomorphism 0. . : R -+ P. (possible since all the factors are 1 , ~ I J 1,j

n 111 and hyperfinite]. Set q. = z q . ., put Nj = {XdiJ(x) : x E R}, and N = e Nj.

i I J

i j=l Then q.N = N., and N is the required subalgebra. # J J

3.6. The fundamental mnetructioa

The discussion of the fundamental construction in Chapter 2 was purely ring theoretic. In the von Neumann algebra framework, where the preferred modules are Hilbert spaces, it is natural to make a construction which, apparently, depends on the choice of a trace on M. We begin by showing that in fact the ring theoretic construction is exactly the same.

First we recall some notions from [Jol] which work for arbitrary finite von Neumann algebras exactly as for factors. Let N c M be finite von Neumann algebras with the same identity. Given a faithful normalized trace on M, there is a unique faithful normal conditional expectation EN : M -, N determined by tr(xy) = tr(EN(x)y) for x E M and

y E N. In fact EN is .the restriction to M of the orthogonal projection

2 % : ~ ~ ( ~ , t r ) 3 ~ ~ ( ~ , t r ) . We denote by (M,eN) the von Neumann algebra on L (M,tr) I

generated by M and %. We let J denote the conjugate linear isometry of ~ ~ ( ~ , t r ) extending the map

x w x* on M.

Proposition 3.6.1.

(i) eNxeN = EN(x)eN for X E M

(ii) JeNJ = eN

(iii) For x E M, x commutes with eN if and only if x x N.

(iv) (M,eN) = JN'J

N ---, (M , eN) (v) The map $J is an injective morphism onto eN(M,eN)eN.

Y - Y ~ N

(vi) The central support of eN in (M,eN) is 1.

(vii) The space MeNM, which denotes the linear span of {x' eNx' : x' ,x' E M}, is

a strongly dense *-subalgebra of (M,eN).

M (cf. [Jol]).

(i) It suffices to check that EN(xEN(y)) = EN(x)EN(y), but this follows from

the N-linearity of EN.

10s Ghapter a: Finite von Neumann algebras 9 3.6. 'me tundamental construction 163

(ii) Follows from ~ ~ ( x * ) = E~(x)* .

(iii) Note that x commutes with eN if and only if left multiplication by x

commutes with EN.' This is clearly so for x E N. On the other hand, if x E M and x

commutes with EN, then x = xEN(B) = EN(x) E N.

(iv) By (iii) N = M n {eN)', so N' = (MIU{eN})' = (M' ,eN). But JM' J = M

and JeNJ = eN, so JN' J = (M,eN).

(v) By (i), the indicated map is an epimorphism. Let R denote the canonical 2 trace vector in .pL (M,tr). If yeN = 0, then yeNn = yR = 0 and y = 0 because Sl is

separating, so g(r is an isomorphism. (vi) Let z be the central support of eN in N'. Thep z E N n N' and

$(z-I) = zeN - % = 0, by definition of a central support, so z = 1 by (v). Now (vi)

follows from (iv) and (ii).

(vii) First note that by (i), the set

n x = {xo + CxieNYi : n E N, x-y.

i=l 1 l E M )

is a *-subalgebra of (M,eN) containing M and eN, so the strong closure of X is

(M,eN). If \

Y = { c x i e N y i : 5,yi E M},

then Y is a two sided ideal in X, so by the Kaplansky density theorem and the joint strong continuity of muliplication on the unit ball, the strong closure Y of Y is a two sided ideal in (M,%). But Y contains the central support of eN, which is 1 by point

(vi), SO Y = (M,eN). #

We now specialize to the case where N and M are direct sums of finitely many 111

factors with minimal central projections {qj; j = 1,. . .,n) and {pi; i = 1,. . .,m)

respectively. By the equality (iv) above, <M,eN> is also a finite direct sum of 111

factors, with minimal central projections {Jq.J: j=l,-. .,n). J

Lemma 3.6.2.

(a) If N C M are type 111 von Neurnann algebras with finite dimensional centers and

N is of finite index in M, then dimC(N1 nM) < m.

(b) If N c M are 111 factors, then dimC(N1nM) 5 [M:N].

(c) If N c M are 111 factors with [M:N] < 4, then N' fl M = C1.

Proof. We first consider the case that N and M are factors. Let H= L ~ ( M ) and - write TrN, for the natural trace on EndN(H). If f is a projection in N' n M, then

TrNt(f) = dimN@) (by definition of dimN)

L dimfMf(fH) (by 3.4.6)

= trM(f)-l (by 3.2.5(h))

2 1

Suppose N' fl M contains k rnutuall~ orthogonal projections f l , . . , fk with z f i = 1.

Then

4

In particular, if N' n M #El. then [M:N] 2 4, and if N' n M is infinite dimensional, then [M:N] = m. Suppose [M:N] < a, and let fl,. . . ,fk be a mazinaal family of mutually

orthogonal projections in N' n M; then [MN] 2 k2 L dimC(N1nM). This proves all the

assertions in the case of factors. Now return to the situation where N and M are finite direct sums of finite factors. The

projections p.q. are central projections in N' n M and p.q.(N' nM) = N;.q. n Mp.q.. SO 1 J 1 J

1 J 1 J

if di-(N1nM) = co the^ must be a pair (ij) for which dim~(N;.~. n Mp.q.) = m. But 1 J 1 J

this contradicts the observation just made for the case of factors, and completes the proof

of (a). #

The next results (3.6.3-3.6.5) depend on ideas of Pimsner and Popa [PPl].

Lemma 3.6.3. Let N C M be finite direct s u m of type 111 factors with N of finite

indez in M, and let tr be a faithhl trace on M. If x E (M,eN), there is a unique y E M

for which xeN = yeN.

Proof. Regard N c M represented on L~(M) . - Let us first check uniqueness. Suppose y,yl E M with xeN = y% = yfeN. If R is

the trace vector in. L 2 ( ~ ) , then

so y' = y because R is separating. To prove existence, we have to show that (M,eN)eN = MeN and we proceed as

follows. As N' is finite, (M,eN) is finite by 3.6.l . i~~ and there exists a faithful normal

conditional expectation F from (M,eN) onto M (see Propositiop-II.6 for the proof of

this latter fact). We claim that F(eN) is invertible in M. Since F is an

M-M-bimodule map, F(eN) belongs to N' n M, which is finite dimensional by Lemma

3.6.2. Consequently, to show that the self-adjoint element F(eN) is invertible, it is

enough to check that xF(eN)x # 0 for any positive element x # 0 in N' n M. But if

then xeNx = 0, since F is faithful. And xeNx = (eNx)*(eNx), so .eNx = 0. .Hence

which implies x = 0 by 3.6.l.v and the faithfulness of EN. This proves the claim that

F(eN) is invertible.

Now we may obtain a formula for xeN. Suppose first that x is in MeNM, namely

that x is a finite sum x a j % b j with a.,b. E M. Then F(x%) = x a . ~ (b.)F(eN) and J J I N J

This formula holds for any x E (M,eN) because both sides are strongly continuous in x

and because M%M is strongly dense in (M,eN) by Proposition 3.6.l.vii. Thus

Theorem 3.6.4. Let N C M be type 111 uon Neumann algebras with finite dimensional

centers and k t t r be a jaithjil normal trace on M for which N' is finite on L2(~ , t r ) .

Then

(i) As a right module over N, the algebra M is projective of finite type.

(ii) The conditional ezpectation EN : M -+ N is very faithful (in the sense of

Section 2.6). n

(iii) (M,eN) = MeNM := { z a j % b j : n 2 1, a b E MI. j=1

j, j

(iv) If a : M -+ M is a right N-module map, then a extends uniquely to an element o f (M,eN) = JN' J on L~(MP).

(v) If x E JN'J then x(M) c M, where M is vaewed as a dense subspace of ~ ~ ( ~ , t r ) .

Proof. (i) Any strongly closed right ideal in N is projective of finite type, and in fact - of the form pN with p a projection in N. (See [Tak], 11.3.12.) We are going to show that M is isomorphic, as a right N-module, to a finite direct sum of such ideals. In the course of doing so we exhibit a basis {vi : 1 s i s n} of M over N with the following

properties: (a) EN(vivj) = 0 if i # j.

- (b) fi := E ~ ( Y ; V ~ ) is a projection in N, v h = vi, and E~(v;x) = fi~N(v;x)l for

l d i ~ n and X E M . (c) Every x in M has a unique expansion

x = x v i y i , with yi E N.

* In fact viyi = viEN(vix).

Since the centrd support of eN in (M,eN) is 1 and since (M,eN) is finite with

finite dimensional center by 3.6.1(iv), there exists a finite set wl,. . .,wn of partial *

isometries in (M,eN) with W.W. l eN and z w j w ; = 1; in particular the w. have J J J

mutually orthogonal range projections. (See [Tak], V.1.34.) As wjeN = wj, there are, by

3.6.3, elements vl,- .,vn E M with w. = v e for all j. We verify that the vi have the J j N

properties (a)-(c). For i # j

* , so EN(v.v.) = o by 3.6.l(v). Sirnilarb, since wiwi is a projection in (M,eN) and I * I J * * i wiwi = EN(vivi)eN, 3.6.l(v) implies that fi := EN(vivi) is a projection in N.

1 9 3.6. 'me Wdarnental construction 167

Furthermore * * vitieN = vieNvivieN = W.W. W.

1 1 1

* = w . = v e I i N7

so that vifi = vi, by the uniqueness statement of 3.6.3. Therefore, since fi E N,

- E v x for X E M . fi~N(v;x) = ~ ~ ( f ~ v ; x ) - N( )

For any x E M, * xeN=CwjwjxeN = zvj%v;xeN

j j 2

= E V j N E (v*x)eN, j

j

and hence x = v.E (v.x), by 3.6.3. To show uniqueness of the expansion, suppose ~ J N ; J

that x =ZviYi with yi E N. Then

using N-linearity of EN and properties (a) and (b) of {vi}. We will refer to a family

{vi} having properties (a)-(c) as a Pimsner-Popa basis of M over N; see [PPl].

Now consider the N-linear map

M - 4 @ f.N 1s jsn

It follows from the expansion x = that Q is injective. On the other hand,

if (y$ E @ f.N and x =XV.~. then by the uniqueness of the expansion, j ', J J *

v.y. = v-E (v.x) for all j. Multiplying both sides on the left by v and applying EN J J J N J * j

gves f.y. = f.E (v.x); since both y. and EN(v;x) are in f.N, that is y. = EN(v;x). J J J N J J J J

Thus (y.) = Q(x) and V is surjective. J

(ii) Let o : M -+ N be a right N-linear map and set a =za(vj)v;. b a l l from

j Section 2.6 that E;(a) : M -, N is defined by E:(a)(x) = EN(ax) for x E M. We have

by N-linearity of EN

= Ek(a)(x),

so that o = Ek(a).

(iii) It follows from 3.6.3 that M%M is a two-sided ideal in (M,eN). But MeNM

mnttbh z v j e N < = ~ W I * J j - - I, so h e N ~ = (M,eN).

j j (iv) If a : M -4 M is right N-linear, , then for X E M ,

4x1 = a ~ v ~ E ~ ( v ; x ) = Za(vj)EN({x); thus o = ZA(a(vj))oENoA(v;), where

J j j X(y) denotes left multiplication by y. The unique Il.llz-cantinuous extension of o to

4 ~ 2 ( ~ , t r ) is z.(Vj)eNV; E (M,eN).

J

(v) Any x E (M,eN) is of the form x a j e N b j by claim (iii). If y E M then

j

Corollary 3.6.5. Let N c M be a pair of von Neurnanla algebras of type 111 having

finite dimensional centers, and suppose that N is of finite index in M . Let tr be any faithjul nonnal trace on M and define eN and EN via tr. Then

M aN M g(M,eN) as N-birnodules, and

~ n d i ( ~ ) (M,eN) as C-algebras,

Proof. Since N C M has finite index, (M,eN) is finite. The isomorphism - End;(~) 1 (M,eN) follows from 3.6.4(iv) or (v); the correspondence is defined by

Zqa$~~A(b~cbj , -zajeNbj.

The isomorphism M eN M 1 E n d i ( ~ ) extending the map a % b - A(a)ENX(b) on '

elementary tensors follows from 3.6.4(i) and (ii) and 2.6.3. One can also verify directly the

100 P""" bnapter a: r 1111ae von lveumann ageoras

isomorphism M eN M 2 (M,eN) by using a Pimsner-Popa basis. #

The next proposition determines one part of the spatial data for the inclusion

M c (M,eN).

Proposition 3.6.6. Let N c M be finite direct sums offinite factors such that N is of

( w e N ) M t jnite index in M, and let tr be any faith@ trace on M. Then AM = (AN) .

Proof. This follows from 3.5.4, 3.5.5, and the formulas JN'J = (M,eN), - JM'J = M. #

To describe M c (M,eN) more precisely, we also have to compute the matrix of traces

T F y e N ) . This is the part of the theory which differs most from the finite dimensional

case presented in Chapter 2. Before proceeding, we summarize our notation: N c M is a pair of finite von Neuman

algebras with finite dimensional centers, with N of finite index in M; the minimal central projections in M and N are respectively {pi : 1 < i 5 m} and {q. : 1 5 j 5 n}. A

J 2 trace t r on M is specified by the row vector 5, si = tr(pi). Let H = L (M,tr). Set

when p;. f 0. We have the t*a& matrix T: with entries c. . = tr (p.q..), and t* 1 J Id piM 1 J . .

index matrix A! with entries

= 0 p.q. = 0, 1 J

r. = if piqi # O, l,j

( M 7 e ~ ) = T:: , namely Our present goal is to compute the entries of TM +,

Lemma 3.6.7. If p.q. # 0, .then 1 J

C . . (i) dimNt. .(piqjH) = A, and

1 , ~ '!,j

(ii) d. . dimN, .(piqjH) = dimqeNt(qjH)- J,' i , ~ J

Proof. By 3.4.6, - dimN. .(PiqjH)

'!,j =-.

and by 3.2.5(h),

dimM. JpiqjH) = trpiM(qjpi)ddimp.M(~i~). IrJ I

But since M is in standard form on H, so is piM on piH, and dimpVM(piH) = 1. 1

Combining these observations,

by 3.2.5.g. Hence (i). (ii) This reads

which follows from 3.2.5(h). #

Notation: For each j, let

the sum being over those i such that p.q. # 0, and let F be the diagonal matrix I J

F = diag((ol,. . . ,cp,). Furthermore, let be the n-by-m matrix

wlapbrn a: r~luce von neumaon-ageoras

( w e N ) Proposition 3.6.8. TM = FT.

Proof. Combining 3.6.7(i) and (ii) we get -

if p.q. # 0, and d.. = 0 otherwise. To eliminate dimq:N,(qjH) we use the fact that 1 J J71

( M,%) T~ is row stochastic,

A? Putting this back in (3.6.8.a) gives d. . = -!.ti if p.q. # 0 and d. . = 0 otherwise, as

1.1 vjci , j I J J,]

desired. #

( M,%) Let us check what that formula TM = F'i? means for finite

dimensional algebras. Suppose that piM g Mat (C) nod qjN g MatUJC). As noted Y

M d before, the inclusion matrix A = AN determines the trace matrix T = T N via

since q.p. is the sum of Aijuj orthogonal minimal projections in piM Setting J 1

= diag(pl,. . . llr,) and "v diag(ull. . .,un), this can be written

T = jtl~"v.

5 3.6. The fundamental construction. 171

x2 I J

"i When p.q. 1 0, we have (9. . = $ = A i j 2; a d when piqj = 0, (T). . = 0 = JJ i,j J J71 J Thus

n Set L = (M,eN), L = @ L.; then L. g Mat,(C) whae rj = ( ~ ' 2 ) ~ =&Aij. Note

j=1 J J J i that

Thus

which is in accord with the relation observed above between the inclusion matrix and the index matrix.

. ..

We now return to the analysis of the general case. 4

As the minimal central projection in (M,eN) = JN'J are precisely {Jq.J : 1 i j 5 n), J

any trace Tr on (M,eN) is specified by a row vector f, with r. = Tr(Jq.J). It will turn J J

out to be useful to calculate the q u a n t i t i Tr(eNJqjJ). h c a l l that JeN = eNJ. Also

observe that

In fact, let 51 denote the trace vector in H = L2(hf,tr), i.e. the identity 1 of M reqarded as an element of H. The l ine i space {x51 : x E M) is dense in H and we have

e Jq.J(x51) = e J .x*n = e x 51 N J

= ~ ~ ( x q ~ ) ~ N q~ = E ~ ( x ) Q ~ ~ N qj

= qjEN(x)n = EN(qjx)fi

= eflj(xfi).

Lemma 3.6.10. Let Tr be any trace on (M,eN) and let r. = Tr(Jq.J). Then J J

(ii) Tr(eNqj) = Tr(eNJqjJ) = r .p . J J

Proof. (i) Since N is in standard form on eNH, so is - q.N and its cornmutant - 3 q e N'q.e on q.e H; hence j N J N J N

1 = dimqe (qjeNH) j N j N

= [trq.N# (qjeN)]'dimq+Nt ( q ~ ) (by 3.2.5(h)-) J J

-1 = [trq.Nl(qjeN)l Qj (by 3.6.8(b).)

J

(ii) Since the map x I+ T~(JX*J) is a trace on the factor qjNJ we have

T~(JX*J) = Tr(JqjJ)trqjN,(x), and in particular, using 3.6.9,

3.7. Markov traces on EndN(M), a generalization of index.

pefinition 3.7.1. Let N c M be finite von Neumann ,algebras with N of finite index in M. We say that a faithful trace tr on M is a Markov trace of ~ o d u l u s p for the pair N c M if it extends to a trace, also called tr, on (M,eN) for which

(3.7.2) p tr(xeN) = tr(x) for x E M.

The extension of tr to (hi,eN) is uniquely determined by (3.7.2). Also it suffices for

(3.7.2) to hold for x E N, since then for x E M

tr(xeN) = tr(eNxeN) = tr(EN(x)eN) 1 1 = ;, tr(EN(x)) = tr(x).

Cf. Lemma 2.7.1.

We restrict our attention to pairs of finite direct sums of finite factors and continue to use the notation of the previous section.

Theorem 3.7.3. A trace o n M specibed by the vector t, si = tr(pi) is a Markov trace

o f m o d d w P ifand only if

Proof. (+). Suppose Tr is a trace on (M,eN) extending the given trace on M and

satisfying the Markov property (3.7.2). Let 'i be the row vector, r. = Tr(Jq.J). By the J J

Markov property we have

C 4- where t = s T i is the vector specifying tr I N . Putting this together with 3.6.10(ii) gives

Hence

/j $ = b ; TLMleN) = /3 : F'f (by 3.6.8)

=if (by 3.7.3.1)

= t T:T.

14- M 1 (P) Given a trace tr on M satisfying t T ~ T = dpline 'i = /T s TNF-

(motivated by 3.7.3.1), and define a trace Tr on (M,eN) by Tr(Jq.J) = Then J 'j.

so Tr extends tr on M (3.5.2(ii)). It remains to show the Markov property, Tr(xeN) = /T1tr(x) for x EN, and by

linearity it is enough to check this for x E Nq Now x I-I Tr(xeN) i4 a trace on the factor j.

Nqj, SO Tr(xeN) = Tr(qjeN)trNqr); hence it ~uffilees t0 show that

1 T*(qjeN) = /T tr(qj) = /T1tj. But by 3.6.10(ii)

1 1 1 buapbar a; rlulrc: vuu l u a u u m lugeuru

Tr(q.e J N ) 5 r j q = (FF)~

= TFF-~F). 3

= T1tj,

as desired. #

_Corollarv 3.7.4. Suppose N c M are jnite direct sums of finite factors, with N of M Pnite i n d a in M . Set T = TN.

(i) If N c M is a connected inclusion, then there is a unique normalized Markov trace on N c M; it is faithhl and has modulw eqwl to the spectral radius of TT.

(ii) If t r is a Markov trace of modulw p on N c M, then the unique eztension of the trace to (M,eN) satishing (3.7.2) is a Markov trace of modulw /3 (for M C (M,eN)).

fEQPfr (i) Since N C M is connected, T is indecomposable and TT is irreducible by a straightforward generalization of Lemma 1.3.2.b. Therefore by Pmon- Frobenius theory, TT has a unique non-negative eigenvector f with z s i = 1. Furthermore si > 0

I

and the corresponding eigenvalue is the spectral radius of TT

(ii) If f is the vector specifying the Markov trace on M, then the extension of the trace to (M,eN) satisfying the Markov condition (3.7.2) is specified by the vector

( MaN> \ ? = ~ l f i TF-l. Let R denote thematrix TM = F"I', with entries

( M,%) Since AM = (A!)~, the m a t h , R (which is to R as T is to T) has entries

That is 'fi = TF-l. But then

; RR = ( r l f TF-~)(FT)(TF-~) = ,rlf TZTF-I = f TF-I (by 3.7.3) = p;.

Hence f defines a Markov trace on (M,eN) byeTheorem 3.7.3. #

Remark. Before going on, let us see how the analysis above agrees with that in Chapter 2 for finite dimensional algebras. Assume that Mpi g Mat (C) and

Y Nq. E Mat,JC). We noted in the remark following 3.6.8 that

J J

where i; = diag(pl,. . . ,pm) and "v diag(ul,. . . ,un). Thus

In this chapter we have been specifying a trace t r on M by the vector fi with si = tr(pi), while in Chapter 2 we specified the trace by f l , where s j is the trace of a

minimal projection in Mpi. The vector f and f ' are related by f = f f i. The

condition given in Chapter 2 for tr to be a Markov trace of modulus is f ' = ,O f ' . But this is equivalent to

4

~ ( T T ) = ( f l ; ) ( ~ l ~ ~ t ; ) = ftAAt;=pf$= pf. #

Definition 3.7.5, Let N c M be finite sums of 111- factors with the same identity and

with N of finite index in M. Let A = A! = (A. .) be the matrix of indicea and M

IrJ

T = T N = ( c ~ , ~ ) be the row stochastic matrix of t r m as above. Form T = '?(A,T), the

A? . matrix whose (j,i) entry is 0 if c. . = 0 and 3 otherwise. Then the index of N in M,

191 i ,j [M:N], is the largest eigenvalue of the matrix TT.

&mark. It is easy to see that this definition agrees with that of Section 3,4 when N and M are factors. We mention again that P. Jolissaint has recently shown that this definition always coincides with the ring theoretic definition given in Section 2.1 and [Jo~].

Corollarv If N c M are as above and [M:N] < 4, then $7'6t [M:N] E (4 cos r/q : q 2 3).

&g& The index is the largest of the numbers [Mz:Nz], where z is a minimal '

projection in Z(M) f l Z(N), so we can assume that M :, N is connected.

By 3.7.4(i), there is a Markov trace tr on M of modulus [M:N]. Then 3.7.4(ii) al1ows:us. to iterate the fundamental construction in the usual way to

obtain a tower

a sequence of self-adjoint projections (ek)k,l with Mk+l = (Mk,ek) for all k, and a

trace t r on UMk satisfying the Markov property k

[M:N]tr(ekx) = tr(x) for x E Mk.

The projections ek then satisfy the usual relations and therefore the restriction on [M:N]

follows from [Jol]; see the argument given in Section 3.4. #

Next we provide some examples. Note that by 3.5.6, to. construct examples it suffices to give the +trices A and T.

Examnle 3.7.7. The simplest new example is where M is a 111-factor, p is a M

projection in M and N = pMp + (I-p)M(I-p). Here the matrix A: is ( I I), and T N

is (t 1-t), where t = trM(p). Thus f = [l;(:] and TT = 2. So [M:N] = 2, \ independent of t! #

M Examole 3.7.8. Consider an inclusion N c M with A = AN = [i :] and

M t 1 t l / t 0 T = T N = [, 1. Then f = and ~f = [11:4 'it]. The characteristic

2 equation is X2 - 3y + 1 = 0, so [M:N] = 4 cos ir/5, independent of t. #

M - .[I I] TM - [" Exam~le 3.7.9. Take AN - , - , with 0 < a,b < 1. Then

f = 'and fi= , The characteristic polynomial is

with u = t H. So [M:N] = 2 + I-, which can be any real number *eater

than .or equal. to 4. # . .

Examole 3.7.10. Let M be a 111-factor, p a projection of trace t in M and

N = pMp + Q, where Q is a subfactor of index X in 1 - 1 - p ) . Then A! = (1 X1l2) and TT# =.(t 14). So f = [$ift] and TT = 1 + A. This is 1 4

2 when X = 1 ,2 ,4 cos ir/5, or 3.

Remark% The index matrices in example 3.7.10 correspond to A3, B3, H3, and GP), 2 respectively, under the corespondence of Theorem 1.1.3, when X = 1, 2, 4cos ir/5; and'3.

This is no accident, as we will see.

Pronosition 3.7.11. Let A = (A. .) be an irredundant m a t h over ( 2 cos(ir/q) : q 2 2) - 1,J

and T = (c. .) is a row stochastic matrix wilh the same pattern ofzero .entries as A. Let 1,J

A? t = ~ ( A , T ) be the mat& whose (j,i)-entW is zero i j c. . = 0 and equal to $ Id i ,j

othekise. If the spectral radius of TT is less than 4, then it equals 4 cos2ir/q for some

q E {3,4,5. + .).

I We can suppose that A and T are indecomposable. By 3.5.6, there is a

connected inclusion M J N of finite direct sums of Ill-factors with A = A: and

T = T:. Thus the result is a corollary of 3.7.6. #

Remark. It would be interesting to find a proof of 3.7.11 within usual matrix theory; hopefully this might give information on the spectral radius of T? even when it is larger than 4.

Pro~osition 3.7.12. (a) Let A be an irredundant m-by-n mat& with non-negative real values. Then

there is a row stochastic m-by-n matriz T with the same pattern ofzeros such that

where p denotes spectral radius and f = T(A,T) is as above.

(b) If A is irredundant with values in (2 cos(ir/q) : q 1 21, then there is a pair N c M of jnite direct sums of 111 - factors with A = AT# and [M:N] = 1 1 ~ 1 1 ~ .

(c) If A is any non-zero mat& over (2 cos(ir/q) : q 5 2) and IlAll < 2, then

ll All E I2 cos(~/q) : q 3).

(a) As A is irredundant, we caa define a row stochastic matrix T = (c. .) by 1,J

178 ' Chapter 3: Finite von Neumann algebras 9 3.7. Markov traces 179

c i j = [ Z + $ - ~ A ~ , ~ , or T = XA, where X is the m-by-m diagonal matrix whose Let 3 be the set of k+l-tuples with il = ik+l; thus a r a ' is bijection of Wk

J (i,i)sntry is [EAiVj]-l. Then the (j,i) entry of f is A i j [ Z A i j ] , i.e., f = dt r l . tr(Ak) = a, = $x (aa + a

J J a€Wk acWk a Thus TT = XAAtX-l, which has the same spectrum as Adt.

M M (b) By 3.5.6 there is a pair N c M with A = AN and T =TN. Then k tr(Ak) t ( a g -1)112 = go = tr(G ), (M:N = p(TT) = llA112. acVk 0'wk

(c) It suffices to consider A irredundant, so the result follows from (b) and

3.7.6. # for all k E PI. When k is even, we have

Of course, 3.7.12(c) was dready known as a consequence of Theorem 1.1.3. Theorem 1.1.3 suggest8 (but does not immediately imply) the following, which is the main result of np(Alk = n p ( ~ ~ ) z tr(Ak) t tr(Gk) 2 p ( ~ ) k ,

this section. where the first equality and last inequality result from considering canonical forms for A

t h Theorem 3.7.13. Let N c M be a connected inclusion of jinite direct sums of and G, noting that the eigenvalues of G~ are positive. Taking k roots and then the

II, - factors. limit as k -+ w gives the result. #

(a) If [MN] < 4, e n A is the matriz associated (in Theorem 1.1.9) fo a Lemma 3.7.15, Let A = be an m-by-n irredzmdant mat& over bicoloration of one of the following Cozeter graphs: : {I E B : r = 0 or r 2 1). Let T = (G .) be a row stochastic matriz PuiIh the same pattern of

1 J

At (L 2 2)7 Be (L 2 3), DL (L t 4), (L = 6,7381, zero entries as A. Let 2 he the n-by-m matriz whose (j,i)-entry b 0 i f c. . = 0 and

1J

F4, G2, Hp, (L = 3341, 12(p) (P = 5 or P Z 7). \

Moreover [M:N] = (lA:$ = 4 cos2~/hY where h i s the Cozeter number. (See tables If p(T"i') i 4 then (lA112 $ p ( e ) .

1.4.5, 1.4.6, and 1.4.7.) Ergpf, We may assume without loss of generality that A is indecomposable. (b) If [WN] = 4, then A! corrmpondr to one of Suppose that there exist indices il,i2,jl,j2 such that the four entria for 1

i

A t ) (L odd, k L I), 861) (L L 2), C P ) (L t 31, p,v E (1,2) are all non-zero; tha& is, the graph f (A) contains a subgraph of the form I / I

D P ) (L L 4), ~ 1 ~ ) (L = 6,7,8), ~ f ) , GF) . - ;I Lemma 3.7.14. (Schwenck, [Sch2]) Let A = (a. .) be a non-negative n-by-n

I>J i4. o I mat& and kt G = (%,J be the motri. with entries q , j = (a. .a. .)'I2. Then

I I

1 J 131 I

IlGll = p(G) < p(A), where p denotes spectral radius. Rearranging the rows and columns, we can suppose il = jl = 1 and i2 = j2 = 2. Denote

, c. x2 && For any k+l-t,uple o = (il,i2,- .,iW1) with 1 I i. $ n, let a-l denote the J the (i,j)-entry of TT by %,i Then .liJ =x v, the sum being over those k I

\ k j ,k reversed tuple a-l = (ik+l,. . . J2,il), and let

I

7 . z '2 with equality everywhere if and only if BU 1 9 1

a = a i ai ..- k ~ , k

a 2, 4k1ik+l. I

1UU c"'i ctnapar a: riulrc: vuu r~eulnauu l u ~ v u ~ s a

By monotonicity of the Perrbn-Frobenius eigenvalue, we have

with equality if and only if A is 2-by*. This in turn is no smaller than

by the observation above, with equality if and only if all the nonzero Xjlk with j j 2 are

equal to one. Truncating the sums defining the entries of the last matrix we see that the spectral radius is at least

with equality if and only if ,Ijfk = 0 for j = 1.2 and k > 2. If we replace the off-diagonal

entries by their geometric mean, we do not alter the spectrum, so the las quantity is equal to 1

, . . .

.. . c 'c where o = 2'2. Finally, this is at least

.: C2,1C1,2 . .

with equality if and only if a = 1. But since p ( ~ f ) 1 4 b hypothesis, we must have equality at every step: A and T are 2-by-2 with A = [: 4 and o = 1. since T is

2 mw-stochastic this implies T = , and llAll = p ( ~ f ) = 4.

If on the other hand f (A) coatains no subgraph of the form

m . 7 . Markov traces

",I -.%,o.

[ 0 if the ith and j rows of A are orthogonal

Observe that ko depends on (ij), and ko(i,j) =' ko(j,i). On the other hand,

7 . = ZXt,. Note that for all pairs (ij), the (ij) entry of AAt is the geometric 1 9 1

k mean of the (i,j) and (j,i)-entries of TT; i.e.,

= I o i f the , ith and jth rows of A are orthogonal

1 li,koAj,ko otherwise.

Hence by Lemma 3.7.14

1 1 ~ 1 1 ~ = p(hAt) 6 P(Tf)- #

Prwf of Theorem 3.7.13. (a) Let T = T! and f = T(A,T). By hypothesis

[M:N] = p(Tf) < 4, so by 3.7.15 we have 1 1 ~ 1 1 ~ s p(TT). Let S be the (convex) set of all row- stochastic matrices of the same dimension and zero-pattern as A and T. For each S a S define S = f (A$), the matrix whose (j,i)-entry is 0 if A. . = 0 and ,IT ./(S)i,j

1,J >J otherwise, and cp(S) = p(SS). Then cp is a continuous function of S by elementary

2 perturbation theory, and (P assumes the value [M:N] = ~ ( T T ) as well as the value 111111 , by 3.7.12(a). But by 3.7.11, the set of values of cp less than 4 is discrete, so by convexity of S, cp is constant and

M The classification of A N then follows from Theorem 1.1.3.

(b) We have 11A112 j ~ ( T T ) = 4, by 3.7.15. If llA112 < 4, then the connectedness 2 argument of part (a) would imply that llAll = ~ ( T T ) c 4, a contradiction. Thus

2 llhll = 4, and the classification follows from 1.1.3 again. #

there is at most one nonzero term in the sum defining %.j,

5 4.1. Introduction 183

CHAPTER 4 In Section 4.2, we study the notion of commuting squares and give a number of ~ ~ m u t @ sqam, subfactors, and the derived tower examples of constructiowhich produce commuting squares. In particular we coqsider the

behavior of commuting squares under the fundamental construction. "

4.1. Introduction.

C1 B1 There are two main themes in this chapter. The first is the approximation of a pair P r o ~ o s i t ion 4.1.2. Consider a commuting square U u with respect to a trace

N M of hypefinite I I ~ factors by pairs cn c Bn of finite din~ensional van Neumann Co Bo

algebras, with tr which is a klarkov trace for the pair Bo c B1 offinite von Neumann algebras &th finite N c M u U d i m ~ ~ ' o n a l centers. Let B2 = (Bl,el) be the von Neumann algebra qbtained via the

B n + l ' kndamental comtruction for Bo c B1, and let C2 = {Cl,el)*. Then

u U

Cn B n c 2 B2

U U a,ld M = (U B,)", N = (U c,)". 1n order for the approximating "ladder1' of finite

C1 B1 dilnensiond algebras to behave well with respect to the fundamental construction and the is ako a commuting square. index, it should behave well with respect to the conditional expectations:

Therefore iterating the fundamental construction will produce an infinite ladder of be tile conditional expectation of Bn onto Cn. We are thus led to the following defillition

commuting squares. Now suppose that we have a pair N C M of finite von Neumann wllicll was first introduced by Popa (Lemma 1.2.2 in [PoplI; see also [Pop2]): . algebras with a Markov trace tr of modulus 0 and a ladder of commuting squares

Definition 4.1.1. A diagram N C M

C1 B1 U u

u u 'n+l B n + l

Co Bo u U

'n B n

of finite van Neumann algebras with a finite faithful norlnal trace tr on B1 is a with N = (U Cn)' and M = (U Bn)". Let (M, e) be the result of the fundamental

mlnlnuting if the diagram construction for N C M and set An = (Bn, e) for each n 2 1. We show that the

E ~ l B1

M c (M, e) C1t---- algebras (An)n20 generate (M, e), and u U is a commuting square with u u Bn An

Bo respect to the Markov extension of the trace to (M, e).

Co - E ~ a

In Section 4.3 we prove a theorem of H. Wenzl on pairs N c M generated by a ladder of commuting squares satisfying a periodicity assumption. (See Section 4.3, Hypothesis

(B), for the exact assumption.)

182

commutes.

184 Chapter 4: Commuting squares and subfactors 5 4.1. Introduction 185 /

Theorem 4.1.3. Suppose N c M is a hyperfinite pair (with a finite faithful trace tr on M) generated by a ladder of commuting squares

of finite dimensional von Neumann algebras. Suppose that the inclusion data for the ladder is periodic, in the sense of Hypothesis B ofSection 4.3. Then

(i) N and M are factors and [M:N] < m.

Let e and (An)n20 be as above and let zn be the central support o f e i n An.

(ii) For large n, zn = 1. Equivalently, An is isomorphic to the result o f the

fundamental constrnction for Cn c Bn.

(iii) For large n, [M:N] = [Bn:Cn] = lit(n))2/11~(n)112, where t(n) and g(n) are

the vectors of the trace on Cn and Bn respectively.

Section 4.4 contains a contruction of (necessarily irreducible) pairs of hyperfinite factors with index less than 4, as follows: Start with a connected pair Co c Bo of finite

dimensional von Neumann algebras with index /3 < 4, and let B1 = (BO, eC ) be the 0

result of the fundamental construction for Co c Bo, with respect to the Markov trace of

modulus 0 on BO. Define q E T by /3 = 2 + q + q", set g = qeC - (l-eCd, a 0

C1 B1 unitary element in B1, and set Cl = g~og-l. Then U U is a commuting square,

Co Bo with . respect . to.the Markov extension of the trace to B1. Let (Bn)n20 be the tower

obtained by iterating the fundamental construction, with Bn+l = (Bn, en) and set

Bn+l

Cn+l = alg{Cn, en} for n >_ 1. Then U U is a ladder of commuting squares,

'n Bn with respect to the Markov trace on U Bn. It turns out that the inclusion data is periodic

and that B = (U Bn)" 3 C = (U Cn)' is a pair of factors with index p.

sequence of projections in the tower construction. We already know another construction of an irreducible pair with index /3, namely {el,e2,- - '1 ' 3 {e2,e3,, - - } " (Theorem 3.4.3).

4 An argument due to C. Skau shows that {el,e2,. - 9 ) ' n (U Mk)" = N (Theorem 4.4.3).

In Section 4.5 we present a construction which yields irreducible pairs of hyperfinite 111

factors, starting with a Coxeter graph of type A, D, or E and a choice of a distinguished vertex wl on r. In particular for r = E6 and wl an end vertex on one of the long

arms of r , we obtain the index value 3 + p, which is at present the smallest known value larger than 4 of the index of an irreducible pair. The construction goes a8 follows.

Give r the bicoloration with wl white and with r white vertices altogether. Let

Mo denote the abelian von Neumann algebra C' and M1 the finite dimensional von -2

Neumann algebra containing Mo such that I' is the Bratteli diagram of the inclusion

I Mo C MI. Form the tower (Mj)j20, starting with the pair Mo c M1 and the Markov

trace t r on M1, and let be the usual sequence of projections. Let M be the

factor ( U M.)" and let N be the subfactor of M generated by P and the e.'s. Set j20 J J

2 /3 = [M1:MO] = llrll ; since ,/3 < 4, Skau's Lemma 4.4.3 applies and M n N' = Mo. Let

& p be the minimal projection of Mo corresponding to the vertex wl and set C = pN and

p B = pMp. Then C c B is a pair of factors with C' n B = p(N1 nM)p = pMOp = Cp; that

is C c B is irreducible. The index of this pair can be computed as follows: Let r also denote the matrix of the bicolored graph I', and let tE denote the unique

Perron-Frobenius row vector for rtI', normalized so that its first co-ordinate (corresponding to the distinguished white vertex wl) is 1. Let A be the Coxeter graph of

type A with the same Coxeter number as I', and with a bicoIoration having at least one white end vertex, which is labelled as the first white vertex. Denote also by A the matrix

t of the graph A, and let 5 be the Perron-Frobenius row vector of A A, normalized so that its first coordinate (corresponding to the chosen white end vertex) is 1. then [B:C] = 115112/11t112. The proof uses Wenzl' s index formula from Section 4.3.

The second main topic of Chapter 4, presented in Sections 4.6 and 4.7, is the and princi~al & of a pair of finite factors N c M of finite index. The derived

tower is the chain of relative commutanis (N' n Mk)k>O, where (Mk)k>O is the tower - - - for the pair N c M. It follows from 3.6.2 that dim (Nr n MI) 5 [M:N]~ for k 2 0.

Let (ek)k>0 be the projections in the tower construction, let Yk denote N' n Mk - and Ak the inclusion matrix for Yk c YkS1. The following summarizes the structure of

Let (Mk)k,O be the tower obtained from a pair N c M of finite von Neumann - algebras with finite dimensional centers, with index /3 c 4 and let (ek)k>l -be the usual -

186 Chapter 4: Commuting squares and subfactors

Theorem 4A.4.

(i) The inclwion Yk c Yk+l is connected.

(ii) YkekYk is an ideal in Yk+l, and if zk+l = z(ek) is the corresponding

central projection in YkS1 then the homomorphism I yk

- 'ke k 'k has inclusion - xzkf 1 t mat& Ak-l.

(iii) For all k, l(Ak(12 < [M:NJ.

(iv) For k > 2, if x E Yk+l and x(YkekYk) = 0, then x ( Y ~ - ~ ~ ~ ~ Y ~ - ~ ) = 0.

(v) For all k 2 1, the following are equivalent:

(a) YkekYk = YkS1.

(b) E ( ~ - ~ ) A ~ - ~ A ~ - ~ = [M:N] E(~-'), where !dk-l) is the vector of the trace

on Yk-1. t

fc) Ak = Ak-1.

I I ~ ~ - ~ I I ~ = [M:Nl'

(vi) If the equivalent conditions of (v) hold for k, then they also hold for k+l.

We call the ideal YkekYk "the old stuff", since it is determined by Yk-l c Yk; the

complementary ideal is called "the new stuff1. Then (iv) says that "the new stuff coma only from the old new stuff", or ( x ~ ~ ) ( l - + ~ ) = 0. The princi~al & of the pair

N c M is obtained as follows: on the Bratteli diagram of the derived tower, delete on each level the vertices corresponding to the old stuff, and the edges emanating from them; the result is a connected bipartite graph with a distinguished vertex *, the unique vertex on level 0. The Bratteli diagram of the derived tower can be reconstructed from the principal graph. The pair N c M is said to be of finite d e ~ t h if the principal graph is finite; the

is the maximum distance from any vertex to * . This analysis, together with the work of Chapter 1, yields a new proof of the restriction

on index values:

Corollarv. (i) Suppose N c M is a pair of 111 factors with [M:N] < 4 . Then 2 (a) [MN] = 4 cos nth for some integer h 3.

. ' (b) The depth of N c M is no greater than h-2. (c) The principal graph of N c M is a Coxeter graph oftype A, D, or E, whose

norm is [ M : N ~ / ~ .

(ii) Suppose N c M is a pair of Ill factors with [M:N] = 4.

(a) If N c M is of finite depth, then the principal graph l'- is a completed Cozeter graph of type A, D, or E, i.e., one of the graphs in Table 1:4.6.

5 4.1. Introduction 187

(b) If N c M is ofinfinite depth, then l' is one of the following:

/ \ / \ / "' (end uertez at distance n from *)

(doubly infinite linear graph) ,- -

(end vertex at distance n fiorn *

Section 4.7 is devoted to computing the derived tower for a number of examples Crossed-products and fixed point algebras for outer actions of finite groups give example with depth 2. The pairs R c R (of Proposition 3.4.4) when P < 4 have principal graph P of type An; for 0 = 4 the principal graph is Am. In 4.7.c we give a general hethod whicf

allows the computation of the derived tower in many examples coming from group actions In 4.7.d we use this method to obtain the derived towers for the index 4 subfactoc R~ c (R @ at^(^))^, where the hypefinite 111 factor R is realized as the weak closur~

m of the CAR algebra @ Mat2(C) in the trace representation, and G is a closed subgroup o

SU(2) acting by the infinite tensor product of its action by conjugation on Mat2(C). 11

this way one obtains as principal graphs all the affine Coxeter graphs of type A, D, and E as well as the infinite graphs Am, and Dm listed above. Finally we compute thc

derived tower for the pair RP c R when /3> 4. This is the most difficult result of tht

chapter, involving a representation of the sequence (ei)iL1 in the CAR algebra due tc

Pimsner and Popa and a theorem of Popa on the tunnel construction (a mirror image oj the tower construction). Ultimately one identifies the pair R c R with the pail P N ~ C (N @ at^(^))' where N is the completion of the CAR algebra with respect to s

certain Powers state. The principal graph is therefore Am,&

188 Chapter 4: Commuting squares and subfactors

4.2. Commuting squares.

We begin with a proposition, inspired by Lemma 2.1 of [Pop2], which gives a number of equivalent conditions for a commuting square.

Proaosition 4.2.1. Consider a diagram

of finite von Neumann algebras and a finite faithful normal trace tr on B1. All conditional

expectations being with respect to tr, the following are equivalent.

6) ECl(BO) C GO.

(ii) ECIEBO = ECO.

(iii) EC EB = EC EB . 1 0 0 0

(iv) EC EB = EBOECl and Bo ilC1 = C,. 1 0

Ec c1 C--l- B1

(v) The diagram U U commutes.

( 4 Ec0(boc1) = Eco(bO)EcO(~l) jar bo E B0 and c1 E C1.

(vii) ECo(bocl) = 0 jor bO E Bo with Ec (bo) = 0 and c1 E C1 with 0

EC (el) = 0. 0 Moreover (i) to (uii) are equivalent with the analogous conditions obtained by

interchanging Bo with C1.

Proof. Let p,q,r be three projections acting on some Hilbert space. The following are clearly equivalent:

(a) P9 = r (b) pq = rq and r d q (c) pq = qp = r.

2 As we may view the conditional expectations as projections on L (Bl,tr), this shows the

equivalence of (ii), (iii) and (iv). Obviously (ii) implies (i).

3 4.2. Commuting squares 189

Assume (i) holds and let bo E Bo. For all co E Co one has

tr(Ec (b0)co) = tr(bocO) = tr(Ec (b0)co). 0 1

As Ecl(bO) E Co, this implies EC (bo) = EC (bo), and (v) follows. As (v) implies (iii), 0 1

conditions (i) to (v) are equivalent. The equivalence of (vi) and (vii) follows from the formula

for bo E Bo and cl E C1.

The next step is to show that (ii) and (vii) are equivalent. Observe first that one has

for any x E B1. Thus (ii) can be reformulated as

EB (x) -EC (x) I C1 for all x E B1. 0 0

Suppose (ii) holds. Then, in particular, bo I C1 for bo E Bo with EC (bo) = 0. 0

Consequently, for all cl E C1 and co E Co, one has

As tr is faithful on Co, this implies Eco(bocl) = 0 and (vii) holds.

Suppose (vii) holds. For all x E B1 and for all cl E C1, one has

which is zero, since the conditional expectations are trace preserving. Consequently (ii) holds.

Chapter 4: Commuting squares and subfactors

Finally, as (iv) is symmetric with respect to Bo and C1, we may exchange Bo and

C1 in any of the conditions (i) to (vii). #

It follows for example from (v) that in diagrams like

the "rectangles" are commuting squares as soon as the "small squares" are commuting squares.

A crucial point about commuting squares is their behavior with respect to fundamental construction defined in Section 3.6.

Pro~osition 4.2.2. Consider a pair N c M of finite von Neumann algebras, a finite faithful nonnal trace tr on M, and the algebra (M,eN) obtained by the fundamental

construction. Assume that M [respectively N] is generated as a von Neumann algebra by a nested sequence (Bj)j20 [resp. (C.). ] o f von Neumann subalgebras in such a way that

J J20 one has for each j 2 0 a commuting square

andset A. = {Bj,eN)'. Then J

( 1 e ~ b e N = ECj@)eN = eNEb:(b) for b c Bj, j 2 0 J

(ii) The algebras (Aj)j20 generate (M,eN) as a von Neumann algebra.

Suppose moreover that tr is a Markov trace of modulw /3 for the pair N c M, and denote the Markov edension of tr to (M,eN) by tr again. (See Definition 3.7.1.) Then

(iii) U U is a commuting square with respect to tr . B~

c A. J I ~ j + l

3 4.2. Commuting squares 191

'j+kc Bj+k

Proof. (i) For each j 2 0 and k L 1, the diagram U U is a commuting C j C Bj

N c M

square, by induction on k. It follows that the limit diagr& U U is also a C j c B

j commuting square, and thus for any b E B one h~ eNbeN = EN(b)eN = EC.(b)eN.

j J Since elements of N, and in particular EC (b), commute with eN, this shows (i).

j 3,

Claim (ii) is obvious. 1 (iii) One has EB (eN) = /3- , because

j+l

1 tr(EBj+l(eN)~) = tr(eNx) = tr(x)

for all x E Bj+l. Consider now yo,yh,y& E B.. J Then

Thus EB (A.) C B. for a dense *-subalgebra A. of A j+l J J J j*

Let x E A.. By the density theorem of Kaplansky, there exists a sequence ( x ~ ) ~ , ~ , J

with xk E A. and llxkll s llxll for all k 2 1, such that x = 1 im xk in the topology J k+ m

defined by the norm 11. 112. It follows that EB (x) = 1 im EB (xk) c Bj. Thus j+l k+m j+1

E (Aj) c B. and this proves (iii). # Bj+l J

C1 B1 Corollarv 4.2.3. Consider a commuting square U U with respect to a trace t r

Co Bo which is a Markov trace for the pair Bo c B1. Let B2 = (Bl,el) be the von Neumann

algebra obtained via the findamental constkction for Bo c B1, and let C2 = {Cl,el)'.

C2 B2 j Then U U is also a commuting square.

192 Chapter 4: Commuting squares and subfactors

Proof. This is the special case of 4.2.2 applied to U - U . # Co C C1 C C2

Remark. Suppose moreover that Bo and B1 have finite dimensional centers. Then

the fundamental construction iterates to give the tower (B ) with BjS1 = (Bj,ej) for j j>O

all j. Define inductively CjS1 = {Cj,ej}' for j 2 1. Then we obtain a ladder of

'j+l Bj+l commuting squares U U . We are going to use this idea to construct

Cj C Bj /-

examples of subfactors below, starting with a commuting square of finite dimensional algebras. The next two lemmas concern conditions which cause the inclusion matrices for the resulting ladder of finite dimensional algebras to be repeated with period 2.

C1 B1 Lemma 4.2.4. Consider a commuting square U U of finite dimensional

Co Bo von Neumann algebras, with respect to some trace tr on B1. Let B2 = (Bl,el) be the

finite dimensional von Neumann algebra obtained via the findamental construction for Bo c B1 and let C2 = alg{Cl,el}.

- -

Suppose that C2 = ClelCl (or equivalently, by 2.6.9, that xxiec0yi x x i e m

is an isomorphism from the algebra (C1,eCo) obtained by the findame6tal co~ t ruc t i on for

Co c C1 onto C2). Then

B2 Bo (i) AC2 = ACo. More exactly, let q, p be minimal central projections in Co, B0

respectively. Let "q u(JC qJC ) and $ = JB pJB be the corresponding minimal 1 1 1 1

central projections in C2, B2 respectively. Then [(Bo)pq : (C ] = [(B2);i :, (C -1. 0)pq 2)pq

Suppose in addition that tr is a Markov trace with respect to Bo c B1. Let (B.). J J'_O

be the tower obtained by iterating the hndamental construction for BO c B1, with

BjS1 = (Bj,ej), and let Cj+l = alg{C.,e.} for all j 2 1. Then J J

B (ii) For all j 1 1, C.e.C. = Ci+l and ABj+l = A j-I. The inclusion matrices

J J J Cj+l Cj-1

C2 C1 t for Cr1 c C. are alternately AC1 and AC = (AC ) . J Co 1 0

5.4.2. Commuting squares 193

Proof. (i) Let f be a minimal projection in (Co)q and let pf = x g i be a - i=l

decomposition of pf into orthogonal minimal projections in (BO)p (so

n = [(Bo)pq : (C ) ]It2). Then (by 2.6.4) £el = u(fec ) is a rninimml projmtion in C$ 0 Pq 0

and f e l b felp (by 3.6.9)

= fpel (because p E Bo)

= pfel (because p E Z(Bo)) -1

n

Thus (fel); is a sum of n orthogonal minimal projections in (B2):?

(ii) We are now supposing that tr is a Markov trace. The statement Cj+l = C.e.C. J J J

is valid for j = 1 by hypothesis. Suppose it is valid for some j. Then Cj+lej+lCj+l is

an ideal in CjS2 containing pejej+lej = e j' where P is the modulus of the Markov

trace. Then Cj+lej+lCj+l 3 C.e.C. J J J 3 1, so Cj+lej+lCj+l = 'j+2'

It follows that for all j, the tower C. J-1 c C. J c Cj+l is isomorphic to

C. C C. C End (Cj), so the inclusion matrices ACj are alternately AC1 and J-1 J Cj-1 'j-1 C~

B. Finally the statement regarding hCJ follows from (i) and induction. #

j

C1 B1

Lemma 4.2.5. Consider a commuting square U U of finite dimensional von

Co Bo B

Neumann algebras, with respect to a trace tr on B1. Suppose A:' = nC: = bt and 0

B1 = AB1 = A for some A. Let B2 = (Bl,el) be the algebra obtained via the Cl

findamental construction for Bo c B1 and let C2 = alg{Cl,el). Then

C B2 (i) C2 = ClelCl, hC: = A, and ACn =

194 Chapter 4: Commuting squares and subfactors

Suppose in addition ffiat tr is a Markov trace with respect to Bo c B1. Let (Bj)j20

be the tower obtained b y iterating ffie hndamental construction, with BjS1 = (Bj,ej), and

set CjS1 = alg{C.,e.) for all j 2 1. Then J J

(ii) The chain Cj-l c C. c Cj+l is isomorphic to C. c C. c EndC. (Cj) for all j. 3 J-1 J 3-1

The inclwion matrices ACj+' are alternately At and A (j 2 0), and the inclusion C i

J

a n alternately At and A (j 2 0).

- for some matrix ill, by Proof. (i) We have C2 =

B C B B 02. Therefore A% = A ~ A ; = n t A + 02nl. On the other hand A: = A ~ A ;

2 1 1 1 = AtA. This is only possible if K = (0), because otherwise i12ill # 0. The remainder of

(i) and (ii) now follows from the previous lemma. #

The next result is that commuting squares are preserved under reduction by certain projections.

- Prouosit ion 4.2.6. Consider a commuting square U U with respect to a

Co Bo

trace tr on B1 and a projection p E Bo n Ci, not zero. Then

pC1 c P B ~ P

U U

pC0 C P B ~ P

is a commuting square with respect to tr I PB,P.

Proof. Let y E pBlp. Then EpBOp(y) = pEB0(y)p b m s e one haa - tr(pEB (y)pu) = tr(pypu) = tr(yu) for all u E pBop. Consider z E pC1, say z = pc,

0 with c E C1. Then

4.2. Commuting squares

and the claim follows. #

Remark. A similar result holds for reduction by projections in Cg.

Next we give some examples of commuting squares involving relative commutants, fixed-point algebras of groups, and crossed-products.

Pro~osition 4.2.7. Let N c M be a pair of von Neumann algebras, let t r be a jnite faithfil normal trace on M, and let S be a self-adjoint subset of N. Then

is a commuting square.

Proof. We may suppose that S is a von Neumann subalgebra of N. Choose x E M. - Denote by C the 11. l12-closure of the convex hull of {mu* : u is unitary and u E S) in

2 L (M,tr), and denote by y the projection of the origin onto C. Then y E M because the 2 ball of radius JJxJ) in M is a )I.)12-closed subset of L (M,tr). Moreover, by the

uniqueness of the projection onto a closed convex set, uyu* = y for any unitary u E S. It follows that y is also in S' .

For any z E S' n M and for any unitary u F. S, one has tr(uxu*z) = tr(xu*zu) = tr(xz), so that ES, n M ( ~ ~ * ) = ES, n M ( ~ ) . Cdnsequently

Es, nM(C) = ES, n M ( ~ ) , and y = ES, nM(y) = ES, n M ( ~ ) . In particular, if x E N, then

C c N and E S t n M ( x ) = y ~ S 1 nN. #

Pro~osition 4.2.8. Let M be a von Neumann algebra given with a jinite faithhl normal trace tr. Let G = H r K be a semi-direct product group which acts on M and preserves tr. Assume that K is a compact group and that the restricted action of K on M is continuow. Denote by MG the algebra of vectors in M jzed by G, and similarly for M~ and MK. Then

M ~ C M >

196 Chapter 4: Commuting squares and subfactors

Proof. For each x E M, one has -

Suppose moreover that x E M ~ . Then k(x) E M~ for any k E K, so that E K ( ~ ) EM^ n M~ = MG. #

M

We leave it to the reader to formulate the details of a proposition involving the diagram

where n indicates now a crossed product. We next describe three examples which are interesting in light of the connections

between the theory of subfactors and that of the braid groups.

Examnle 4.2.9. Let el,. . -,en be a sequence of projections acting on some Hilbert

space such that P eiejei = e.

1 if li-jl = 1

e.e. = e.e. if li-jl 2 2 1 3 J 1

for some real number 0 1 (see the last remark of Appendix IIc). Let tr be a normalized faithful trace on the algebra generated by the identity and the e.'s, J and assume that the

Markov relation

holds (see Section 3.4). Then the diagram

is a commuting square.

5 4.2. Commuting squares 197

Proof. Let us show that EB (x) E Co for any x E C1. This is obvious when x E Co. - 0

By Proposition 2.8.1, one may then assume without loss of generality that x = yenz with

1 1 y,z E Co. As EB (en) = B (see the proof of 4.2.2.iii), one has EB (yenz) = y r z E Co. 0 0

Examnle 4.2.10. Let N c M be a connected pair of finite von Neumann algebras with fidite dimensional centers, of finite index (Definition 3.5.3). Let tr be the normalized Markov trace on N c M (Corollary 3.7.44, and let P = [M:N] be its modulus (Definition 3.7.5). Then tr has an extension to (M,eN) which is again a Markov trace of modulus

p on M C (M,eN) (Corollary 3.7.4.ii), and that we denote by tr again.

Suppose moreover that /3 ?. 4, write P = 2 + q + q-l, define

g = qeN - \

and observe that g is a unitary which commutes with N. Then

&%-' c (M,eN)

U U

N c M is a commuting square.

Proof. Let x E g ~ g - l . If y = g-lxg E M, one has -

Since EM(eN) = we have

EM(x) = P EM(eNyeN) + (1 - (q+l)B1 - (q-l+l)B1}~

= P EM(EN(y)eN) = EN(y). #

Remarks.

(1) Up to scalars, g and 6'' are the only unitaries in alg{l,eN} for which the

above construction works. Observe that g is precisely the element involved in the braid roup representation of [Jo~].

(2) This example is the basis for the examples of Section 4.4 below.

198 Chapter 4: Commuting square and subfactors

Example 4.2.11. Let N c M be a pair of factors, of finite index /I, and let t r denote

the normalized trace on M. Assume that there exists a projection eo E M such that

eo and N generate M

tr(eoy) = /I tr(y) for all y E N.

Let (Mj)j20 be the tower and let (ej)j21 be as usual. (See Section 3.4; of course

M1 = M.) Let Mw denote the von Neumann algebra generated by U M.. Then jio

{I,eo,el,. . .I ' c Mw

U U

Q: c N is a commuting square.

Proof. We want to check that tr(xy) = tr(x)tr(y) for all x E {I,eo,el,. . -1' and for - all y e N. Because of the density theorem of Kaplansky (see the proof of Proposition 4.2.2.iii), we may check this for all x E alg{P,eo,. . .,en) and for all n > 0. If n = 0, this

follows from the hypothesis on eo. To end the proof, we may assume that n 2 1 and that

the claim holds up to n - 1.

For aOb, E alg{l,eo,. . . ,en-l) and x =zaaenba, n

which is by induction

This shows that the claim holds up to n. #

Remark. It would be interesting to have a systematic classification of commuting squares

c1 c i ~ , , t r )

U U

Co Bo

4.3. Wenzl's index formula

of finite dimensional von Neumann algebras.

4.3 Wenzl' s index formula.

In this section we prove a formula due to H. Wenzl [Wen 21 for the index of a pair of factors generated by a ladder of commuting sqyarw. The set up is as follows: We are given a chain (B.). of finite dimensional von Neumann algebras and a faithful tracial

J 320 state tr on Bw = UB Since the GNS representation s of tr (on 1,) is faithful, we

j j'

regard Bw as a subalgebra of B = n(UB.)', a finite hyperfinite von Neumann algebra. j J

We suppose we have a chain (Cj)j20 of finite dimensional von Neumann algebras such

that 1 E C. c B. and: J J

\ C j + l C B j + l Hypothesis (A). For each j, U U is a commuting square.

C j c B j

Then C = (UCj)" is a von Neumann subalgebra of B. In the periodic case which we j

consider below, tr is the unique tracial state on UC. and UB so that C and B are j~ j j '

factors. If E : B -+ C and E. : B. -, C. denote the conditional expectations with respect to

J J J C c B

Ej; that is U u is a commuting square for each j. Let A = (B,e) C. c B.

J J be the result of the fundamental construction for C c B with respect to tr, and let

, Aj = {Bj,e)' for each j. Then A. is an E.-extension of B in the terminology of J J j'

Section 2.6. Hence if (B.,f.) is the result of the fundamental construction for C. c B J J 3 j'

then the formula o.(xa.f.b.) = x 5 e b i (ai,bi E B.) defines an isomorphism from J 1 J 1 J

i i (Bj,fj) onto the two sided itieal B.eB. generated by e in A., by 2.6.9.

J J 3

Lemma 4.3.1.

: (i) The central support z. of e in A. is a.(P); this is also the central support of J 3 J 7 the ideal B eB

j j'

(ii) !im z. = 1 in the strong operator topology. t j+w J

200 Chapter 4: Commuting squares and subfactors 5 4.3. Wenzl's index formula 201

Proof. (i) This is straightforward, since the central support of fj in (Bj,fj) is 1, by - Proof. (i) That B and C are factors follows from the uniqueness of the trace on

3.6.l(vi). ) is a finite factor. In any case A is

semi-finite, so has a faithful normal semi-finite trace Tr; we have to show that Tr(1) < m.

= [AeXd = 1,. That is, z. increases to P. # Now eAe = Ce is isomorphic to C, which is a finite factor, so e is a finite projection J

and Tr(e) < W. Fix some j 2 jo and let qi be a minimal central projection in C and j'

Next we introduce a very strong periodicity assumption on the inclusion data for the ng minimal central projections in (B f ) C j + ~ C B j + l

j' j

u . ladder of inclusions U

C j c B j

Tr(eCi) Tr(eqi) -=- (using 3.6.9) Hypothesis (B). We assume there is a jo 2 0 and a p 2 1 and a suitable ordering of Tr ( t i Tr(Ci

the factors in the B.'s and C.'s such that for all j 2 jo: J J

j+l is the same as that for B (i) The inclusion matrix for B. c B

Similarly for C. c Cj+l and C j+p C j+p+~ .

tions in <.A. (by 2.6.4) while Ci is the J 1 J

Let d. = min{v~)/((A!~.dj))~). Then J i J J

primitive.

(iii) The iiiclusion matrix A. for C. c B. is the same as that for Cj+p C Bj+p. J J J

Tr(e) = Tr(ez.) J = x ~ r ( e < ~ ) 2 d x ~ r ( ? & ) = d.Tr(z.). i

j i

J J

Since ;(j+'~)/$ converges to a Permn-Frobenius vector for Pi, it follows that

eigenvector for @ and similarly for the vectors ~ ( j ) . j'

at Tr(e) > d Tr(P), and Tr is finite.

Since p = [B:C] < mi the normalized trace tr on A has the Markov property: Lemma 4.3.2. Assuming hypotheses (A) and (B), tr(ex) = ~ l t r ( x ) for x E B. It follows from this and 2.6.4(c) that the weight vector of tr (i) B and C are factors and [B:C] < CO. on B.eB. = z.A. is r1 t(j). (ii) A . < B : ( 1 for all j, the inequality holding component- J J J J

J J - (ii) It follows from 2.4.l(b) and 2.6.9 that the inclusion matrix of B. c A. is of the by-component. J J

(iii) If zk is the central support of e in Ak and $. denotes the spectral radius of J form [;I for some j' A! J being the inclusion matrix of B. c z.A By the remark

Q., then for j 1 jo, J J j.

J above, the weight vector of t r on A. has the form (/T1 f(j),?(jl), so that

4) B:Cl-lt(j)AtA $-LT',(~+~P)), J tr(n-zj+tp) = ( t - [ j S' j

202 Chapter 4: Commuting aquares and subfactors

(iii) First tr(zj) = (fllt(j), AjA.?(j)), since the trace and dimension vectors on z A

1 j j are ripectively F1t(j) and A>$J, Hence

Now apply this formula to z and use that t(j) = t(j+'p)~e = $8 t(j+'~) and that? j + t ~ J J

Aj+tp = 5 to get

(4.3.2.2) tr(lzj+tp) = (t(j+tp) - gl t ( j+ tp) AJ+tpAj+tp' ! ;(j+tp))

= - gl t ( j IAtA +-t ;(j+e~)) # j j , j

Theorem 4.3.3. Assuming hypotheses (A) and (B),

[B:C] = llt(j)112/11f(j)112 = l l ~ ~ 1 1 ~ for all j 2 j,.

Proof. Fix j 2 jO and consider the formula (4.3.2.2) for t r ( l ~ ~ + ~ ~ ) . Since - l i m tr(lzj+tp) = 0 and since @ s(~+'P) converges to a strictly positive vector while e+, J t(j) - F1t(j)A!A. is a non-negative vector, we must have t(j) - F1t(j)A!A - 0.

J J J j - Therefore z - 1 by 4.3.2.1, and furthermore t(j) is a Perron-Frobenius eigenvector for

t j -

2 t A.A. with eigenvalue /3, whence llA.11 = p. Finally A. is the inclusion matrix for J J J J

Bj c Aj and /T:t(j) is the trace vector of A . consequently ~ ( j ) = F 1 t ( j ) k is a j' J Perron-Frobenius eigenvector for A .A! and

J J

Remarks.

(1) If it is known a pn'ori that B and C are factors with [B:C] < W, Wenzl can obtain the index formula assuming only periodicity of the inclusion data for (C.). J JLO'

(2) We used periodicity of the inclusion data for (Bjj10 only to obtain that B is a

factor.

§ 4.4. Examples of irreducible pairs 203

4.4. Examplea of irreducible pairs of factors of index less than 4, and a lemma of C. Skau.

We have sbown in Chapter 2 that there are connected pairs N c M of finite 2 dimensional von Neumann algebras of index 4 cos (r/q) for any integer q > 3. One of

the main results of [Jol] shows that there are pairs of factors with the same indices. In the present section, we give a construction for such pairs which has been sketched in [Jo~]. These pairs are necessarily irreducible by 3.6.2(c). (Recall that N c M is irreducible if the only elements of M which commute with N are the scalar multiples of the identity.)

We also present a theorem of C. Skau regarding irreducibility of certain subfactors. Consider a connected pair N c M of finite dimensional von Neumann algebras with

2 inclusion matrix A, set p = [M:N] = IlAll , and assume that 2 s ,O < 4. Let tr denote both the normalized Markov trace of modulus fl on this pair (see Theorem 2.7.3) and its ~ a r k o v extension to (M,eN) Set g = qeN - (l-eN), with p = 2 + q + q-l, and

consider the commuting square

of Example 4.2.10. Define inductively for each j 2 1:

(i) The conditional expectation B. -4 Bj-l, denoted by e. when viewed as an J J

2 operator on L (B.,tr). J

(ii) The algebra Bj+l = (Bj,ej), together with the Markov extension of tr from

B. to BjS1, again denoted by tr. J

(iii) The subalgebra Cj+l = alg{C.,e.) generated in Bj+l by C. and e J J J j'

These data satisfy Hypothesis (A) of section 4.3 because of Corollary 4.2.3. Before checking that these data also satisfy the periodicity Hypothesis (B), we need a preliminary proposition of independent interest.

Pro~osition 4.4.1. There ezists an endomorphism @ of B with @(B.) = Cj+l for J

j > 0, and consequently with @(B) = C. Set eo = eN and go = g. For j 2 1, set

gj = (q+l)ej - 1, so that gjgj+lgj = gjSlgjgjS1 for j 2 0. Then

for all x E Bm = B 6 J > O j. g

204 Chapter 4: Commuting squares and subfactors

Proof. Let j 2 0 and let x E B.. The formula for @(x) makes sense, because x E B - J j commutes with ek, and thus also with gk, for k 1 j+l. Observe for example that

gj-l E B. so that one has, by using the braid relations: J

On .U B., the map @ is clearly a *-endomorphism which preserves the trace and also- 510

the norm. Consequently this map extends to a (a-weakly continuous) *-endomorphism of B, denoted by @ again. As u C. is strongly dense in C, the only thing left to be

j20 J

proved is that @(B.) = Cj+l for j 2 0. J

For j = 0, this holds by definition of C1. For j 2 1 one has

and consequently, using the formula for @(gk),

= a16{Cl,gl,.. -q) = alg{Cl,el,. . .,ej) = c ~ + ~ J

as wanted. #

&%& For any complex number w of absolute value 1 and for j 1 0, set gj(w) = - (1 - e.), and for x E U B., define J J j20 J

The same argument as above shows that this defines an endomorphism \ of B, and the

map

5 4.4. Examples of irreducible pairs 205

is pointwise strongly continuous. Moreover = id and @ = @. Thus C is 4

connected to B by a continuous family of subfactors. I t would be interesting to compute the index [B : Qw(B)] as a function of w. We do it below for w = q only.

Now we may check that the Hypothesis (B) of section 3 holds for the data of the present section, with jo = 0 and a period p = 2.

First, the inclusion matrices of B. c Bj+l are alternately h t and A by Proposition J

2.4.l.b, and those of C. c Cj+l are A and nt by the Proposition above. J

Second, as N c M is a connected pair, the inclusion matrices for B. c BjS2 and J

C. C Cj+2 are primitive for j 2 0 by Lemma 1.3.2. J

= Ad(gogl..~gj-l) for j 2 1, the pair C. c B. is the image of J J

B, C B. by an inner automorphism of B so the inclusion matrices for C. c B. are J-1 J j' J J

alternatively A and h t (for j 2 0).

\ Thus the Hypothesis (B) holds.

Theorem 4.4.2. With the notation above, if P( 4 then the pair C c B of factors of type 111 is irreducible, and its index satisfies

Proof. The index value follows from Wenzl's Theorem 4.3.3, and the irreducibility - follows from 3.6.2(c). #

Remark. The construction of the pair of factors C c B with [B:C] = P still makes

sense if p = 4, but the pair need not be irreducible. For example take M = N @ Matr

Then

g 0 ~ s i 1 C (M , eo)

u u

N c M is isomorphic to

206 Chapter 4: Commuting squares and subfactors

w m Furthermore C c B is isomorphic to N @ P @ (@ Mat2) c N @ Mat @ e Mat2), so

1 (1 C' n B r Mat2.

Let now N c M be a pair of finite direct sums of finite factors, as in Chapter 3. Assume that N is of finite index in M and let tr be a Markov trace of modulus P on this pair. We consider as usual the tower

M 0 = N c M 1 = M c ... c M ~ c M ~ + ~ c .-.

the projections (ek)k21 with el = eN and the Markov extension of the trace on the finite

von Neumann algebra Moo obtained by GNS-completion of U M j20 j'

Theorem 4.4.3. (Skau's lemma). If /3 s 4 then {el,e2,. # ) ' n Mw = N.

Proof. Set = {el,e2,. .)' n M,. As N C fl is obvious, we have to show the - opposite inclusion. For each k 2 I , let Fk be the conditional expectation of Mw onto

{ek,ek+l,, . -1' Observe that FkFe = Fhn(k,e). We have to show that

F1(Mw) C N.

It is enough to show that F2(MJ c M, because this and Proposition 3.6.15 show

that FIF2(M,) c N.

Suppose we know that Fk+1(ek) E C for each k 2 1. One has then for t 2 1 and for

a,b E Me

F2(eeb) = F2Fe+l(web) = F2(aFt+l(ee)b)

= Fetl(ee)F2(ab) E F2(Mt)-

This implies F2(Me+l) c F2(ML), and this implies in turn by induction that

F2(Mw) C F2(M1) = M, so that the proposition is proved.

We still have to prove that FkSl(ek) E E. The diagram

8 4.5. More examples of irreducible pairs 207

is a commuting square by Proposition 4.2.7. As the pair {ek+l,ekf 2,- . '1' C {ek,ek+l,. . '1' is isomorphic to the pair R c R of Lemma 3.4.5, R we may write the commuting square above as

Let E denote the conditional expectation from R onto R' n R, and recall that Fk+l P is the conditional expectation from Mw onto RbnMm. As e k e R , one has

Now, if p < 4, the relative commutant R' n R is trivial and E is just the trace. (by P .6.2(~) when P < 4 and by [Jol], 55.3 when P = 4.) Thus Fk+l(ek) € C, and the proof

& Remark. The condition P s 4 is necessary for Skau's lemma. For any pair N c M of 111-factors with [M:N] > 4, we claim that

1 We will see in Section 4.7.f that there is a projection eo and a subfactor P of N such b

ki that M = (N,eo) and M is obtained by the fundamental construction for P c N. Then

k by 4.7.5,

contains a non-scalar element x. Then x E {el,e2, . .) rI hR but x N by Example

t 4.2.11.

4.5. More examples of irreducible pairs of factom, and the index value 3+3'12.

Consider a Coxeter graph I' of finite type in one of the classes A,D,E, with a bicoloration involving m black vertices and n white vertices, and with a distinguished white vertex wl. We shall associate to these data an irreducible subfactor C of the

hyperfinite factor B, of type 111 and we shall compute the index [B:C].

208 Chapter 4: Commuting squares and subfactors

In particular, if l? = E6 with the vertex wl chosen as

we shall find [B:C] = 3 + 3ll2. At the time of writing, this is the smallest known value larger than 4 of the index of an irreducible subfactor.

Let Mo denote the abelian von Neumann algebra Cn. Let M1 be a f i z e

dimensional von Neumann algebra containing. Mo such that r is the Bratteli diagram of

the inclusion Mo c MI.' As I? is connected, there is a unique normalized Markov trace tr

on the pair MO c M1. Form the tower (Mj)j10 and let (ej)jl be the usual sequence of

projections. Let M be the factor of type 111 obtained by completion of U M. with j,o J

respect to its unique positive normalized trace and let N be the subfactor of M generated by P and the e.'s.

J Let h be the Coxeter number of r and set

Since /3 < 4, Skau's Lemma 4.4.3 applies and M n N' = Mo.

Let p be the minimal projection of Mo corresponding to the vertex wl and set

(Observe that pN = pNp because p commutes with e. for each j 2 1.) Then C C B is J

a pair of factors with C' n B = p(NfnM)p = pM0p = Cp; that is C has trivial

commutant in B. Our goal is to compute [B:C]. Number the vertices of r so that the distinguished white vertex is wl. Departing

somewhat from previous practice let I? also denote the matrix of the bicolored graph I?, which has m rows and n columns. Let denote the unique Perron-Frobenius row vector for rtr, normalized so that its first coordinate (corresponding to the distinguished white vertex) is 1. Thus [ > 0, e rtI' = B e , and t1 = 1.

Let A be the Coxeter graph of type Ah-l, with a bicoloration having at feast one

white end vertex; choose one white end vertex and label it as the first white vertex. Denote also by A the matrix of the graph A, and let 5 be the Perron-Frobenius row

5 4.5. More examples of irreducible pairs

vector of A ~ A , normalized so that its first coordinate (corresponding to the chosen white end vertex) is 1.

Theorem 4.5.1. [B:C] = l15112/11e112.

Proof. Define No = N1 = C and Nj+l = {P,el,. . -,e.)' for j 2 1, so that N = m. - J j J

Since

is evidently a commuting square, so is

for all j, by 4.2.3, and induction. For each j, let C. = pN. and B. = pM..p. Then since J J J J

p E Mo C N! for all j, Proposition 4.2.4 implies that J

is a commuting square for all j. Evidently B = UE. and C = m. We will show that j j J

the inclusion data for these commuting squares are periodic with period 2 for large j. First we need to describe the Bratteli diagram for the chain (Bj)jlO. The Bratteli

t h diagram for (M ) has n vertices each of dimension 1 in the 0 floor, and alternate j 2 0 stories given by and rt. To obtain the diagram for (Bj)j20, take instead the

dimension vector $(O) = floor (that is introduce n-1 phantom vertices of

L-,

dimension 0 on the Oth floor), and again form alternate stories by and rt. Compute the dimension vectors on each floor by $(2j) = (rtr)j,dO) and = r(rtr)j$O).

21d Chapter 4: Commuting squares and subfactors

Finally erase all vertices of dimension 0 and all edges emanating from such a vertex. If to

is the maximum distance from wl to any vertex of I', then for j 2 to-1

I' if j is even

For example if I' is E6 with the distinguished white vertex at the end then the diagrams

are:

If tr is the normalized trace on M, note that Tr(+) = tr(.)/tr(p) is the unique, normalized trace on yBj, and if E. : Mj + MjVl is the tr-preserving conditional

J

expectation, then E.(B.) = Bj-l (since p E Mo) and E. I is Tr-preserving. Finally J J J Bj

for a E Bj, e.pae.p = E.(a)e.p, so {B.,e.p}" is an E.-extension of B in the J J J J J J J j'

terminology of Section 2.6. We have

and the inclusion matrix for B. c {B.,e.p}" is of the form J J J

B. il is some katrix, by 2.6.9. But if j 2 to, then = I?, and consequently

J J

5 4.5. More examples of irreducible pairs 2'1 1

the chain (B ) is in fact (isomorphic to) a tower obtained by iterating the j heo

fundamental construction. The Bratteli diagram for (C.). is the same as that for (Nj)j20, and is obtained

J ~ 2 0 from' A, the Coxeter graph of type Ah-l, in exactly the same way as that of (Bj)j20 is

obtained from I'; see section 2.9. In particular if j > h-2 then

It now follows from Lemma 4.2.4 that for j 2 jo = max{to,h-2) the "horizontal" B. B

inchion matrices are also periodic, A J-l = A j+'. Therefore by Wenzl's Cj-1 Cj+l

Theorem 4.3.3, [B:C] = ilt(j)l12/llf(j)l12. for j > jo, where t(j) and are respectively

the w&ht vectors of Tr on C. and B Now for 2j 2 jO, d2 j ) [resp. ~ ( ~ j ) ] is a J j'

t Perron-Frobenius eigenvector for A ~ A [resp. I' I'] and so is proportiopal to 5 [rap. 8. It remains only to establish the correct normalization of ~ ( ~ j ) and t (2~) . We have

so the first component of k(2j) is ~j and $(2j) = ~ j t , Similarly t(2j) = pljfi, .and thus [B:C] = ))t(2j)1)2/))g(2j)))2 = )lfi)12/))t)12. #

Pro~osition 4.5.2. The possible values of the indez in Theorem 4.5.1 are as follows:

For I' of type At ( l 2 2): sin2[kr/(e+l)]/sin2[r/(l+1)], k = 1,2, *' ,[(l+l)/2].

For I' of type Dl ( l ? 4): 2 sin2[klrj(2e-2)]/sin2[rr/(2e-2)], k = 1,2,. . , ,l-2,

For I' of type E6:

For I' oftype E7: seven values, the smallest being approzimately 7,759.

For I' of type E8: eight values, the smallest being approximately 19,48.

Proof. The calculation, based on the data of 1.4.5, 1.4.8, and 1.4.9, is straightforward, and is left to the reader.

212 Chapter 4: Commuting squares and subfactors

Remark: The only one of the values between 4 and 5 is 3 + 8 g 4,732. The values between 5 and 10 are:

[sin2 3r/lj/[sin2 r/lj for l > 7, 8cos2 r / l for e > 6, 3 + f i g 5 5 , 3 6 (D6)'

6 + 243 2 9,464 (E6),

[sin2 4=/10]/[sin2 r/lO] g 9,472 (A3),

ca. 7,759 (E7).

310 9. ii3 L 4.6. The derived towcr and the Coxctcr invariant.

The results of this section, with the exception of 4.6.3(vi), were all known to V. Jones before the inception of A. Ocncanu's work on subfactors. Nonetheless, our exposition has been strongly influenced by conversations with Ocneanu, to whom we would like to record our gratitude.

The proof given in [Jo 11 for the restrictions on the possible values of the index of a pair N C M of 111-factors proceeded by constructing the tower associated to N C M and

then examining the algebra generated by the projections ei in the construction, as in

Chapters 2 and 3. I t was of great importance that {ellea,. . .,en}' is finite dimensional

for each n > 1. Here we will describe another chain (Yk)k>O of finite dimensional

algebras associated to a pair N c M, such that Yk contains {el,e2,+. .,ek-l}', but is in

general strictly larger. I t turns out that the chain (Yk)k,O is determined by a certain

(possibly infinite) graph, called the princi~al & of the pair N c M, which is a conjugacy invariant of the pair. In case [M:N] < 4, the principal graph is a Coxeter graph of type A, D, or E whose norm is the square root of index; this provides another proof of the restrictions on the index values, as well as a conjugacy invariant fiaer than index itself; these results were announced in [Jo~] .

Consider a connected inclusion N c M of finite von Neumann algebras with finite dimensional centers, with N of finite index in M. Let

be the tower obtained by iterating the fundamental construction, as in Chapter 3, with Mk+l = (Mk,ek) for all k. Write P = [M:N], and let tr be the unique trace on UMk

k with the PMarkov property with respect to each inclusion Mk-l C Mk.

5 4.6. The derived tower 213

Definition 4.6.1. The derived tower aM/dN is the chain of algebras

C = N ' n N c N ' f l M c N ' n M 2 c . * . .

Lemma 4.6.2.

(i) 3' n Mk is Pnite dimensional for all k. k

(ii) If N and M are factors dim(N1 nMk) i [M:N] . (iii) With respect to tr, ,

is a commuting square for all k.

Proof. (i) It follows by induction on k, using condition 3.5.4(iv) that N is of finite - index in Mk for all k; hence by 3.6.2(a) N' n Mk is finite dimensional.

k (ii) If N and M are factors, then [Mk:N] = [M:N] , so the inequality follows from

3.6.2(b). (iii) Follows from 4.2.7. #

We assume henceforth that N, and M are factors and we denote N' 0 Mk by Yk,

E ~ k - l by E ~ , and the inclusion matrix for yk c Y ~ + ~ by A ~ . Since ekxek = Ek(x)ek

= E (x)ek for X E Yk (by 4.6.2(iii)), {Yk,ek)" is an E -extension of Yk, in Yk-l yk-l

the terminology of Section 2.6. Let Xk+l = (Yk,ey ) be the algebra obthined by the k-1

fundamental construction for Yk-l c Yk, with respect to tr, for k 2 1. Give Xk+l the

not necessarily normalized trace Tr defincd by the weight vector ~ l ; ( ~ - l ) , where B(") is the weight vector of t r on Yk-l, and the minimal central projections of Yk-l

correspond to those of Xk+l as in 2.4.1. By 2.6.9, ukSl(xey x ' ) = xekxl defines an k-1

injective algebra homomorphism from Xk+l into {Yk,ek)' c YkS1 with image

YkekYk; it is evident that uk+l is a *-homomorphism. We will refer to U ~ + ~ ( X ~ + ~ )

as "the old stuff" in Yk+l, because it is determined by Yk-l t Yk. It turns out that the

old stuff is an ideal, whose complementary ideal we call "the new stuff".

214 Chapter 4: Commuting squares and subfactors

Theorem 4.6.3.

(i) For k 1 and a E Xk+l, tr(%+l(a)) = Tr(a).

(ii) For k 2 1, U ~ + ~ ( X ~ + ~ ) = Yk%Yk is an ideal in Yk+l.

(iii) For k 2 1, uk+l(lX ) is the central support o f % in Yk+l. k+l

(iv) For all k, the inclusion Yk c Yk+l is connected.

(v) For all k, l l ~ ~ 1 1 ~ r #[M:N].

(vi) ("The new stuff comes only from the old new stufi") For k 2 2, i f x IE Yk+l

and X U ~ + ~ ( X ~ + ~ ) = 0, then xuk(Xk) = 0.

(vii) For all k 2 1, the following are equivalent:

(a) flk+l(xk+l) = Yk+1.

(b) Tr is normalized on Xk+l.

(c) is an eigenvector o f A ~ - ~ A ~ - ~ with eigenvalue b.

(dl Ak =

I I ~ ~ - ~ I I ~ = iMZN].

(viii) If the equivalent conditions of ( 6 ) hold for k, then they also hold for k+l.

(i) We have to show that Tr(eyk-lx) = tr(ekx) for x IE Yk Since

Weyk = Tr(ey Ek(x)) and tr(ekx) = tr(ekEk(x)), it is enough to prove the - k-1

equality for x E Yk-l, and since both x I+ Tr(ey x) and x I-+ tr(ekx) are traces on k-1

Yk-l, it suffices to prove it for x a minimal projection in Yk-l. But if x is a minimal

projection in some direct summand of Yk-l, then by 2.6.4(c), e x is a minimal Yk-l

projection in the corresponding direct summand of Xk+l, whose trace Tr(ey x) is by k-1

definition /T1tr(x) = tf(ekx).

(ii) We must show that if a,b E Yk and x IE Yk+l, then xae b E U ~ + ~ ( X ~ + ~ ) . As k

in the proof of 3.6.3, xaek = &+l(xq)ek, and by the N-N bilinearity of Ek+l,

EkSl(xwk) E Yke SO =ekb E YkekYk.

(iii) This follows from (ii) and the fact that e has central support 1 in Xk+l. yk-l

(iv) An equivalent statement is that Ak is indecomposable for all k. This is evident

for k = 1 since Yo = C. By (ii) and 2.6.9, the matrix Ak+l has the form

r has no row of zeroes; hence if Ak is indecomposable, so is Ak+l.

5 4.6. The derived tower 215

(v) If $1 denotes the dimension vector of Yj, then for fued i and for i? > 0,

dim Yi+2p = ll$(i+2p)l12 2 (~(A~A~)$~)l(i)ll~.

2 Suppose that for some i, [M:N] r llAill , and choose c > 0 such that (~A~ll'(l-c) > [M:N].

Let 3 be a Perron-Frobenius vector for h;hi (which exists due to (iv)), normalized so

that $(i) > 3 (component-by-component inequality). Then for p > 1

dim Yi+2p , l l(~:~~)~:11~ = 11$114P11a12, whence

(dim Y )l/i+2p 2 [f$]1ii+2p.

1 1 ~ ~ 1 1 ~

Since the right side converges to 1 as p increases, it follows that for some k,

(dim yk)lIk 2 (1-r)llhi(12 > [M:N]

which contradicts 4.6.2(ii). (vi) Because of (ii) we can suppose x is a central projection in YkS1; then xek = 0

impliea xek-l = ek-lxekek-l = 0, so that also xaek-lb = 0 for all a,b E Yk.

(vii) Conditions (a) and (d) are equivalent since the inclusion matrix for Yk C XkS1

is A&, and (a) is equivalent to (b) by (i). If (d) (and thus (b)) hold, then the vector of

tr on Y ~ + ~ is / ~ l ; ( ~ - l ) and

Thus (d) implies (c). I£ (c) holds, then l l ~ ~ - ~ ) ~ > fix and since tbe opposite inequality is

; true by (v), we have that (c) implies (el. If (e) holds, but not (d), than hk is a non- r :

negative matrix of the form I , with , # 0, so , l ~ ~ 1 1 ~ > 1 1 ~ ~ - ~ 1 1 ~ = [M:N],

?, contradicting (v).

Chapter 4: Commuting squares and subfactors

(viii) This follows from (d) and (e) of (vii) together with IlAll = 1 1 ~ ~ 1 1 . #

Theorem 4.6.3 gives an interesting qualitative picture.of the Bratteli diagram for the derived tower aM/BN. One is led to the following concepts which have been emphasized by Ocneanu:

Definition 4.6.4. A pair N c M is said to have finite d e d if there is a k for which

the conditions of 4.6.3(viii) hold. In this case the smallest such k is called the &p& of N c M.

Definition 4.6.5. The princi~al & of N c M is the bipartite multigraph constructed as follows: On the Bratteli diagram of the derived tower dM/ON delete on each floor the vertices belonging to the old stuff, and the edges emanating from these vertices.

Since the new stuff is connected only to the old new stuff, the resulting graph r is connected. The principal graph r has a distinguished vertex *, the unique vertex on floor 0, and the distance of any vertex from * is the number of its floor. The pair N c M has finite depth if, and only if, I? is finite, in which case the depth of N c M is the maximum distance of any vertex from *. The Bratteli diagram of aM/aN can be reconstructed from r (given together with the distinguished vertex *).

For purposes of illustration, let us give an example of what might be the Bratteli

diagram of aM/aN and the principal graph for a pair N c M of finite depth. (We are not claiming that this example actually occurs; this is a much more delicate question!)

$4.6. The derived tower 217

The pair N c M would have depth 4 according to our conventions. The Bratteli diagram of aM/aN for a finite depth pair N c M will always exhibit a growth in complexity up to a certain level, after which the remaining structure is obtained by reflecting. Note that if N c M has depth k, then the Bratteli diagram for YkWl C Yk is

isomorphic to the principal graph. We can now record the following consequence of Theorem 4.6.3 and the work of

Chapter 1. ,

Corollarv 4.6.6. Suppose N c M is a pair of 111-factors with [M:N] < 4. Then 2 (a) [M:N] = 4 cos s /h for some h ? 3.

(b) The depth of N c M is no greater than h-2. (c) The principal graph o f N c M is a Cozeter graph of type A, D, or E, whose

norm is [ M : N ] ~ / ~ .

Proof. If N c M were not of finite depth, then one would have a sequence Ai of non-

negative integer matrices with 11$112 L l l ~ ~ + ~ l l ~ r [M:N] < 4, which is impossible by the

classification of Chapter 1. If k is the depth of N c M, then for j 2 k-1 the graph of A j 2

is isomorphic to the principal graph r, and 1 1 ~ . 1 1 ~ J = llrll = [M:N] < 4. Therefore, by 2

Chapter 1, is a Coxeter graph of type A, D, or E and [M:N] = 4 cos s/h, where h is the Coxeter number. Furthermore k r diam(r) s h-2. #

This completes the proof of the restriction on index values outlined in [JoS].

Corollarv 4.6.7. Suppose N c M is a pair of 111-factors with [M:N] = 4.

(a) If N c M is of finite depth, then the principal'graph r is a completed Cozeter

graph o f type A, D, or E, i.e., one ojthe graphs in Table 1.4.6. (b) If N c M is of infinite depth, then r is one ojthe following:

Chapter 4: Commuting squares and subfactors

(doubly injnite linear graph)

(end vertez at distance n t o rn *)

Proof. (a) Follows from 4.6.3 and 1.4.3. - (b) Let rk denote the subgaph of I' containing vertices of distance no greater than

k from *; then rk is also isomorphic to the Bratteli diagram for Yk-l C Yk, 80

llrk12 < 4 for all k. Thus each rk is a Coxeter graph of type A, D, or E; furthermore

is obtained from rk by addition of one or more vertices at distance k+l from *. It is easy to verify that the only possibilities are those listed. #

We will see in Section 4.7 that subfactors of finite depth and of infinite depth do occur. In particular all possibilities allowed by 4.6.7 do actually occur except possibly A,,, and

D,,,; we will also see why A is also labelled U. m,m

Finally, we cannot resist saying a few words about the truly exciting results of Ocneanu, who has added to the structure described here something we have completely neglected in our treatment, namely the involutions Ji coming from each basic

construction in the tower Mi. He shows that they combine to define an endomorphism of

aM/aN and, together with the eils and the principal graph, seem to complstely determine

N c M in many cases. In particular, he can show that there are osly finitely many subfactors of the hypefinite 111-factor (of index < 4) for each Coxeter graph, up to

conjugation by automorphisms, and he determines which Coxeter graphs are allowed,

5 4.7. Examples of derived towers

4.7. Examples of derived towera.

4.7.a. Finite group actions. We shall analyze the derived tower for a pair N c M,

when N is the fixed point algebra M~ for an outer action of a finite group G on a 111

factor M, and also when M is the crossed product N r G of N by an outer action of a finite group G on N.

(i) N = M ~ . In this case we know from [Jol], that (M,eN) = M2 can be identified

with the crossed product M r G, so that Y2 = M2 n N' contains the group (von

Neumann) algebra CG. The inclusion matrix for C C CG is [no = l,nl,. - ',nk], where

the ni are the degrees of the irreducible representations of G. Thus 2 2

llAlll 2 1 ni = I GI. On the other hand, by [Jol], [M:N] = IG I . Hence by 4.6.3,

Y2 = CG and N c M is of depth 2. Note that the derived tower is independent of M or

the action of G; for example, in case G = S3 the Bratteli diagram for OM/aN is

Remark. According to Ocneanu, depth = 2 ,and N' n M = C characterizes fixed point algebras of outer actions of finite dimensional Kac algebras.

(ii) M = N r G. In this case (M,eN) is known to be the crossed product of M by

the "dual action" of G (see [NT]). To be more concrete, denote by u the canonical g

2 unitaries of the crossed product. On L (M), each of the projections e = u e u* onto g g N g

the closure of Nu = u N is evidently in (M,eN) n N' = Y2. By the same reasoning as g g

in case (i), the (commutative) algebra which they generate is equal to Y2, and one always

has the following Bratteli diagram for aM/aN .

Chapter 4: Commuting squares and subfactors 3 4.7. Examples of,derived towers 221

Remarks. (1) Ocneanu's endomorphism allows one to reconstruct the multiplication table for G, once the elements have been put into bijection with the factors on the third line!

(2) By choosing G = 2/22 one obtains the Coxeter graph A3 as the principal graph,

and G = 2/32 gives D4 (for either the fixed point algebra or crossed product case). One

can also check that the pair N . Fi2 c N . Fi3 or MG3 c Mb2 has principal graph As.

(6, denotes the symmetric group.)

4.7.b. The An Coxeter graphs. Let {eiIB0 be a sequence of self-adjoint

projections satisfying the relations

eieialei = ~ l e ~

e . e . = e e for li-jl22, I J j i

for some P > 0. Realize the hyperfinite 111 factor R as R = {l,eo,el, ...)' and write

RP for {P,el,e2, ...) " . We have computed that [R:R ] = P (Proposition 3.4.4). 2

P When P = 4 cos ir/n for some n 1 3, it follows easily from the proof of Skau's

lemma (4.4.3) that the principal graph of R c R is the Coxeter graph Anql. In fact, P write e-l,e-2,... for the projections occurring in the tower construction for R c R, so

P that Mi = {l,e-i+,l ,..., el,e2 ,... )" (i 1 0). The proof of Skau's lemma shows that

N' r l Mi = {l,e-i+l,...,e-l)', and we saw in Section 2.9 that the chain (N' n Mi)i2l has

the appropriate Bratteli diagram. (The statement of Skau's lemma does not apply since {e-i+l,...,eo,el)' is not isomorphic to the result of the fundamental construction for

unless i n-3.)

Since Skau's lemma also is valid for P = 4, the same argument shows that the principal graph for the pair R c R when P = 4 is P

4.7.c. A eeneral method. The following result is useful as it allows the computation of the derived tower in many examples coming from group aictions. A more powerful result has been discovered and exploited independently by A. Wassermann [Wa].

Lemma 4.7.1. Let N c M c P be von Neumann algebras, cp a faithjil normal state o f P, e a projection in N' n P, and G a group of automorphisms of P preserving N, MI Q, and e. Suppose:

(i) eMe = Ne, (ii) dxe) = cp(e)cp(x) V x E M, (iii) tr = cp is a trace, I P

(iv) NG c MG c pG are 111 factors with [ M ~ : N ~ ] = [ P ~ : M ~ ] = dB)-'. G Then there is an isomorphism @ of ( M , e ) onto pG such that @(eNG) = e

NG G and @(x) = x for all x E M . ..

m. Let us first show that we = E G(x)e for x E MG. By hypothesis, exe = ae N

for some a E N, and for all g E G we have ae = exe = g(we) = g ( x ) = g(a)e. But then de)cp((a-g(a))*(a-g(a))) = rp((a-g(a))*(a-g(a))e) = 0, so a E NG since cp is faithful. Also if y E N ~ , then cp(e)tr(yx) = tr(yxe) = tr(eyxe) = tr(yexe) = tr(yae) = cp(e)tr(ya), so a = E G ( ~ ) .

N Now by 2.6.9 and 3.6.4 there is a *-algebra isomorphism @ of (M G G . eNG) Into

G such that @(xe Gy) = xey for x,y E M . The map @ is trace preserving (since N

tr(e G) = 1 ~ ~ : ~ ~ l - l = tr(e)), so normal and unital. We need only show that is N

G surjective. But the image of @ is a 111 factor containing M (since @ is unital) as'a

G G G subfactor of index tr(e)-' = [P- :M 1, so the image is P . #

222 Chapter 4: Commuting squares and subfactors 3 4.7. Examples of derived towers 223

This lemma will be used repeatedly in 4.7.d and 4.7.f to calculate derived towers by calculating it in a simple situation aqi then passing to the fixed point algebra of some group action. See also [PP3]. Wassermann calls the lemma the invariance principle, since in cases where [M:N] makes sense it should also be rp(e)-'.

4.7.d. Some examales of derived towers for index 4 subfactors. We realize the hyperfinite 111 factor R as the completion, with respect to the unique tracial state tr,

m of the infinite tensor product of Mat2(C), R = (@ Mat2(C))-. Any closed subgroup G of

m SU(2) acts on @ Mat2(C) by the infinite tensor product of its action by conjugation on

Mat2@). The action preserves the trace, so extends to an action on R. The group G

acts in the same way on R @ Mat2(C), so one has a commuting square

Now RG and R ~ ~ ( ~ ) contains in its unitary group a copy of the infinite symmetric group Gw, khich acts ergodieally on R. So (RG)' n R = C l , and, in

G particular, R is a factor. The projection

1 el = 21R @ {ell @ 22 - el2 @ e21- e21@ el2 + e22 @ ell)

satisfies elxel = ER(x)el and (R @ Mat2(C))el(R @ Mat2(C)) = R @ Mat2(C) @ Mat2(C).

Hence, using 2.6.9' and 3.6.4, the result of the fundamental construction for R c R @ Mat2(C) can be identified with R @ Mat2(C) @ Mat2@), the projection eR being

identified with el. Hence the tower for R c R @ Mat2&) is identified with

The projections e i being. identified with IE @ d-l(e1), with o the shift endomorphism w

of @ Mat2(C). Note that the projections ei .are SU(2)-invariant.

In case G is finite, it follows from the multiplicativity of the index that [(R @ ~ a t ) ( c ) ) ~ : RG] = 4, and then 4.7.1 gives an isomorphism of

((R @ ~ a t ~ ( C ) ) ~ , e ~ ~ ) onto (R @ Mat2(C)) @ ~ a t ~ ( C ) ) ~ taking eRG to el. If G is

infinite, we have to do a little more work to reach the same conclusion. Identify R (G-equivariantly) with Rg @ Mat2(C), where Ro g R. Set

1 eo = lRO @ {ell @ en - e12 @ eZl - eZ1 @ el, + e22 @ ell). Thus (R @ Mat2(C))eo

G = Reg (by 3.6.3 or by direct computation). If x E (R @ Mat2(C)) , then there is an

xO E R such that xeo = xoeo, and

so (R @ ~ a t ~ ( C ) ) ~ e ~ = RGeO. Therefore,

the last equality because (R @ ~ a t ~ ( C ) ) ~ is a factor. If e denotes the orthogonal G 2 G projection of L ~ ( ( R @ Mat2(C)) ) onto L (R ), then e eo e = $ e. We claim that also

eo e eo = eo. Because RG eo RG = (R @ at^(^))^, it suffices to check this equality on G vectors xeoyO, where x,y E RG and 0 is the trace vector in L ~ ( ( R @ Mat2(C)) ). But

It follows that eo N e in ( ( R e ~ a t ~ ( C ) ) ~ , e ) and since e is a finite projection by

3.6.1(v), it follows that eo is finite in ((R @ ~ a t ~ ( C ) ) O , e ) . But 1 is the sum of finitely

many projections, each equivalent to a subprojection of eo, so ((R @ at^(^))^, e) is a

finite factor. Therefore [(R @ ~ a t ~ ( C ) ) ~ : RG] = tr(e)-l = tr(eo)-l = 4.

We can now conclude from 4.7.1 and induction that the tower for RG C (R @ ~ a t ~ ( c ) ) ~ is

224 Chapter 4: Commuting squares and subfactors

k k and since ( R ~ ) ' n R = lC, we have ( R ~ ) ' n R @ (@ Mat2(()) = IR @ (@ Mat2(C)) and

k k ( R ~ ) ' n (R @ (@ ~ a t ~ ( C ) ) ~ = lR @ (@ at^(^))^.

Thus we have identified the derived tower for R~ c ( R e at^(^))^ with the sequence of

finite dimensional von Neumann algebras Yo = C, Y1 = ( ~ a t ~ ( c ) ) ~ , Y2 =

[Mat2(C) @ at,(^)]^, etc.

It is fairly evident that these algebras are just the cornmutants of the tensor powers of the fundamental representation of G on C2 determined by its inclusion in SU(2). We

can now use the McKay correspondence between finite subgroups of SU(2) and affine Coxeter graphs (see [Slo] or [Jo 41) to calculate the Bratteli diagrams or principal graphs when G is finite. The correspondence is as follows. Let I? be the matrix with rows and columns indexed by irreducible representations of G, whose (i,j)-entry is the multiplicity

2 of j in the tensor product of i with the fundamental representation of G on C . Then r is the adjacency matrix of an afSne Coxeter graph of type A:), DA1), EA1), EP), or

EQ~). (In fact, A:) corresponds to a cyclic group, DA1) a dihedral group, ~ t ) the

tetrahedron group, E P ) the cube group, and Ek1) the icosohedron group.) The method

of constructing the Bratteli diagram is clear from the representation-theoretic interpretation of I?: use r (= r t ) as the inclusion matrix and start with the dimension vector [1,0,0,. . .lt on the Oth floor (as for example in the calculation of Bratteli diagrams in Section 4.5). The resulting principal graphs are AL1) (n 2 2), DA1) (n 2 4),

and Eil), Eil), EQ~) (see Table 1.4.6). G For G the maximal torus T, the principal graph for R c (R @ at)(^))^ is the

graph A of Corollary 4.6.7. For G the infinite dihedral group Dm the graph is Dm, m,m

and for G = SU(2), thegraph is Am.

Of course, the method we have described here is quite general and also applies in dimension greater than 2.

4.7.e i'he tunnel. construction. We describe the tunnel construction of V. Jones ([Jo 11) for a pair N c M of finite factors with finite index. This is a sort of mirror image inside N of the tower construction of Chapter 3. The essential observation is that there is

2 a representation of M on L (N,tr) extending the standard representation of N, 2 although not a canonical one. Start with the representation of M on H = L (M,tr) and

choose any projection p € M' with trM, (p) = [M:N]-l. Then dimN(pH) = 1 by

3.2.5(e), so that the N-mo$ule pH is isomorphic to ~ ~ ( ~ , t r ) by 3.2.4(a); if

5 4,7. Examples of derived towers 225

2 u : L (N,tr) -I pH is a unitary N-module map then x I+ u*xl pH u is the desired . - representation.

2 Represent the pair N c M on L (N,tr) and let J denote the conjugate linear 2 isometry of L (N,tr) extending the map x x* on N. Write M-l for JM'J and let

e,, be the projection of ~ % ( ~ , t r ) onto ~ ~ ( ~ - ~ , t r ) . Then [N:M-l] = [N' :MI] = [N:M],

and JMJ = MIl = (N1,e0), SO M = JMLIJ = (N,eo). That is, the pair N c M is the

result of the fundamental construction for the pair M-l C N.

Iterating this construction, one obtains a decreasing "tunnel1' of 111 factors

along with projections {eo,e-l,e-2,' a ' ) such that (M-k,ek) = M-k+l. The projections

{ei : i 5 0) satisfy the usual relations with ,O = [M:N].

The tunnel construction has been exploited systematically by Ocneanu in his classification of subfactors.

4.7.f The derived tower for R 3 R, when P > 4. In this section we will compute P

the derived tower for R 3 RD when ,O > 4; compare Section 4.7.b for the cases P < 4

and /3 = 4. The computation uses a beautiful representation of the eils due to Pimsner

and Popa [PP 11 as well as a theorem of Popa on the tunnel construction for certain pairs of factors. We begin with a preliminary lemma from [Jo 11.

Lemma 4.7.2. Suppose M is a 111 factor containing a projection f such that there is

an isomorphism 8 : fMf -4 (1-f)M(l-f). Let N = {x+B(x) : x E fMf). Then N is a subfactor with [M:N] = tr(f)-l + tr(l-f)-l = tr(f)-ltr(1-f)-'.

Proof. The algebra N is isomorphic to fMf, so is a subfactor. Since f € N' n M, - [M:N] = dimN(H) = dimN@) + dimN((l-f)H), where H = L ~ ( M ) . Since Nf = fMf,

and similarly for 1-f. #

The following theorem is due to Popa, who has kindly showed us the proof and allowed us to present it here.

Chapter 4: Commuting squares and subfactors 5 4.7. Examples of derived towers

Theorem 4.7.3 (Popa). Suppose M is a 111 factor containing a projection f, with

tr(f) < $, such that then is an isomorphism 0: f 1 - 1 - f ) . Let

N = {x+B(x) : x E N f ) . (By 4.7.2, [M:N] = tr(f)-' + tr(1-f)-' := P; note that P > 4.)'

Let {eo,e-l,e-2,+ - ') be projections in the tunnel construction for the pair N c M, set

R = {eo ,e - l ,~~~) ' and RP= {e-l,e-2,.,.)'. Then

(1) f E R b n R ,

(2) fRf = R f and (1-f)R(l-f) = R (1 f). P P -

- 1 Proof. Write t = tr(f), so 8 = t-I + (1-t)- . Since EN(eo) = P1,

is a commuting square, and ENER = EREN = E Ri

We first claim that Rb n R ; E l . Let p be any projection with p 5 f and 1

tr(p) = = t(l-t), and set q = 1-f-8(p), so tr(q) = r1 as well. Let v be a partial isometry in M such that v*v = q and vv* = p and define

e = (I-t)p + tq + ATJTfJ (v+v*).

Then e is a projection. One can check that for any x E M,

1 In particular, EN(e) = r 1. By [PP 11, there is an automorphism of M leaving N fixed

pointwise and carrying eo to e, so we can assume eo = e.

Since f E M n N' , ER(f) E R n (NnR)' = R n Rb . Suppose that ER(f) is a scalar,

that is, ER(f) = t l . Then

1 which is impossible since t < 2 Thus ER(f) E Rb n R\C1.

Now let fo be any projection in R' n R with 0 < tr(fo) 5 $. Ultimately we will P show that f = fo. Since

1 1 1 1 f + 1-t = P = [R:R$ + (by the proof of 3.6.2),

0 ' - 0

it follows by calculus that tr(fo) ) t. Assume tr(fo) > t. If g := fo A (1-f) = 0, then

fo = fo - fo "1-f) N fo V (I-f)-(l-f) 5 f, so tr(fo) < t, contradicting our assumption.

Since EN(fo) E N n R n Rb = RP n Rb = El, it follows that EN(fo) = tr(fo). Thus

$1 Z tr(fo)l = EN(fo) Z EN(g) (since fo 2 g)

= (1-t)(rl(g)+g) (since (I-f) > g).

1 Since g # 0, this implies 3 ) 1-t, contradicting the definition of t. It follows that

tr(fo) = t.

Define fl = f - f A fo and f2 = s((1-f)fo(l-f)), where s(-) denote support. Remark

that s((f0-fAf0) (1-f) (fo-fAf0)) = fo-fAfo

because (fo-fAfo) E; f = 0. Hence

Write h = B(fhf0). We have

t l = EN(fo)

= ~ ~ ( f f ~ f + (n-f)fo(n-f))

z EN(fAfo + (I-f)fo(l-f) since EN((l-f)fof) = 0

Chapter 4: Commuting squares and subfactors

= t(fAfo+h) + (l-t)(o-l(l-f)fo(l-f) + (I-f)fo(l-f))

> th + (1-t)(P-f)fo(n-f).

Hence th 2 th + (1-t)h(l-f)fo(P-f)h,

so h(1-f)fo(l-f)h = 0. It follows that

so B(fAfo) 5 I-f-f2. On the other hand, multiplying the equation

by I-f-f2 gives

which implies that B(fohf) > (I-f-f2) Ilence O(fohf) = P-f-fZ

Finally we compute

Solving gives (1-2t) tr(fohf) = (1-2t)t, or, since 1-2t > 0, tr(fohf) = t = tr(f) = tr(fo).

It follows that f = fAfO = fo and f E R n Rb , as was to be shown.

If x E R, then t-lEN(xfx) = fxf + O(fxf) lies in R fl N = R p , and

f d = t-lEN(cfc)f E Rp f. Thus fRf = R f. Similarly (I-f)R(l-f) = RP (I-f). # P

Corollarv 4.7.4. [Jol] Let P > 4, and let R 3 R be the pair of factors in Proposition P 3.4.4. Then R n Rb ; C l .

&& Let M be the hyperfinite 111 factor, and let f E M be a projection of trace

t = $ - , so that t-'(1-t)" = /3. Then there is an isomorphism 8 : fMf - (I-f)M(l-f). Define the factor N and the projections {eo,e-l,e_2,e. .) u in Theorem

§ 4.7. Examples of derived ,towers 229

4.7.3. Then f E {eo,e-l,e-2,. .}' n {e-l,e-2,-. .)'. But the the pair R 3 Rp is

isomorphic the the pair {eo,e-l,e-2,. . }' 2 {e-l,e-2,. . '1'. #

We now describe a representation of the ei's due to Pirnsner and Popa [PP 11. Let A

be the infinite tensor product of Mat2(C), that is, the inductive limit of the algebras n @ Mat2(C), each imbedded into the next via x I---, x @ 1. It is well known that A has a 1

unique c*-norm and that K, the c*-completion of A, is asimple c*-algebra. Fix /3 > 4 and let X E ]0,1[ satisfy 2+X+Xe1 = P. Define a state r] on Matp(C) by

q(x) = tr [[i !]XI a n d let 9 be the corresponding Powers state [PI on ?i defined s ,I

on A by q(xl@ x2@, . .@xu) = ll q(xi) Let N be the weak closure of K in the i=l

GNS-representation corresponding to 9, the Powers factor of type HIX.

There is an obvious shift endomorphism on A,

which preserves 9 and so extends to N. Define el E Mat2(C) @ Mat2(C) by

under an appropriate identification of hIat2(C) @ Mat2(C) with n;l&t4(C). Let

ei = &-'(el) for i 2. It is a bit tedious but straightforward to check that {ei : i 2 1)

satisfy p eiei+lei = ei,

eiej = ejei, 1 i-j 1 > 2,

and that cp al is the Markov trace of modulus P. The circle group acts on I g{1,e1,e2,...l

A by the infinite tensor product of its action z I-+ Ad [" ,-I O I on Mat2(C); this action

preserves 9 , and extends to an action on N. One checks that the projections ei are

T-invariant. Equivalently (since the T-action is actually the modular automorphism group

230 Chapter 4: Commuting squares and subfact~rs

for (o) the eils are in the normalizer of (o, (o(eiy) = dyei) for all y E N. We shall

apply Lemma 4.7.1 with M = N @ Mat2(C), P = N @ Mat2(C) @ Mat2(C), G = T, , and

the state (o @ q @ q on P, in order to compute the derived tower for R c R. The key P result is the following, from [PPl].

Theorem 4.7.5. With the notation above, {el,e2,. .)' k the $zed-point algebra N T

for the T action on N.

/

A proof of this result using the ConnesStormer relative entropy is in [PPl]. We shall give a simplified proof due to Popa which is based on his Theorem 4.7.3.

First we observe that {l,el,e2,. . .) together with the projection f = ell E Mat2(C) T 1 X generate N . In fact felf = ell @ e22 and (I-f)el(l-f) = e22 @ ell, so that

2 alg{l,f,el) contains the self-adjoint unitary u = e i j @e. as well as the diagonal

J ,i i , j=l

algebra {dl @ d2 : di is a diagonal 2-by-2 matrix), and contains in particular 4 f ) =

1 @ e l l By induction, alg{l,f,el,e2,. . ,) contains the infinite tensor product of the 2-by-2 k diagonal algebra, as well as {cr (u): k?O). It is a well known fact of invariant theory that,

m r

2 on the Fock space $ (i c2) over C , the mmmutant of the direct sum of the tensor k=O

powers of the representation a I+ Z-l of T on t2 is the algebra generated by the [" I permutations {$(a) : k?O ) together with the infinite diagonal algebra. This implies

T that 5 = ~ * { l , f , e ~ , e ~ , - . -) and N = {l,f,el,e2,. . .)'. Therefore it suffices to prove

that f E {el,e2,. + .)". T T T T The equality N = {l,f,el,e2,. . . )' implies that fN f = a(N )f and (1-f)N (I-f) =

T T so that 8: xf - x(l-f) (x E or(N )) is an isomorphism of fN f onto , T and o(.N ) = {x + B(x) : x E fNTf ). Thus the pair 4 ~ ' ) c N' is of the

type discussed in Lemma 4.7.2, and therefore [ N ? ~ ( N ~ ) ] = tr(f)-l + tr(1-f)-l = 2 + X + A-l = p.

We can now vexify that the conditions of Lemma 4.7.1 are satisfied by the algebras 2 o (N) c a(N) c N, the state cp, the projection el, and the group T, and therefore we can

T T identify the inclusion 4 N ) c N as coming from the fundamental construction applied to 2 T a (N ) c a ( ~ ~ ) , the projection . el being identified with e T . Applying the

0 (N k T endomorphism o repeatedly, we have that o (N ) = ($+'(N~), ek+l) is isomorphic

to the result of the fundamental construction for d + 2 ( ~ T ) c ak+'(~'), and the

5 4.7. Examples of derived towers 231

T T projections {el,e2,. .) axe those of a tunnel construction for a(N ) c N . Therefore the

conclusion f E {el,e2,. . '1' follows from Popa's theorem 4.7.3. #

There is no further difficulty in computing the entire derived tower for R c R. Using P 4.7.1 and 4.7.5, the tower for the pair R c R can be identified with P

T It is well known that (N )' n N = C1. In fact, the infinite symmetric group em acts

ergodically on N (by permutation of the tensorands), and this action is implemented by T k unitaries in N ; the image of in N' is the group generated by { o (u) : k?O ).

k k T k T Therefore (N')' n (N @ (@ Mat2(C)) = 1 @ (@ MatZ(C)) and (N )' rl (N @ (@ Mat2(C)) =

k 1 @ (@ at^(^))^. That is, the derived tower is precisely the sequence of fixed-point,

k algebras for €he tensor product action of T on @ Mat2(C) (k>O). The Bratteli diagram is

Pascal's triangle [Bra]:

and the principal graph is

Note that it differs from the graph for P = 4 (Section 4.7.b).

APPENDIX I Classi£ication of Coxeter graphs with spectral radius

just beyond the Kronecker range

1.1. The reaulta.

The first purpose of this appendix is to complement section 1.4 and to classify finite connected graphs satisfying 2 < llrll < Am where we set

The results are those of Cvetkovi6, Doob and Gutman [CDG]. Then we also classify Coxeter e ra~hs with norms satisfving the same ineaualitv.

The following graphs enter the classification. First, the T 's already introduced P,W

in Section 1.4. Next, for an integer m 2 3, we set ~-,1

As usually for Coxeter graphs, edges to which the associated integer is 3 are left unmarked, so that

Given integers p, q, r, m, m' with 2 < p < q 2 3, r 2 5 and m,ml > 3, we also define the H-shaped Coxeter tree

with p+q+r-3 vertices. The constant Am makes its first appearance in a result of Hoffman (Proposition 3.7 of

[HofI):

$1.1. The results 233

Pro~osition 1.1.1. Let r be a jnite connected graph which is not a cycle and which contains a cycle. Then llrll > A,. Moreover Am is the largest constant for which this

holds.

One important result about the set E of norms of graphs (see section 1.5) is also

essentially due to Hoffman [Hofj, and can be stated in terms of a sequence (A ) of q22

numbers defined as follows. Let v be the largest real root of the polynomial 9

We set

If vm = $(5lI2+1) is the golden mean, namely, the positive root of &I, it follows by

induction on q from the identity L (v) - L (v) = vq-'(2-v-1) that q+l q

As Am = vY2 + vi1I2 one has also

Theorem 1.1.2. The accumulation points of E n [O,XJ are precisely the X ' s for Q

q = 2,3,. . . ,m and X3 is man'mal among real numbers c such that E n [O,c] is well

ordered. Moreover, (i) llAkll and llDkll both increase strictly with k and converge to X2 = 2. One has

E n ]2,d[ = 4 where d = !IT2 711 is the square root of the largest root o f the polynomial 9 9

5 4 p - 9p + 27p3 - 31p2 + 12p - 1.

(ii) For q > 3 and p with 2 5 p 5 q:

llT 11 increases strictly with r and converges to X 2,q,r q'

llT3,3,rII increases strictly with r and converges to Am,

llH 11 decreases strictly with r and converges to X p,q,r Q'

234 Appendix I: Classification of Coxeter graphs

To state the corresponding classification of Coxeter graphs, we set

F$+ 0- 2 40-4,-

Fk 0--0--0--0-. 4 . -4- k vertices

Hk 0---0~-. 5 . e - 0 4 k vertices

Theorem 1.1.3. Let I' be a jnite connected Coxeter graph with 2 < IlI'll i Am. Then I'

is one of the following.

F:+ with norm approximately 2,053;

Fk with k 1 6; the norms llFkll increase and tend to Am;

Hk with k 5; the nonns JIHkJJ increase and tend to Am;

T2,q,r with 3 i q 5 r, and r t 7 i f q = 3, and r 2 5 if q = $ the norms increase

and tend to X . 9'

T2,q,r(4) with 3 < q 5 r and r large enough; the n o m s decrease and tend to X . 4'

T3,3,r with r 1 4; the n o r m increase and tend to Am;

T3,4,4 with norm apf;roximately 2,053;

Hp,q,r(m,m') with 2 5 P i q 2 3 and r large enough, with m,ml E {3,4} and

m = 3 i f p = 2; the n o m s decrease and tend to X q'

The classification of matrices X E Mfin(E) with llXll i Am (up to pseudo+quivalence)

follows from Theorems 1.1.3 and 1.1.3.

Remarks. 2 (1) The number llF:+112 = ]IT3 4112 is the largest raot of p3 - 6p + 8p - 2.

9 I

L v L4 ( "1 (2) The polynomials e, L3(u), are irreducible in Ulu]. We have not

L2 ( 4 checked whether and LZp+l(u) are irreducible for larger p's.

(3) The list of theorem 1.1.3 is neither including nor included in other lists of Coxeter graphs "just larger" than those of theorem 1.1.3, such as the list of hyperbolic graphs in [Che] and [Kos], or the list of trees with Lorentzian associated quadratic form in [Mxl].

(4) The consideration of infinite graphs brings no surprise [Tor].

The proofs which follow are very elementary, though rather tedious. Many of the partial results have been checked numerically with a computer. It is a pleasure to thank F. Ronga, G. Wanner, and E. Hairer for crucial help with the computations.

5 1.2. Characteristic polynomials

1.2. Computations of characteristic polynomials for ordinary graphs.

Most computations below are by induction, based on the following.

Lemma 1.2.1. Let I' be a graph and let v be a vertez of I?. Let I'- be the graph

obtained from I? by deleting v and all edges ending at v; let I'+ be the graph obtained

from I? by adding a vertez v+ and an edge between v and v+. Denote by P, P- and

P+ the characteristic polynomials of I', I'- and I'+. Then

P+(X) = XP(X) -P-(A).

w. By standard expansion ofdeterminants; more similar lemmas in [Schl]. #

One example is provided by

Let P E U[A] be the characteristic polynomial of some graph I'. For our computations, it helps to consider larger rings

with

3 3 For example, we write P(A) = p + pw3-2 to mean P(A) = X -3X-2. The function

is strictly increasing and bijective. When P E U[A] is given by a function f of ,u, this has the following consequence: for po E [l,m[, one has f ( b ) = 0 if and only if P(XO) = 0,

with A. = po + pi1; and llI'll = Xo if and only if po is the largest root of f. This will

be used constantly below.

236 Appendix I: Classification of Coxeter graphs

A For any integer k 2 1, we denote by P k the characteristic polynomial ~f Ak.

Promsition 1.2.2. One h m P;(A) = (yp-l)-l(pl'+l-pcckkl) =

~ ( X - ~ C O S ( T ~ / ( ~ + ~ ) ) ) and BAkll = 2m(d(k+l)) . li jik

m. From the previous lemma one has the difference equation /

P:+~(A) - AP:(A) + PEl(A) = 0

which holds also for k = 1 if p i ( \ ) = 1. The indicial equation being $ - Ap + 1 = 0,

k the general solution of the difference equation is Pk(A) = Cp + ~ p - ~ , where C and D

are constants in ~[p,p-l] independent of k. Adjustment of C and D to fit P; and

P: provide the formula with p's. Roots of P;(A) = fk(p2-l)-1(p2k+2-~) are given

by p2 = exp(2ilrj/(k+l)), namely by

for j = 1,. . . ,k. The largest of these is 2cos(s/(k+l)). #

Observe that the roots of P: are 2cos(mj/h) where h = k+l is the Coxeter

number of Ak and where ml,. .,mk are its Coxeter exponents 1,2,. . .,k.

For any integer k 2 2, we denote by P t 7 1 the characteristic polynomial of AL')

(the cycle with k+l vertices) by AP-') the graph with k+2 vertices

and by P:,'~' its characteristic polynomial.

Lemma 1.2.3. One has P:ll(X) = ,$+I+ p-k-l- 2 and I I A ~ ~ ) I I = 2. One has

5 1.2. Characteristic polynomials

so that I I A L ~ ~ ~ ) ~ ~ decreases strictly with k and converges to Am = (51/2+2)1/2.

&&. Expansion of the determinant defining pf 'l along the first column gives A P ~ ~ ' ( A ) - A P ~ ( A ) as a sum of two k-by-k determinants; expansions of these along their

first lines give

and the formula for P The largest root of pl\'l is given by p = 1, namely by

X = 2 (see also lemma 1.4.1). The formula for P : ~ ~ ~ ~ follows by lemma 1.2.1. Set pm = (251/2+1))1/2 = v1l2 m

4 2 (with vm as in 1.1). If p = pm one has p -p -1 = 0 and

Consequently the left-hand side has a root larger than pm, and 1 1 ~ P ~ l ) l l > Am = pm+ pi1

for all k 2 2. Let p > y then 4-p2-1 > 0 and ~ : ~ ~ ' ~ ( p + p - ~ ) > 0 for k large enough.

Consequently l i m l l ~ ( ~ ~ ~ ) 1 1 = Am. It is clear that I ~ A ~ ~ ~ ) I I is strictly decreasing when k k+m k

increases; this is a particular case of the next lemma. #

Proposition 1.1.1 follows from this lemma and from 1.4.2.

The following lemma is reproduced from [HS]. Let L = (u,w) be an edge of a connected graph I?; we denote by f or simply by f the graph obtained from I? as

(u,w) follows: delete 4 add a vertex v, add two edges e- = (u,v) and L+ = (v,w). For

example,

Say that L is internal if there &st an integer n 2 1 and a sequence of vertices xO,. ., xj = U, xjS1 = w,. . ., xn such that all xi are distinct (except possibly xo = x,),

where the degrees d(xi) satisfy

238 Appendix I: Classification of Coxeter graphs

d(xO) 2 3, d(xl) = . . . = d ( ~ ~ - ~ ) = 2, d(%) 1 3

and where is adjacent to xi for i = 1,- . . ,n.

Lemma 1.2.4. Let I' be a connected graph with an internal edge C = (u,w); assume that r is not one of the DL1)'s. Then ~ ~ f ( ~ , ~ ) l i < IlI'll.

&f. Let xo,, . ,x, be as above. We shall assume that xo # xn (the case xo = 5, substantially easier, is left as an exercise, and is also done with details in [HS]). Let - x - ~ , . . . x be vertices adjacent to xo other than xl; by hypothesis m 2 2. Let [ be ' -1 a Perron-Frobenius vector for the adjacency matrix Y of r; one has Y[ = IlI'll[; also Ilrli > 2 because there exists an integer k such that DL1) is a proper subgraph of I'.

We write instead of f we denote by 9 the set of vertices of f and by 9 its (~,w);

adjacency matrix. Suppose one can find a positive vector 2 E llii with $2 $ (II'/I& it will

then follow from Perron-Frobenius theory that llf 11 < IlI'll; see lemma 2 and remark 4 in 5 XIII.2 of [Gan]. We distinguish three cases.

Assume first that there exists t E (1,- . -,n-1) with [+, 5 4 for i = 0,. . .,n. There is

no loss of generality in assuming xt = u and x ~ + ~ = W. One has

Define i to be the vector with coordinate corresponding to v given by iv = f t and with

other coordinates as those of [. One has

with at least one inequality being strict by (*). Hence s llr~~i and Pi # IlI'lli. For the two next cases, to < ei for i = 0,. . . ,n, and we assume xo = u, xl = w

without loss of generality. m

Assume now that x 4 2 6. One has i=l

8 1.2. Characteristic polynomials

Define 2 by iv = to, the other coordinates being as those of t. One has

with at least one ingquality being strict by (**). The conclusion follows. m

Assume finally that xf, < k. One has to I IlI'llC, for i = l , . . . ,m. since i=i

m 2 2, this implies 2t0 m g i III'IIC~-~ and consequently i=i

rn Define 2 by $ = x4 and 4 = to, the other coordinates being as those of [. One

i=i

has

by (***). The proposition follows. #

One may also apply lemma 1.2.4 to Coxeter graphs such that, notations being as above, edges ( X ~ - ~ , X ~ ) for i = 1,. . ,n are marked with m = 3.

For q,r with 2 5 q r, we denote by pT 2 ,q,r the characteristic polynomial of the

graph T2,q,r. For k 2 4 and k 2 6, one has respectively P! = ~f ,2,k-2 and

= 'f,3,k-3.

240 Appendix I: Classification of Coxeter graphs

a

2 Pro~osition 1.2.5. With v = p one has

P;,,,(\) = (pP-1)-1{prq+1[vyvq-2+. * .+I)] - p-r+q-1[uyu~+2+. . .+I)]).

In particular,

-l k-l "1) = n ( w c o s ( m . , h ) ) P;(U = (P+P )(P +P 1s jdk J

with h = 2k-2 and ml,* . ,mk = 1,3, . . ,2k-3,k-1;

P ~ ( A ) = J'J(A-2cos(m./l2)) with m. = 1,4,5,7,8,11; Id j<6 J J

P:(A) = n( A-2cos(mj/18)) with m. = 1,5,7,9,11,13,11; Id jd7 J

P:(A) = H ( A - 2 c o ~ ( ~ / 3 0 ) ) with m. = 1,7,11,13,17,19,23,29; Is js8 J

so that

W f . The formula for P:,~,~ follows from 1.2.1 and 1.2.2. Rmts of

( p + l ) ( k l + p k l ) are given by p = +i and p = exp{&(1+2j)] for

j = 0,1,. . ,2k-3. As one value of p and its inverse correspond to the same value of A, roots of P: are given by A = 0 and A = 2cos{&(1+2j)) for j = O,l,. . ,k-2; the

D product formula for P t follows. T From the formula for P or from a direct computation,

E E The product formula for the P 's may be checked as follows. Firstly, the roots of P 6

112 E are A = 4 and A = .t Secondly, for P,, set A = 2~08 6, a that 7

5 1.2. Characteristic polynomials 241

A2 = 2(l+cos 26) and

6 4 2 3 A - 6A + 9A -3 = 8cos (28) - 6~0~(28) -1 = 2cos(68) -1;

1 the roots of P; are given by A = 0 and cos(66) = Z, namely by A = 0 and

A = 2cos(b + ji) for j = O,1,2 f ,4,5. Finally, set Q(1) = A4 -7A3 +14A2 -8A +l; a

straightforward computation shows that

where aS0(T) = n ( T - T j is the cyclotomic polynomial with mots the primitive 30 th

1s js8 roots of unity T - exp(irn.2~/30), where m. = 1,7,. . .,29. As T. + 2 + T:'

j - J 3 J J 2 2 = 4cos (m.lr/30), the roots of Q are 4cos (m.r/30) for m. = 1,7,11,13, and the formula

J J J E for P8 follows. #

Corollarv 1.2.6. For q 2 3, the sequence (IIT2,q,rll)r>q is strictly increasing and

converges to A '4 = v1J2 q + u:l2, where u 9 (a the largest root of the polynomial

Similarly, the sequence ((IT3 7 9 r(()r23 is strictly increasing and converges to Am.

m. The sequences are strictly increasing by lemma 1.4.2. One has -p-r+q-l L (vl) > 0 for p > 1 (and u = p2 > l) , so that llT2 rl l > Aq for all r 2 q

q 7 7

by proposition 1.2.5. The argument for limllT2 = Aq is as in the proof of 1.2.3. r+m "

For the graphs T3 ,, one computes first 9 1

and proceeds then in the same way. #

For integers p,q,r with 2 < p < q 2 3 and r 2 5, we denote by P the P 99,r characteristic polynomial of the graph H = Hp ,(3,3) defined in 1.1.

P,q,r A,

/'--

242 Appendix I: Classification of Coxeter graphs

Pro~osition 1.2.7. With L as in corollav 1.2.6, one has 4

-1 -1 r-pq-2 P;,~,~(A) = (W { P L~(U)L~(~)-,~-~+~+~+~L~(U-~)L~(V-~)~

2 where V = ,u . The sequence (IIHp,q,rll)r25 is strictly decreasing and converges to

= 2 1 2 + 4 4

6 &&. By lemma 1.2.1,

pH (A) = APT ~ 1 4 , r 2 ,p,q+r-4 - p$p;,p,r-4

By proposition 1.2.2 and routine manipulation

P;,~,~(u = (~r,u-~)-~{~(,u)-~(,u-~)l with

2 As ,urn is the largest root of ,u -~-,u-~, one has for p 2 prn

which is positive for r large enough; otherwise said JIH 11 < Am for r large enough. P,q>r 2 One has also F(,u) = pr-W4f(~) with v = p and

f(y) = $+q+2- 2$+q+1- $+q+ 2 ~ + q - 1 + $+q-2+ $+I- 2 -1

+,q+l -y4-1- ,,P- ,q+ 1

The formula given for pH (A) follows, and the other claims are checked as in earlier P,q,r proofs of the present section. #

5 1.3. Proofs of theorems 1.1.2 and 1.1.3 243

1.3. Prooh of theorewI.1.2 and 1.1.3.

We know from proposition 1.1.1 that the spectral radius of a connected graph which contains strictly a cycle ia strictly larger than Am. There are other conditions which imply

the same inequality.

Lemma 1.3.1. Let I' - be one of the graphs

"777" 0 0 0

Then llI'll ) A,.

Proof. By proposition 1.2.7 one has - -1 -1 2 -2 -2 2

~ y , ~ + ~ , ~ + ~ ( 4 = (P-P ) , u - )Lk+l(v)-(~ -,u ) ~ ~ + ~ ( v - l ) }

It follows that

The first factor is positive and the second is '

H for k 2 3. Consequently P2,k-l,k+3(Arn) ( 0 and llH2,k+l,k+311 ) A, for k 23.

One has similarly

244 Appendix I: Classification of Coxeter graphs

and ((H2,k+l,k+21( > A, for k 1 3. This follows also from the first computation and from

1.2.4. By a direct computation, the norm of the last graph in the lemma is

($5+17112)112 8 2,136, and is thus larger than A,. #

Pro~osition 1.3.2. Let r be a connected graph. Each of the following conditions implies llrll > A,.

(i) I? contains a vertez of degree d 2 5; (ii) I' contains n vertez of degree 4 and i. # DY); (iii) r contains three vertices of degree 3; (iv) r contains the graph

with k 2 8 vertices.

&Q& (i) The adjacency matrix of o&o is Y = [ O with / \ xt O 0 0

x = (1 1 1 1 1) E M ~ , ~ ( I N ) . AS X X ~ = 5 one h a l l ~ l l = s1J2 > t '7

(ii) The adjacency matrix of o ~ + - is Y = [ O with X = [i i :]. 0 xt O

as xxt = 5 i] has norm k5+13lI2) , A: one has IIYII > .\,. (iii) If I' contains threevertices of degree 3, then r contains

for some integers with 2 j p j q. If p 1 3, one has

and if p = 2, q 2 3 one has

by the previous lemma. If p = q = 2, one has also

5 1.3. Proofs of theorems 1.1.2 and 1.1.3 245

llrll 1 llr2,211 ' A,

by the same lemma. (iv) The characteristic polynomial Pk of the graph in (iv) is given by

It follows that the norm of these graphs decrease and converge to A, when k increases.

#

Corollarv 1.3.3. Let r be a connected graph with

Then I? is either some Tp q r or some H 1 3 P,99"

Proof. Clear from the previous proposition. #

The adjacency matrix of

The characteristic polynomial of X X ~ is

5 4 3 2 p -9p +27p -31p +12p -1.

Its largest root y a r 4,0264; consequently llT2 , 9 , I1 = d with d = p;l2. 2,0066.

Pro~osition 1.3.4. Let r be a connected graph with 2 < llrll s d. Then !? = T2,3,7.

246 Appendix I: Classification of Coxeter graphs

W. By 1.3.3, the graph I' is some of T or of H because d < Am. By p,q,r P,9,1'

1.2.7, it cannot be some of H because d < ,I3; indeed X3 = +$I2 where v3 P,4,1' 3 is the largest root of v -PI, so that v3 c~ 1,325 and X3 m 2,0198. Hence r = T

P,4J with 2 i p < q j r. Moreover the triples

(2,2,r) with r > 2 (2,4,4)

(2,3,r) with r i 6 (3,3,3) -

are ruled out, because IlI'll > 2. This and lemma 1.4.2 imply that r must be one of

T2,3,7 T2,4,5 T3,3,4.

By 1.2.5 one has

and by direct computation

It follows that I? = T2 7. # I ,

Remarks. (1) The proof of theorem 1.1.2 is now complete. (2) Proposition 143.4 calls to mind various extremal properties of the triple (2,3,7).

One is that T2 is a hyperbolic Coxeter graph of highest possible rank, namely 10; see 3 9

[Kos]. Another one is about Hurwitz' group, of presentation

which is "the largest" group of automorphisms of a Riemann surface with genus g 2 2; see the discussion in [Magl], page 103. This group has a quotient which is the simple group PSL2(U/7U) of order 168, namely the finite simple group which is neither cyclic nor

alternating and of smallest possible order. (See [Bur], in particular, section 166 and note

N.)

§ 1.3. Proofs of theorems 1.1.2 and 1.1.3 247

Given f = Xn + alXn-I +. . a + an E U[X], let n(f) be the product of the roots of f

outside the unit disc. Lehmer [Le 1) has asked about f with n(f) minimal. The best

polynomial found by Lehmer (and known today) is L(X) = XI0 +X9 -X7 -X6 -X5 -X4 -X3 SX $1. Let P(X) = X5 -9X4 +27X3 -31X2 +12X -1, so that P;~,~(A) = p(h2). Then I( \ ) = X5p(X + l/h). We know this from

Misiurewicz [Mis].

Lemma 1.3.5. One has llT3,4,511 > Am > llT3,4,411.

- 0 X Proof. The adjacency matrix of T3 is [ ] with " xt 0

t The characteristic polynomial of XX is

2 If pm = m = s1I2 +2 one has P(pm) = -4 + 5lI2 < 0; hence

llT3, = 1 1 ~ ~ ~ l l ~ / ~ > The adjacency matrix of T3 is [Z: :] for 9 9 9 9

The characteristic polynomial of Z Z ~ is

3 2 Q(P) = (~-2) f (~) , f(p) = P - 6 ~ +8P-2.

One has

f(0) < 0, f(1) > 0, f(2) < 0, f(51t2+2) = -2+5lt2, 0.

/--

Hence all roots of Q are in lo,~@[ and llT3 411 = I I Z Z ~ I I ' / ~ < ad # 1 I

248 Appendix I: Classification of Coxeter graphs

Pro~ogition 1.3.6 Let I' be one of the T 's. Then 2 < llrll s A, if and only i f P14,r

(p,q,r) is one of the following:

(2,3,r) with r 2 7 (3,3,r) with r 2 4 (2,4,r) with r 2 5 (3,4,4). (2,q,r) with r > q 2 5.

m f . It follows from lemma 1.4.1, corollary 1.2.6 and lemma 1.3.5. # - The proof of theorem 1.1.3 for ordinary graphs is now complete. One may add minor

refinements. For example, IIH2,q,rll < Am implies r 2 q+2 by 1.3.1 (and a direct

computation if q = 3); but this is not sharp, because one has for example llH2 d Am 9 9

if and only if r 1 7.

We observe that X3 N 2,0198 is in the closure of E but not in E. One of the

estimates in the proof of 1.3.4 shows that X3 # IIT2,4,5(( . Similar estimates show that X3

is strictly smaller than

(IT2 61( = the largest root of X8 - 9X6 + 27X4 - 31X2 + 11 N 2,0237 9 9

as well as

]IT2 511 = the largest root of X6 - 6X4 + 8X2 - 1 x 2,0285 1 ,

Thus if follows from Theorem 1.1.3 that X3 f! E, and in particular that E is not closed. 1 Similarly, Theorem 1.1.3 shows that, if Xw E El then Am = IlI'(l , with I' = H

P,W for some triple of integers satisfying 2 5 p 5 q, 3 5 q, and q < r. A.E. Brouwer (private communication) has checked that this is not the case, so that Xw $ E. We have not

checked whether X E E for q 2 4. 4

Lemma 1.3.7. For s E lR+ with s > 1 and for 1 E W with 1 2 I, consider the

symmetric matrices

5 1.3. Proofs of theorems 1.1.2 and 1.1.3

in Md{O,l,s)). When 1 increases,

IIY1lI increasa and converges to s+s-l

2 2- -112 11Zd1 increases and converges20 s (s 1) .

m f . Observe that Y1 and Z1 differ by one and two entries only from the example

in section 1.2. Let P1 be the characteristic polynomial of Ye and let Q1 be that of Zt AS in

proposition 1.2.2, one has PeS1 = XPrPbl, so that

When 1 increases, the largest root llYlll of Pe increases and converges to a limit y 2 which is the largest root of pP2-PI. As pP2-PI = p ( p s ) one has y = s+s-l.

Similarly the largest root llZlll of Q1 increases and converges to a limit z which is 2 2 the largest root of pQ2-Ql = p(p +I+ ), so that r = (s2-1)'12 + (s2-1)-lI2.

The proof is that of lemma 3.5 in [Hof]. #

Pro~osition 1.3.8. Let I' be a Coxeter graph. ( i) Suppose that the underlying graph of I' is a segment. Then 2 < llI'll i A, i f and

only if I' is one of the following.

5 H1 0-1-0-. . -4- with 1 vertices, 1 2 5

4 F1 0--0--~--0-. . .-0- with 1 vertices, 1 2 6

F:+ o c 0-4 with 6 vertices.

(ii) Suppose that 2 < llrll s A,. Then either I? = H1 as in (i), or I' is a Coxeter

graph where marked edges are marked with m = 4.

Proof. The has characteristic polynomial -

250 Appendix I: Classification of Coxeter graphs § 1.3. Proofs of theorems 1.1.2 and 1.1.3 25 1

Its norm is near 2,0608, and thus strictly larger than A,. As the norm of 0% is

2cos(lr/m) < 2 for all m 2 3, a Coxeter graph with 2 < llrll d A, cannot have any edge

marked with some m > 7. 6 The graph G$'): o&- is of norm 2. The graph 0---0--4 has

4 2 characteristic polynomial A -5A +3 and its norm is 2,074 s ($5+131/2))1/2 > a The

graph o A A and the graph o ~ < both have norm 5lI2 > a; Finally the

6 graph 0 - 4 - has norm ($5+191/2))1/2 > Am. Hence a Coxeter graph with -

2 < llrll < A, cannot have any edge marked with m = 6. 5 The graph 0-4- has characteristic polynomial A4 -$7+5lI2)A2 + I; its

norm is near 2,095, and thus strictly larger than A,. Consequently, if a Coxeter graph r with llrll 6 A, has an edge marked with m = 5, this edge must be a free edge (namely

one at an end of r) . The previous lemma shows that, for m 1 4, the norms of

om,--. . . 4 e vertices

2 2 increase with 1 and converge to 4cos (lr/m)(4cos (~ / rn) - l ) -~ /~ . For m = 5, these are the H i s and the limit is A,; observe that llH411 = 2. Now the

norms of 5 4

0 4 - - 0 - . . -0-4 1 vertices

decrease and converge to Am by lemma 1.2.4; the same holds for

and this ends the proof of (ii). The same argument as in 1.3.7 shows that the norms of Fe increase and converge to

the largest root x of pP2-P1 with

4 2 As pP2-P1 = ~ ( p -p -1) one has x = A,. Observe that F5 = ~ 1 ~ ) has norm 2. As

above, the norms of

4 K 0 - - 0 l vertices

decrease and converge to A,.

Consider now a Coxeter graph r with underlying graph a segment and which has

exactly one edge marked with m = 4 (all other edges, being unmarked, correspond to m = 3). If this is a free edge, r = Be for some t! ;! 2 and llrll < 2. If this is next to a

free edge, r = Fe as considered above. In the other cases, either I? = F:+ or I'

contains

3 2 Direct computations show that 11~:+]1~ is the largest root of p -6p +8p2 and that

l l ~ ~ 1 1 ~ is the largest root of ( ~ - 2 ) ( ~ ~ - 5 ~ + 3 ) . Hence IIF:+~~ < A, and 1 112 112

llL711 = (2(5+13 1) , A,.

Consider finally a Coxeter graph r with underlying graph a segment and which has at least two edges marked with m = 4. If there are the two free edges, r = Ce for some

t ! ? 3 and IlI'll = 2. In the other cases one has llrll ? llKell > A, for some t! ? 4. #

Pro~osition 1.3.9. Let I' be a Cozeter graph, the underlving graph of which is some

T~,q , r with 2 < p i q 5 r. Then 2 < llrll < A, if and only if l? is one of the Coxeter

graphs on the list ofI.3.6, or T2 ,(4) for some q 2 3 and for r large enough (see 1.1 for ,q, a picture of T2 ,(4)).

79, \

PFoof. Suppose first that I? has underlying graph Dk for some k ;! 4. The norm of

BP) o&--. . .-o+ l+l vertices, e ? 3 /--

is 2. The norms of

252 Appendix I: Classification of Coxeter graphs

decrease and converge to Xa by lemma 1.2.4. The graph

has characteristic polynomial X(X4-5X2i- 3) and its norm, equal to that of 6 o----0--4--o, is larger than Am. Hence I' cannot satisfy 2 < llI'll s A,.

Suppose now that the Coxeter graph I' has underlying graph T with 3 s q d r. 2,q,r

From lemma 1.2.1 and proposition 1.2.5, the characteristic polynomial of

is given by

For (q,r) # (3,3), it is easy to check that the value for X = Xw of this polynomial is

strictly negative; consequently the norm of the graph is strictly larger than Xw. Lemma

1.2.4 shows that llT2 ,(4)11 decreases and converges to X when r increases. 9

Also the norm of

4 0------0------0-. '-0- k vectors, k 2 7

decrease and converge to Am. The proposition follows. #

A final computation of the same kind for Coxeter graphs H (m,ml), also based on P,9J

1.2.4, ends the proof of theorem 1.1.3.

APPENDIX 1I.a * Complex semisimple algebras and finite dimensional C algebras

, Let M be a complez algebra. An involution on M is a R-linear map cu : M --t M satisfying

cu(xy) = (Y(y)cu(x) ~ ( i x ) = -i(Y(x) 4 4 4 ) = x

for all x,y E M; such an involution is positive if moreover

or(x)x = 0 implies x = 0.

In particular, let M = EndC(V) for some finite dimensional complex vector space V.

Let ( I ) be a hermitian product on V. For x E M, denote by x* the adjoint endomorphism, defined by (x*(~ q) = ((1x7) for all (,q E V. Then x H x* is the standard example of a positive involution on M. Let a be any involution on M. It follows from Skolem-Noether's theorem that g x ) = gx*g-l for some g E GL(V), and a(cu(x)) = x implies that g - l * is central, hence a scalar. One may always choose g hermitian: g* = g and all eigenvalues of g are real. Then cu is positive if and only if all eigenvalues of g have the same sign. If tr : M -+ 6: is the usual trace, we leave it to the reader to check that cu is positive if and only if tr(ol(x)x) > 0 for all x E M; hence "positive" has the same meaning above as in [Wei].

Pro~osition 11.1. Let M be a jnite dimensional complex algebra. (a) If M has a positive involution, then M is semisimple. (b) If M is semisimple, then M has a positive involution x H x*. Moreover, for any

other positive involution cu on M , there ezists an invertible element g E M with .g* = g and with positive spectrum, such that 4 x ) = gx*g-l for all x E M .

(c) Let M and x H X* be as in (b) and let p be a central idempotent of M . Then *

P = p . (d) Let N c M be a pair of semisimple algebras and let P be a positive involution on

N . Then there ezbts a positive involution cu on M which extends P.

Proof. (a) Let x H x* be a positive involution on M. Assume that the Jacobson - radical J of M some xl # 0; then x2 = xlxl is not zero, so that

x3 = xix2 # 0, and so this contradicts the fact that J is nilpotent. Hence J = 0

and M is semisimple. (b) If M is semisimple, M is a direct sum of factors, and thus has a positive

involution by the example before the proposition.

254 Appendix 1I.a.

Let V be as in this same example. A straightforward application of Skolem-Noether' s theorem is that any involution cu on End(V) x End(V) which exchanges the factors is of the form a(x,y) = (gy*g-l, g*x*g*-l) for some g E GL(V); such an involution is not positive because ol(l,O)(l,O) = (0,O). Now let M be an arbitrary semisimple algebra. What precedes implies that factors in M are invariant by any positive involution; claims (b) and (c) follow.

(d) Assume first that M is a factor, so that one may set M = End(V) as above. Let U(N) be the unitary group {h E N I ,B(h)h = I), which is compact, and let dh denote its Haar measure of total mass 1. Define a new scalar product on V by

Then ((D(y)[l q)) = ((51 yq)) for all 5,q E V and y E N; for y E U(N), this follows from invariance of Haar measure, but any Y E N is a linear combination of unitary elements. Define the involution a on M by ((a(x)tI q)) = ((elxq)) for all (,q E V and X E M .

n In the general , case, one has M = @ End(Vi). One defines

j=1 a(xl,. . .,xm) = (a1(x1),. . . ,am(xm)) where each ai is defined from (N End(V.))

1

as in the factorial case. #

The proposition holds also for real algebras; see [Wei], Proposition 1 and corollary.

Exam~le 11.2. Let G be a finite group. Its group algebra C[G] is made of complex functions on G, the product being convolution

There is a natural involution defined on C[G] by

This is positive, because x*x = 0 implies

and thus x = 0. Thus proposition 1l.l.a provides one proof of (one formulation of) Maschke' s theorem.

* Semisimple algebras and C -algebras

Exam~le 11.3. Consider the two-dimensional truncated polynomial algebra M = c[x]/(x2), with basis {l,x) and with product defined by x2 = 0. Then M has two involutions which map 1 to 1, and x to x and -x respectively. Of course M is not

semisimple and these involutions are not positive.

Recall that a c*-algebra is a complex algebra M (possibly infinite dimensional, possibly without unit) furnished with an involution x I+ x* and a norm x I+ llxll such that

(i) M is a Banach space for 11 (I; (ii) llxyll 6 llxllll~ll for all X7Y E M.

2 (iii) ~(x*x(( = llxll for all x E M. In particular, (iii) implies that the involution is positive. If M is finite dimensional, (i) is of course automatic, but moreover the norm is uniquely determined by the involution:

Pro~osition 11.4. (a) Let M be a semisimple complez algebra (offinite dimension) and let x I+ X* be a positive involution on M. For x E M, dejine

2 * Iixll = Supit E R+ : t -x x is not invertible on M).

Then x llxll is a norm which, together with x I+ x*, make M a c*-algebra. (b) Let M,M' be two c*-algebras and let rp : M 4 M' be an injective linear map

such that

Then cp is an isometry:

IIcp(x)ll = llxll for all x E M.

w f . (a) If M = End(V) with V a hermitian vector space, one may also define the norm by

Then ~lx*xll = 1 1 ~ 1 1 ~ as in lemma 1.2.4, and the two definitions of llxll coincide by the spectral theorem for hermitian matrices. The general case follows from that of factors.

(b) This claim is proved by elementary functional calculus, and comes early in any book about c*-algebras, for example on page 13 of [Arv]. This claim would not hold without the completeness requirement (i) in the definition of a c*-algebra. #

Let M be an algebra (possibly infinite dimensional) with a positive involution x n x*. Recall from section 2.5 that a on M is a linear map tr : M 4 C such that tr(xy) = tr(yx) for all x,y E M; say it is positive if tr(x*x) > 0 for any x E M. In case

256 Appendix I1.a.

dimc(M) < m, this definition is the same as that of Section 2.5 by I1.l.c. It follows from

the Cauchy-Schwarz inequality that a p~sitive trace is faithful if and only if tr(x*x) > 0 for all x # 0 in M.

In the context of c*-algebras and w*-algebras, it is usual that "trace" means "positive trace." We shall take up this habit in Chapter 9.

Then, in the same way as for measures on locally compact spaces, positivity makes it possible (and quite useful) to consider infinite traces; we shall return to this in Section 3.2.

Let N be a sub-C*-algebra of a c*-algebra M. A conditional expectation E : M 4 N is self-adioint if ~ ( x * ) = E(x)* for all x E M; it is positive if, for every -'

x E M, there exists y E N with E(x*x) = y*y. It is an important point that the construction of Proposition 2.6.2 provides positive conditional expectations from positive traces.

Pro~osition 11.5. Let N C M and tr be as in Proposition 2.6.2 (with K = C); assume moreover that M is given a positive involution x H x* and that the faithful trace tr is positive. Then the faithful conditional expectation E : M 4 N of Proposition 2.6.2 is self-adjoint and positive.

Proof. Observe that tr(x*) == for any x E M; indeed, this is clear when x is positive, and one has in general x = x -x +ix -ix with xl,. . x

1 2 3 4 , positive. Since tr is positive, both tr and t r 1 are faithful. We check then that E is

self-adjoint. For any z E N one has

and the claim follows because tr is faithful. I N Consider now x E M. As ~ ( x * x ) is self-adjoint, there exist positive elements

zl,z2 E N with E(X*X) = z;zlz2z2* and zlzi = 0. If z2 # 0 there would exist a * self-adjoint idempotent e E N with zlzle = 0 and ez2z2e # 0, so that

tr((xe)*(xe)) = tr(~(x*x)e) = -tr((z2e)*(z2e)) < 0

contradicting the positivity of tr. Hence E(X*X) = z;zl. #

It can be shown in general that a positive conditional expectation is faithful if and only if E(x*x) > 0 for any non-ero x. See 9.2 in [Str]. This is evident for conditional

* Semisimple algebras and C -algebras 257

expectations defined by a faithful positive trace, for E(x*x) = 0 implies that tr(x*x) = tr(~(x*x) = 0, and therefore x = 0 by faithfulness of the trace.

In the eontezt of c*-algebras and w*-algebras, it is usual that "conditional ezpectation" means "positive conditional ezpectation". We will adopt this habit in

Chapter 9. - Conditional expectations are very important in the study of operator algebras; see, e.g.

[Strl, 59.

In the remainder of this appendix, we comment on C* versions of the various constructions of Chapter 2. All of the results of sections 2.2 and 2.3 have C* versions. For example, in 2.2.3 and 2.2.5, let F be the factor EndC(V), where V is a finite

dimensional complex Hilbert space, M a C*-subalgebra, and in 2.2.5 take q to be a

self-adjoint projection. Then CF(M) and qMq are also C*-subalgebras. Versions of

2.3.9 and 2.3.10 are valid for c*-algebras, with the isomorphisms respecting the involutions. We have already remarked and the end of Section 2.3 that one can associate to a Bratteli diagram a chain of c*-algebras, via the path model.

Now let N c M be a pair of finite dimensional c*-algebras and set L = ~ n d $ ( ~ ) .

Then L has a unique C* structure such that the inclusion of M in L is a *-isomorphism. In fact, from the Bratteli diagram for N C M c L, we can impose such a C* structure on L, using the C* version of 2.3.9, or the path model. A more natural way to go about this, hdwever, is to take any positive faithful trace tr on M, and to give M the hermitian inner product (xly) = tr(xy*). Then the left regular representation of M on (M,tr) is a faithful *-representation, and the right regular representation of N is a *-anti-representation. Hence L := p(N)' is naturally a c*-algebra, by the C* version of 2.2.3. (Here the prime (') denotes centralizer in EndC(M,tr).)

Furthermore, the (faithful, positive) conditional expectation E : M -+ N determined by the trace t r is also the self-adjoint projection of M onto N, with respect the trace inner product. (See the proof of 2.6.2.) Therefore the subalgebra (M,E) of EndC(M,tr)

generated by X(M) and E is a *-subalgebra, so equal to its own bicentralizer by the C* version of 2.2.3. Bu he centralizer of (M,E) is easily seen to be p(N), so that (M,E) = (M,E)" = p(N r = L. Finally, if z is a central projection in L orthogonal to the ideal X(M)EX(M), then for all y,x E M, one has 0 = (EX(y)z)(x) = E(yz(x)); hence z(x) = 0 by faithfulness of E, and so z = 0. Thus L = X(M)EX(M). This is the C* version of Corollary 2.6.4.

Suppose now that N c M is a connected pair and choose for tr the unique positive Markov trace of modulus B = [M:N] (Theorem 2.7.3). Then the unique extension of tr to L with the Markov property, fir@) = tr(x) for x E M, is also positive, since the

1 weights of the tram on L are /3- times the corresponding weights on N. Iterating the fundamental construction thus yields a chain of finite dimensional C* algebras carrying a

258 Appendix I1.a.

faithful positive trace tr and a sequence of self-adjoint projections (Ei)i>l satisfying the

conditions of Proposition 2.7.5. We have already noted in Sections 2.8 and 2.9 that, for K = C and for P > 4 or = 2 4cos ( d q ) for some integer q > 3, the algebras BAk constructed there have a C*

structure making the generators {e.} self-adjoint projections. Furthermore, 2.8.5 or J

2.9.6(e) implies that Bp,k is, up to *-isomorphism, the unique c*-algebra generated by

the identity and self-adjoint projections {el,' - .ek-l) satisfying the relations of 2.8.4 (ii),

with a faithful trace tr satisfying fir(we.) = tr(w) for w E alg {l,el,. . -ejwl}. J

APPENDIX II.b The algebras Ag,k in statistical mechanics

The first recorded occurrence of the algebras of Section 2.8 arose in a work of H.N.V. Ternperley and E.H. Lieb (1971) about microscopic models for magnets. The purpose of this appendix is to introduce the reader to this circle of ideas. For a systematic account see

It is an idea going back to Daniel Bernoulli (1738) to compute the pressure of a gas in a container from the change in momentum of the molecules impinging on the walls. During the last century, physicists made this computation on several occasions and compared the outcome with the equation of state of a perfect gas pV = nRT. One result of this comparison is that the mean kinetic energy of a simple (i.e., monatomic) molecule in a gas at temperature T is

where k denotes the Boltzmann constant, Let us now sketch the argument of J.C. Maxwell (1860) for finding the velocity

distribution of the molecules in a gas. Let x,y,z be Cartesian coordinates in the velocity space and denote by f(x)dx the probability of finding the first coordinate of a molecule in the small interval (x,x+dx). As all directions are equivalent, the density of probability of finding a velocity vector at (x,y,z) is given by f(x)f(y)f(z), and this should be a function of the absolute value c = (x2 + y2 + z2)ll2 alone. As Maxwell writes:

iation with respect to x

Differentiation with respect to y or z leads to

g y$ = constant

Appendix ILb.

for appropriate constants C and A. As f should be integrable, A < 0, say A = -G2 for some (Y > 0. Moreover C = c1 because f(x)dx = 1. It is now easy to

compute the mean kinetic energy of a molecule of mass m:

so that $ = & by comparison with (11.6). Finally, for a simple molecule (monatomic (Y

1 2 gas), the energy is E = 2mc , so that the density of probability of finding a molecule with

energy E is given by

[&I 312exp(-~/k~). (11.7)

For all this, see [Som], Sections 22-23 and [BR].

More generally, consider abstractly a physical system with set of possible states S, and denote by E(s) the energy of the system when it is in the state s E S. The canonical

postulate is that the probability of the system being in the state s is proportional to exp(-E(s)/kT). That this is a sensible postulate can be understood either from general a priori considerations involving additivity with respect to subsystems (see [Gib], Chap. IV), or from the particular case above (11.7). Then, it is clearly quite important to understand and compute Gibbs' partition function, which is the sum-over-states

As Gibbs has shown, the relevant physical quantities can be expressed in terms of Z. For example the average energy is

The one dimensional Ising model.

We follow the beginning of Chapter 2 in [Bax], ahd consider a linear array of n equally spaced atoms

The algebras Ap,k in statistical mechanics

Assume first that each atom can have one of two possible spins. Then a $& of the

array is a sequence o = (oi)lsiSn with oi E {fl). Let us assume moreover that only

states with periodic boundary conditions are allowed ol = on. Assume also that energy

comes from interactions of opposite spins at neighboring sites, say

for some constant J. (One has J > 0 because energy in minimal if all spins are alike.) The computational problem is to evaluate

for large n, where the constant K is J/kT.

Here comes a clever trick. Given a matrix A = the (k,l)-entry of A'

is

. . .

/- Consequently we set

If S+ denotes the set of states with al = on = +1 and Zn,+ its contribution to Z,

one has

which can be recognized as one diagonal entry of A". The same holds for S , and

Appendix I1.b.

It is now quite easy to compute the eigenvalues of A ~ , and thus to handle Zn for large

n. Alternatively, we might assume that each atom has one of p possible spin states, so

that S is now the set of sequence (a$lgjgn with a. E {I,..-,p} and, say, with J

ul = un. (Standard notation for the number of spins is q instead of p, but this would

conflict here with the notation of Section 2.10.) Assume that neighboring atoms interact in one of only two possible ways, depending on their spins being equal or not. Thus, up to an irrelevant additive constant

K where 6 is the Kronecker zero-one symbol. Now let A be the pby-p matrix with e on the diagonal and 1 at any 'other place. Then

which is again easy to handle. E. Ising proposed his model in 1925, and was successful with computations for one

dimensional systems. Later, similar ideas have been used for systems with one or two dimensions by several physicists, including L. Onsager (1944) and R.B. Potts (1952). For our purpose, the main point to remember is the trick of using the trace of large powers of an appropriate matrix to evaluate the partition function when the number of atoms is large.

From square lattice Potts' model to A B,n'

We fpllow here part of chapter 12 in [Bax]. In a square lattice, consider a rectangle with m atoms on the base and n on the side

The algebras AB,k in statistical mechanics 263

- m atoms - Each vertex is again thought of as one atom with one of p possible spins. A state in S is now a double sequence k k

s = (aj)lSjgm,l<k<n with u. J E {l,. . .,p). (No boundary

condition.) We assume that energy comes from either horizontal or vertical nearest neighbor

interactions. More precisely, the horizontal contribution of some row with spins ul,. . . ,am to the partition function Z is

and the vertical contribution of two adjacent rows with spins al,. . .,arn,rl,. . .,rm is

r- Introduce the pm-by-pm matrices V,W with entries

m As ll 6(ak,rk) = 0 if a # T, the matrix V is diagonal and (VW) is the product of

k= l 0,T

(11.8) and (11.9). Consider now a state s = (ul,. ,,$) E S, where each ak denotes a sequence of m

spins. Write 0 = u1 and T = 2. Then ( ( V W ) ~ - ' V ) ~ , ~

Appendix I1.b. The algebras in statistical mechanics

k represents the contribution to Z of the states s = (u.) J l~jsm,lsksn With

1 1 u = (al,. .,om) and T = (<, . . . , u i ) . Thus Z has again been expressed in terms of

iterates of matrices. But now, the two so-called transfer matrices V and W do not commute, and further

devices have to be used. One is to define pm-by-pm matrices Ul,. . . ,U2m-l by -

j # i (U2i)u,r = ~ - ~ ~ ~ 6 ( ~ ? ~ + ~ ) n 6(q,Tk).

1s ksm

In particular Ugi is diagonal while U2i-l is of the form

where 1 denotes the p-by-p identity matrix and where the ith factor on the left is the matrix g with all entries equal to p-'I2. The transfer matrices may be written as

where I is the pm-by-pm identity matrix. For our point, the crucial fact is that the matrices U1,. . .,U2m-1 satisfy the relations

2 112 ui = p ui U.U.U. = U. if li-jl = 1

I J 1 1

U.U. = U.U. if li-jl 2 2 1 J J 1

for i j = 1,. .,2m-1. In other words, p-lI2ui define a representation of AP12m-l for

P = P. m

It is readily seen that R = lT U2i-l is a pm-by-pm matrix with all entries equal to . i=l

P - ~ ~ ~ . For any pm-by-pm matrix X, denote by D(X) the sum of d l entries of X,

and observe that RXR = p - m / 2 ~ ~ ( ~ ) .

Then z = c((VW)~V)

and the computation of the partition function Z is now a problem inside the algebra

Ap,2m-l. This happens to help, though a full solution is still missing, and constitutes

indeed "one of the most tantalizing unsolved models" (see [Bax], page 337).

APPENDIX 1I.c More on the algebra for non-generic P

Consider a field K, a parameter P E K*, an integer k 2 1, and the cor&sponding algebra A defined in Section 2.8. If (Pj)j20 are the polynomials of Proposition 2.8.3,

P,k we have shown in Proposition 2.8.5 that A is a multi-matrix algebra as soon as

P,k pj(T1) + o for j 5 k-1.

Theorem 11.10. If p j ( ~ l ) # 0 for j s k-1 and if p k ( r l ) = 0, then AD,k+l k not

semi-simple.

For example A1,3 is not semisimple, as checked in Example 2.9.1, or ,$ is not ,4

semisimple if K is not of characteristic two. More precisely, under the hypothesis of the theorem, we show hereafter that A

P,k+ 1 contains a onedimensional two-sided ideal of square zero. Most of the proof is contained in a lemma of H. Wenzl [Wen 11 that we recall below.

As for notations, given two idempotent u and w in some algebra B, we write u r w if uw = u = wu. This is evidently a partial order on idempotents of B. In case a family { u 1 , , . } of idempotents in B has a least upper bound, we denote it by

J U 1 V U 2 V ... v u

j' With P ,k as in the theorem and (Pj)j20 as before, we define inductively elements

61,*. .,% E by

Lemma 11.11. One has for m E 11,. . . ,k):

2 pm(B1) ('1 (cmJm) = 1 t m m ' 6

Pm-l(F 1

More on Ap,k for non-generic P

Furthermore, if m s k-1: (iii) 6m+1 is an idempotent.

(iv)) tj6m+1 = 6m+1~j = 0 for j = 1,. . .,m.

(v) SmS1 = 1 - el V . . . V Em.

Proof. Recall that the integer k is fixed in this discussion; we may assume that k 2 2. As the lemma is obvious for m = 1, we proceed by induction on m. Thus we assume that m 2 2 and that claims (i) to (v) hold for 1,2,. . ,m-1.

2 Set (r = ( ~ ~ 6 , ) = tm(6m-1 - P ~ - ~ ( s ~ )

1 6m-l'm-16m-l)'m6m. As Jm-i is a Pm-l(T )

linear combination of monomials in tl , . . . , L ~ - ~ , it commutes with 6,. Using (v) for the

values m-l and m-2, one has 6m-16m = 6,. From these two facts we compute

This shows (i), and (ii) is similar.

2 P,(B~) From (i) one has (6,em6,) = 6 t 6 and (iii) follows. 1 m m m

Pm-l(B

For j E 11,. .,m-l}, we know by induction that t.6 = 6 t . = 0, so that J m m J

ejSmS1 = 6m+1~j = 0 by definition of 6m+1. Moreover

by (i), and similarly 6m+ltm = 0 by (ii). This shows (iv), so that 1 - bmS1 is an upper

bound for {el,. . .,em).

Let C J,f3,k+l be an ide potent which is an upper bound for {el,. - . , E ~ ) . As k 1 - 6m+1 is a linear combination of (non-trivial) monomials in el,. . . ,tm, one has

268 Appendix I1.c.

(1-6 m+l )C = 1 - 6,+1 = C(l-6m+1), namely 1 - bmS1 s C. Thus 1 - 6m+1 is the

least upper bound for {el,. . . ,cm). #

For Theorem 11.10, we are interested in Ap,k+l. Observe however that 61,. .,fik are

Lemma 11.12. Set y = Jktk4, € A = AP,k+l. Then the two-sided ideal AyA is

reduced to Ky, and is in particular of dimension at most 1. Moreover yAy = 0 and AyA --,

has square zero.

Proof. Let w be a standard monomial in A, as in Proposition 2.8.1. If w # 1, then - yw = wy = 0. Indeed, if w starts with c., with 1 i j i k-1, then yw = 0 by claim (iv)

J of the previous lemma. Also the proof of claims (i) and (ii) shows that

so yw = 0 also for any word beginning with ck, Similarly wy = 0 for all non-trivial

words w. By Proposition 2.8.1 and Corollary 2.8.6, this shows that AyA = Ky. As y2 = 6,cky = 0, it follows that ywy = 0 for any standard monomial w,

including w = 1. This implies 7.47 = 0. #

To prove Theorem 11.10, it is enough to show that y # 0, namely that the radical of AP,k+l contains the nonzero ideal AyA of Lemma 11.12.

If k = 2, this is straightforward. Indeed y = (1-e1)e2(1-el) is not zero because

{l,tl,t2,c1e2,c2c1) is a basis of Al f. (In example 2.9.1, y- 0 under the map from

A1,3 to T, so that the radical of A is strictly larger than AyA.) But for k 2 3, we 1,3

need more lemmas.

Lemma 11.13. (i) For any j 2 2, the subspace AP,j~jAP,j of AP,j+l is the linear span

ojthe reduced monomials distinct from 1 (see Proposition 2.8.1). In particular

(ii) The dimension of A @ A as a vector space over K is not more than '

f l j Ap,j-l P,j

dim(Ao,j+l) - 1.

More on for non-generic ,b

For any i E {l,...,j) one has

c. 1 = d-i€iei+l . . cj-l€jcj-l . . .

in A and claim (i) follows. P,j+19 . The vector space of (ii) is spanned by elements of the form wl @ w2 where wl and

w2 are standard monomials in A P,j' If w2 does not involve cjWl then wl @ w2 = w w @ 1. Otherwise w2 = wicjelwi 1 2

where wi E Ap,j-l ends with ce for some Q. < j-2 and where wi = 1 or

wi = tj-z . en for some n E {1,2,. . . ,j-2). Then wl @ w2 = wlwi @ E ~ - ~ . . - tn. One

has Q. < j-1 in case wi = 1 and Q. < n otherwise, because w2 is a standard word.

In all cases wl @ w2 = p G l @ G2 where r is an appropriate integer, and with

G1 cjG2 a standard monomial in A P,j+l' Consequently the vector space of (ii) is spanned

by a set containing dim(AP,j+l) - 1 elements. #

Lemma 11.14. For any j 2 2 the natural map

is a K -linear isomorphism.

Proof. The map is clearly well defined (because each of el,. . ,cj-2 commutes with - e.) and onto. The conclusion follows from the previous lemma. J

#

Lemma 11.15.

(i) The element 4, is not zero in A P,k' (ii) The element q @ 61, is not zero @AP,k-lAil,k.

(iii) The element y ojLemma II.12 is not zero in AP,k+l.

m. (i) Denote by Bi, twc-sided ideal generated by cl,...,ck-l. The

definition of 4, shows that 1 - 4 E B1;. If 4 were zero, this would imply B1; = AP,k' in contradiction with Lemma 2.8.4.iii, so that 4 fi 0. (Observe that % is a central

idempotent in A so that 4, is the same as dk in Lemma 2.8.4). P,k'

270 Appendix I1.c.

(ii) By Lemma II.ll.iii, the image of $ @ by the K-linear map

is 6k' Thus (ii) follows from (i).

(iii) This follows from (ii) and from Lemma 11.14. #

The proof of Theorem 11.10 is complete.

In the same way as the braid group on three strings is an epimorphic image of the braid group on four strings, the assignment

extends to an epimorphism Al,4 -4 It follows that A is not semi-simple. 194

However, we do not know in general when A is semi-simple for j > k+2. It would P,j

be interesting to compute the discriminant of A P,j'

Following Wenzl [Wenl], we shall record another useful consequence of Lemma 11.11,

Theorem 11.16. Consider a real number fi # 0 and an infinite sequence (cj)j21 of

orthogonal projections on some complez Hilbert space such that the usual relations hold

2 '. = c. 1 1

P ricjci = ci i f 1 i-j 1 = 1

r.r. = r . ~ . if li-j1 2 2 1 J J 1

together with

If rl # 0 then one has

2 either P = 4 cos (dq) for some integer q 1 3 or p 2 4.

Observe that Pr1c2r1 = el obviously implies that 1. Before we prove the

theorem, we need two more lemmas.

More on AD,k for non-generic p 271

LemmalI.17. Consider an integer k L 3 with p2(/.T1) # 0,. . .,Pk-l(rl) # 0, so that

61," *,$ are defined and satish the relations of Lemma 11.11. If = $ then

rk-l 5 If moreover k 2 4 then $-3cirj = 0 whenever i,j 2 k-2 and (i-j ( ? 2.

Proof. Set p = ~ ~ - ~ ( $ - ~ - 6 ~ - ~ ) . By (iv) and (v) of Lemma 11.11 and the hypothesis - = Sk, one has

tkkl 6k-1 = ~k-1 $ = 0

rk-1$-2 = 6k-2rk-1'

Hence p is a projection and p = ~ ~ - ~ 6 ~ - ~ 1 By (i) of the same lemma

As rk-l and 6k-2 commute \

If p # 0 this implies P ~ - ~ ( T ~ ) = 0 and contradicts the hypothesis. Thus p = 0,

namely ck-16k-2 = 0, and ck-l 5 1 - Sk-2.

Assume k 2 4 and set q = 6k-3~k-26kl which is a projection because $-3, ck-2 and

ck commute. One has q = ~ ~ r ~ - ~ ( 6 ~ - ~ - 6 ~ - ~ ) because

by the first part of the present proof. The same computation as above shows that

APPENDIX 111 Hecke group and other subgroups of PSL(2P)

generated by paxabolic pairs

All facts exposed in this Appendix are well known, and most of them may be found in Magnus' survey [Magal. Our purpose is twofold: firstly to expose background for the examples in Section 3.3; secondly to offer a self-contained proof of the following result, which we believe is due to Fricke and Klein. It provides spectacular comparison with Kronecker's result (our Theorem 1.1.1) and with the results described in Section 3.4,-but we do not know whether it is a superficial curiosity or if there are interesting hidden explanations. We denote by :] the class in PSL(2,R) of a matrix [i k] E SL(2,R).

Theorem 111.1. Let X be a strictly positive real number and let I'i be the subgroup of

PSL(2,R) generated by [i $1 and [: y ] . Then Ti is discrete if and only if

either X = 2 cos for some integer q >_ 3 or X 2 2. 9

Moreover Ti has elements of finite order in the first case and I'i is free on two

generators in the second case.

For comparison with works of Fricke and Klein, see $11.2.11-12 in [FKl] and 11.1 in [FK2]. Among the many places where variations and complement on what follows can be found, we shall quote [Eva], [Kna], [Leu], [Mat], [Puz]. [Rus].

Let GC denote the group PSL(2,C), acting by fractional linear transformations on

the Riemann sphere E = C U {a}. Recall that [i :] is parabolic if it is not one of the two

matrices a1 and if (,+dl2 = 4, namely if the transformation z o (az+b)(cz+d)-' of has exactly one fixed point. The study of non commuting parabolic pairs reduces to that of pairs as in Theorem 111.1, as the next proposition shows.

Prouosition 111.2. Let A,B be two parabolic matrices in SL(2,C) and let X E C be such that

4 tr(A8) = (2+X2)tr(4tr(B).

Then X = 0 if and only if A and B commute. Moreover, i f X = 0, there exist C E SL(2,C) and z E C* with

Hecke groups and other subgroups of PSL(2,R)

On the other hand, i f X f 0, there exists C E SL(2,C) with

Proof. Let A,B be the classes of A,B modulo {tl}, and let v,w be the fixed points of A,B in E .

If v = w, a conjugation by reduces the situation to the case v = w = m, so

that one has

for some x,y E c*. After a second conjugation by , one obtains matrices as in the

theorem. It is now straightforward to check that 4 tr(AB) = 2 tr(A)tr(B) and A8 = BA. If v f W, a by the appropriate scalar multiple of [ -;] reducgl the

-w

situation to the case v = w = m, so that one has b

for some x,y E c*. ~f c = [: 7 then

2 and one may choose z so that z x = z-2y; thus, we may now assume that x = y. A computation shows that 4 tr(A8) = (2+x2)tr(~)tr(B) and that & f B4 in particular x2 = X2. If x = A, the proof is nished; if x = -A, it is enough to choose iz instead of

25. # /a/

For any X E c*, let ri be the subgroup of GC generated by

AA = [; and BX = [i y ] and let I'X be the subgroup of GC generated by AX and

Appendix I11

As JAXJ = B: one has I'j c rX. As J2 = 1, the index satisfies [ r X : r j ] i 2. The

cases [FA : r j ] = 2 and rX = I',i occur both, depending on X (see below).

Corollary IIL3. For n,X E c*, the groups and I'i are conjugate in Gc as soon 4 4 a s h = A .

- 2 2 2 2 w If n = X , it is clear from the proposition. If n = -A , it follows from the

fact that r; is generated by AX and 13i1 and from theequality

It would be nice to know whether the converse is known, but we have not found this in the literature, even under the assumption X E R+.

The next proposition shows the claim of Theorem IIL1 for X 2 2. We denote by {1,J) I the group of order 2 generated by J, by AX the infinite cyclic group generated by AX,

and by H * K the free product of two groups H,K.

Proposition 111.4. Let X E C with I X I > 2. Then

and these groups are discrete in GC.

We write A for1 AA during the proof. Set

1%

S A = { z € C : IF& [&I I > 1)

S J = { z E C : 121 < 1).

One has J(SA) c SJ (the 8-shaped domain below) and An(sJ) c SA for all n E I with

n # 0. Let W be a non empty reduced word spelled out with the letters A and J. Then W # 1 in GC, as results from the four following cases; nl,. - .,nk are integers in I\{O).

Hecke groups and other subgroups of PSL(2,R) 277

n n (2) W = JA 'J JA k~ + JWJ # 1 by (1) + W # 1.

(3) W = An1JAn2 . . . JAnkJ 4 A m w A m # 1 by (1) for m s I with

Z Z I Hence rX is the free product of AX and {1,J). It follows that r j is AX * BA and is

of index 2 in rX. Let U be a non empty open subset of SJ disjoint from J(SA) and such that

J(U) n SA = 4. Then g(U) n U = 4 for all g E rX\{l) , as it is clear for each of the four

cases above. Hence rX has a domain of discontinuity in c, and this

discrete in GC. #

We denote by G the group PSL(2,R) acting on the Poincar6 half plane 7, and we normalize the Haar measure on G as in 53.3.d.

Proposition 111.5. Consider an integer q 2 3 and set X = 2 cos(lr/q).

(i) The group rX is discrete in G. It has a fundamental domain in 'P of finite 2 2 area r ( l - so that the volume of G/rA is also s(1 - -

9).

Appendix I11

(ii) rX is the free product o f the group generated by AAJ, which is cyclic o f order

9, and o f {l,J). In particular, all conjugacy classes in rX\{ l ) are infinite.

(iii) rX = PSL(2,U) if q = 3 and A = 1 .

(iv) ri = rX if q is odd and [ rX : r i ] = 2 if q is even.

@& Let AX be the (hyperbolic) triangle in 7 with vertices eirIq, W, -e-i*/q.

The transformation AX maps the half-line (-.?-idq,~) onto (eidq,w), and J

exchanges the edges ( i r q , i ) and (i,Jrlq). By Poincar6's theorem (see [Mas] or [Rha]), AX is a fundamental domain for rX acting on 7, and rX is discrete in G.

This and Gauss-Bonet formula show (i). PoincarB's theorem also shows (ii), namely that rX has a presentation with generators AX, J and with relations ( A ~ J ) ~ = 1 = J ~ .

Statement (iii) is standard: see for example [Ser2], Chapter VII. Let us demonstrate (ii) in a different way (see $so [Eva]). Set KA = AAJ = [[I: i],

which is a rotation of order q around eirIq. Consider the action of rX on the boundpry

= R U {m) of 7 in E . Then KA maps ]o,A-~] onto ]w,O]. Set SK = ]O,m] and

SJ = ]w,O]. Then J(SK) = S j and K'(s~) c SK for r E {1,2,. . .,q-1). It follows as in

the proof of Proposition 111.4 that rA is the free product of {1,K,. . .,Kq-I} with {l,J}.

(Digression: let p = e i d q and p-I be the two fixed points of KA in C. Then p and

p-I are given by f ( I + (X2 - 4)'12), as in Section 1.2.)

One has A ~ B ~ ' = K:. If q is odd, the group ri generated by AX and BX

contains also KA and J, so that rjl=rX. If q = 2 p is even, then

r i = A: * ( 1 , ~ ~ ~ . . . , ~ q - ~ 2 ) x Z * (Z/pZ) as shown in [LyUj. #

1 Heck groups and other subgroups of PSL(2,I) 279 1 i

As a digression, let us observe that this family of groups shows dramatically that rigidity B la Mostow-Margulis (see [Zim]) does not hold in G = PSL(2,lR). Indeed, if

q - + m so that X = 2 cos(ir/q) 4 2 one has a sequence of groups having fundamental

domains with one cusp which to a group for which any fundamental domain has more than one cusp, and moreover all these groups are pairwise non isomorphic. Also if X , 2 tends to 2, one has a family of groups with infinite covolume in G which tends to the group r2 with covolume r.

5 The "only if" part of Theorem 111.1 is yet to be proved. We consider again X E C*

1 rather than X E IR:, as long as it does not complicate the arguments. 1

Pro~osition 111.6. Let X E C with 0 < I X I < 1. Then rX is not discrete in GC.

Define inductively Cn = [: t] E Ti by

Thus q = X2"-', and in particular IPnI -r 0 if n -+ m.

2n Observe that 6n+1 - 1 = X 6,, so that 6, is a polynomial in A, with coefficients

in {-l,O,l) and with constant term 1. Hence lJn-l ( < & and (6n+l-1 1 < &, so that 6 ,4 1 if n -I m. It follows that yn -.r X and oc, 4 1 (because Cn is

1 parabolic for n 2 3) if n 4 W. Thus Cn 4 [: y ] if n -+ and FA is not discrete in

1 GC. # I

Comllarv 111.7. Let P = [t a] E Gc and let A E C with 0 < 1 cX 1 < 1. Then the

subgroup generated by P and AX is not discrete in GC.

i

i Proof. As ~~(APAP-I) = 2 - X2c2, Proposition 111.2 shows that the group is conjugated to PiXc, and the claim follows from Proposition 111.6. #

The groups of Proposition 111.5 are called Hecke ~ U D S [Hec] and have received considerable attention. For some authors, Hecke groups include rX when X E R and

A 2 2; see page 333 in [Leh].

280 Appendix 111

1 Pro~osition 111.8. Consider a real number r c 3, and set X = 2 cos(m). Assume

either that r is irrational or that r = with (p,q) = 1 and p 2 2. Then rX is not 9 discrete in G.

The group rX contains

= J A ~ = [-! -2 cAs(m)].

0 1 --. As the eigenvalues of ms(nr)) are -eim and -e-jm, this CX is a rotation of

angle 2m around -e-lm, in the sense of hyperbolic geometry. If r is irrational, the subgroup of G generated by CX is not discrete in G.

I f r = with (p,q) = 1, there is some power DX of CX which is the rotation of

angle around Set o = dq, so that DX is the rotation of angle 20 around 4 -e-jP@. set also

= (sin po)lI2 -(a po)(s i n po)-lI2 [ 0 s i n(pa)-112 I cos o sin o

'2o= [-sin o cos o].

As E(i) = -e-IP" and as Rgo is the rotation of angle 2o around i (mark the factor 2!),

one has

Assume now that p 1 2. One has

sin a ' IGXI = 2 s i n o w I sln p o 2 s i n o m 8 20-cos 2o s i n o - T i E 3

1 As r < 3 one has q > 6 and o < r/6, so that - < 1. Then Corollary 111.7 shows

that thesubgroup rX generated by DA and AX is not discrete in G. #

For X E C with X - 2 cos(lr/q) not zero and small enough, it is known that rX is not

discrete in GC. See [Magal.

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INDEX

Adjacency matrix of a graph: 1.3 Fundamental construction Ap,k: 2.1.7, 2.8, 1I.b for multi-matrix algebras: 2.1, 2.4

and Hecke algebras: 2.1.9,2.11 for finite von Neumann algebras: 3.1, 3.6

Agen,k ' Atr,k(Mo c M1) : 2.1.8, 2.7, 2.8 Generic /3 : 2.1, 2.7

Graoh:

6 A'+ Bicommutant theorem: 2.2.3 1 Bicoloration number of a graph: 1.4 Bore1 subgroup: 2.10.a

1 Braiding relations: 2.1 Bratteli diagram: 2.3 Bruhat decomposition: 2.10.a

Catalin numbers: 2.7

and the fundamental construction: 2.6.4

Connected pair of algebras: 2.1 Coupling constant: 2.2, 3.2 Covolume of a lattice: 3.3.e Coxeter emnents: 1.4 Coxeter graph: 1.1, 1.4 Coxeter invariant: 4.6 Cusp form: 3.3.e

Depth: 4.1, 4.6 Derived tower: 4.1, 4.6 Dimension of a projection: 3.2 dimM(H) : 3.2 Discrete series: 3.3.a

E = U(N) : 1.1, 1.5 %extension: 2.6.6 ei - relations: 2.1.6, 2.7.5

Factor: 2.1, 2.2, 3.2 Factor of type 111: 3.2 Faithful conditional expectation: 2.6 Faithful trace: 2.1, 2.5 Finite depth: 4.1,4,6 Finite factor: 3.2 Finite index: 3.1, 3.5 Finite representation of a pair: 3.5 Floor of a Bratteli diagram: 2.3 Formal dimension: 3.3.a Full factor: 3.4

labelled: 1.1 marked: 1.1 norm: 1.3 principal: 4.1, 4.6 spectral radius: 1.3 spectral spread: 1.4

Hecke groups: 3.1, I11 Hecke algebra H(G, G,) : 2.10.a Hecke algebra Hq.k : i.1, 2.10.b, 2.11

Inclusion matrix, index matrix: 2.1, 2.3, 3.1, 3.5

Index: 2.1, 3.1, 3.4, 3.7 of semi-simple pairs: 2.1.1 of subfactors: 3.4

Index of pairs of finite von Neumann

algebras: 3.7.5 Infinite conju acy class (icc) group: 3.3.b Involution: 1fa Irreducible subfactor: 3.4 Ising model: 1I.b

0( = {0,2} U { 2 ~ 0 s ( l r / k ) ) ~ > ~ : 1.1 - K = a given field: 2.1 K, : 2.3

U

[] = [ f ] - [iFl] : 2.8 '

ronecker s theorem: 1.1.1, 1.2.1, 1.2.2

Lattice (in a Lie group) : 3.3.b A! : 2.1, 2.3, 3.1, 3.5

(Aq)q22 1.171.1

Markov relation: 3.1 -- Markov trace: 2.1, 2.7, 3.1, 3.7

and index for multi-matrix algebras:

~ a t r i x adjacency: 1.3 aperiodic (=primitive) non-negati~e:

1.3 of a bicolored graph: 1.3 equivalent: 1.3 indecomposable: 1.1, 1.2

Index

index matrix = inclusion matrix: 2.1, 2.3, 3,1, 3,5

irredundant: 1.3 norm of a matrix: 1.1 parabolic: I11 pseud~quivalent : 1 .l, 1.3 reducible: 1.3 trace matrix: 3.1, 3.5 transfer matrix: 1I.b

Modulus of a Markov trace: 2.1, 2.7, 3.1, 3.7

Monomial in A,., : 2.8

4s ) : 1.1 Natural trace: 3.2 von Neumann algebra: 3.2 Normalized trace: 3.2

Pk E Z[A] : 2.8 Parabolic matrix: I11 Partition function: 1I.b Path model: 2.3.11

and the fundamental construction: 2.4.6, 2.6.5

and the tower construction: 2.7.6 Perron-Frobenius theory: 1.4 Peterson inner product: 3.3.e Pimsner-Popa basis: 3.6.4 Popa's theorem: 4.7.3 Positive conditional expectation: 1I.a Positive involution: I1.a Positive trace: 2.1, 2.5, ILa Principal graph: 4.1,4.6

Rank of a module: 2.1 Reduction b a projection: 2.2, 3.2 R e y a r sub!actor: 3.4 Raw vector: 2.1, 2.5

Self-adjoint conditional expectation: 1I.a Skau's lemma: 4.4.3 Skolem-Noether theorem: 2.2.6 Square lattice Pott' s model: 1I.b Story of a Bratteli diagram: 2.3

Temperley-Lieb algebras ( A ): 2.1, P,k

. I.. 2.7, 2.8, 2.11, 1I.b 3 + 3''' : 4.5.2 -. Tower: 2.1, 2.4 Trace: 2.1,2.5, 3.2, 3.5,11.a Trace matrix: 3.1, 3.5 Transfer matrix: 1I.b 8 + 51/2)1/2 : 1.1

unnel construction: 4.7.e

Ultraweak topology: 3.2

Very faithful conditional expectation: 2.6

Weights of a trace: 2.5 Wenzl' s representations: 2.10.d

index formula: 4.3

Young diagrams: 2.10.c