Operator Algebras and Their Applications

671

Operator Algebras and TheirApplications

A Tribute to Richard V. Kadison

AMS Special SessionOperator Algebras and Their Applications:

A Tribute to Richard V. KadisonJanuary 10–11, 2015

San Antonio, TX

Robert S. DoranEfton Park

Editors

American Mathematical Society





San Antonio, TX


Editors

Richard V. Kadison

671





San Antonio, TX


Editors

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Catherine Yan

2010 Mathematics Subject Classification. Primary 46L05, 46L10, 46L35, 46L55, 46L87,19K56, 22E45.

The photo of Richard V. Kadison on page ii is courtesy of Gestur Olafsson.

Library of Congress Cataloging-in-Publication Data

Names: Kadison, Richard V., 1925- | Doran, Robert S., 1937- | Park, Efton.Title: Operator algebras and their applications : a tribute to Richard V. Kadison : AMS

Special Session, January 10-11, 2015, San Antonio, Texas / Robert S. Doran, Efton Park, editors.Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Con-

temporary mathematics ; volume 671 | Includes bibliographical references.Identifiers: LCCN 2015043280 | ISBN 9781470419486 (alk. paper)Subjects: LCSH: Operator algebras–Congresses. — AMS: Functional analysis – Selfadjoint

operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – General theory of C∗-algebras. msc | Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neumann(W ∗-) algebras, etc.) – General theory of von Neumann algebras. msc | Functional analysis –Selfadjoint operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – Classificationsof C∗-algebras. msc | Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neu-mann (W ∗-) algebras, etc.) – Noncommutative dynamical systems. msc | Functional analysis –Selfadjoint operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – Noncommu-tative differential geometry. msc | K-theory – K-theory and operator algebras – Index theory.msc | Topological groups, Lie groups – Lie groups – Representations of Lie and linear algebraicgroups over real fields: analytic methods. msc Classification: LCC QA326 .O6522 2016 | DDC512/.556–dc23 LC record available at http://lccn.loc.gov/2015043280Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/671

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10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16

To Karen Kadison

Contents

Preface ix

List of Participants xi

Exactness and the Kadison-Kaplansky conjecturePaul Baum, Erik Guentner, and Rufus Willett 1

Generalization of C∗-algebra methods via real positivity for operator andBanach algebras

David P. Blecher 35

Higher weak derivatives and reflexive algebras of operatorsErik Christensen 69

Parabolic induction, categories of representations and operator spacesTyrone Crisp and Nigel Higson 85

Spectral multiplicity and odd K-theory-IIRonald G. Douglas and Jerome Kaminker 109

On the classification of simple amenable C*-algebras with finite decompositionrank

George A. Elliott and Zhuang Niu 117

Topology of natural numbers and entropy of arithmetic functionsLiming Ge 127

Properness conditions for actions and coactionsS. Kaliszewski, Magnus B. Landstad, and John Quigg 145

Reflexivity of Murray-von Neumann algebrasZhe Liu 175

Hochschild cohomology for tensor products of factorsFlorin Pop and Roger R. Smith 185

On the optimal paving over MASAs in von Neumann algebrasSorin Popa and Stefaan Vaes 199

Matricial bridges for “Matrix algebras converge to the sphere”Marc A. Rieffel 209

Structure and applications of real C∗-algebrasJonathan Rosenberg 235

vii

viii CONTENTS

Separable states, maximally entangled states, and positive mapsErling Størmer 259

Preface

Richard V. Kadison has been a towering figure in the study of operator alge-bras for more than 65 years. His research and leadership in the field have beenfundamental in the development of the subject, and his influence continues to befelt though his work and the work of his many students, collaborators, and mentees.

This volume contains the proceedings of an AMS Special Session OperatorAlgebras and Their Applications: A Tribute to Richard V. Kadison, held on January10-11, 2015, in San Antonio, Texas. The table of contents reveals contributions byan outstanding group of internationally known mathematicians. Most of the papersare expanded versions of the authors’ talks in San Antonio.

This volume features expository papers as well as original research articles,and Dick Kadison’s influence can be seen throughout. All the articles have beencarefully refereed and will not appear in print elsewhere. All of the contributorsare esteemed members of the mathematical community, and for this reason wehave elected to simply present the papers in alphabetical order by the first-namedauthor.

We the editors thank everyone who participated in both the AMS Special Ses-sion and the preparation of this volume. Without the hard work of the authors andthe referees, as well as the editorial staff of the American Mathematical Society, thisvolume would have never seen the light of day. We especially thank Christine M.Thivierge for her invaluable assistance and patience. In addition, we thank GesturOlafsson for his permission to use the photo of Dick that appears in the volume,and also Bogdan Oporowski for his nice editing work on the picture.

Finally, we express our great appreciation for Dick Kadison. The subject ofoperator algebra, and indeed mathematics itself, would have been much different,and poorer, without his contributions.

Robert S. Doran

Efton Park

ix

List of Participants

Roy M. AraizuSan Jose State University

Joe BallVirginia Tech University

Paul BaumPennsylvania State Unversity

Alex BeardenUniversity of Houston

David P BlecherUniversity of Houston

Remi BoutonnetUniversity of California, San Diego

Michael BrannanUniversity of Illinois,Urbana-Champaign

Joel CohenUniversity of Maryland

Ken DavidsonUniversity of Waterloo

Bruce DoranAccenture

Bob DoranTexas Christian University

Ronald DouglasTexas A&M University

Ken DykemaTexas A&M University

Edward EffrosUniversity of California, Los Angeles

Søren EilersUniversity of Copenhagen

George ElliottUniversity of Toronto

Adam FullerUniversity of Nebraska, Lincoln

Liming GeUniversity of New Hampshire andChinese Academy of Sciences

Elizabeth GillaspyUniversity of Colorado, Boulder

James GlimmStony Brook University

Jan GregusAbraham Baldwin Agricultural College

Benjamin HayesVanderbilt University

Nigel HigsonPennsylvania State University

Richard KadisonUniversity of Pennsylvania

David KerrTexas A&M University

Magnus LandstadNorwegian University of Science andTechnology

David LarsonTexas A&M University

Zhe LiuUniversity of Central Florida

Jireh LoreauxUniversity of Cincinnati

xi

xii LIST OF PARTICIPANTS

Terry LoringUniversity of New Mexico

Martino LupiniYork University (Canada)

Ellen MaycockAmerican Mathematical Society

Azita MayeliCity University of New York

Matt McBrideUniversity of Oklahoma

Niels MeesschaertKU Leuven (Belgium)

Ramis MovassaghMIT and Northeastern University

Paul MuhlyUniversity of Iowa

Magdalena MusatUniversity of Copenhagen

Pieter NaaijkensLeibniz Univeritat Hannover

Judith PackerUniversity of Colorado, Boulder

Efton ParkTexas Christian University

Geoffry PriceUnited States Naval Academy

Sorin PopaUniversity of California, Los Angeles

Ian PutnamUniversity of Victoria

Timothy RainoneTexas A&M University and Universityof Waterloo

Kamran ReihaniTexas A&M University

Marc A RieffelUniversity of California, Berkeley

Min RoUniversity of Oregon

Mikael RøordamUniversity of Copenhagen

Jonathan RosenbergUniversity of Maryland

Christopher SchafhauserUniversity of Nebraska, Lincoln

Mohamed W. SesayHoward University

Juhhao ShenUniversity of New Hampshire

Fred ShultzWellesley College

Roger SmithTexas A&M University

Baruch SolelTechnion (Israel)

Myungsin-Sin SongSouthern Illinois University

Erling StørmerUniversity of Oslo

Wojciech SzymanskUniversity of Southern Denmark

Mark TomfordeUniversity of Houston

John VastolaUniversity of Central Florida

Henry WarchallNational Science Foundation

Gary WeissUniversity of Cincinnati

Alan WigginsUniversity of Michigan, Dearborn

Wei ZhangPurdue University

Contemporary MathematicsVolume 671, 2016http://dx.doi.org/10.1090/conm/671/13501

Exactness and the Kadison-Kaplansky conjecture

Paul Baum, Erik Guentner, and Rufus Willett

With affection and admiration, we dedicate this paper to Richard Kadisonon the occasion of his ninetieth birthday.

Abstract. We survey results connecting exactness in the sense of C∗-algebratheory, coarse geometry, geometric group theory, and expander graphs. Wesummarize the construction of the (in)famous non-exact monster groups whose

Cayley graphs contain expanders, following Gromov, Arzhantseva, Delzant,Sapir, and Osajda. We explain how failures of exactness for expanders andthese monsters lead to counterexamples to Baum-Connes type conjectures:the recent work of Osajda allows us to give a more streamlined approach thancurrently exists elsewhere in the literature.

We then summarize our work on reformulating the Baum-Connes con-jecture using exotic crossed products, and show that many counterexamplesto the old conjecture give confirming examples to the reformulated one; ourresults in this direction are a little stronger than those in our earlier work.Finally, we give an application of the reformulated Baum-Connes conjectureto a version of the Kadison-Kaplansky conjecture on idempotents in groupalgebras.

1. Introduction

The Baum-Connes conjecture relates, in an important and motivating specialcase, the topology of a closed, aspherical manifold M to the unitary representationsof its fundamental group. Precisely, it asserts that the Baum-Connes assembly map

(1.1) K∗(M) → K∗(C∗red(π1(M))

is an isomorphism from the K-homology of M to the K-theory of the reducedC∗-algebra of its fundamental group. The injectivity and surjectivity of the Baum-Connes assembly map have important implications—injectivity implies that thehigher signatures of M are oriented homotopy invariants (the Novikov conjecture),and thatM (assumed now to be a spin manifold) does not admit a metric of positivescalar curvature (the Gromov-Lawson-Rosenberg conjecture); surjectivity impliesthat the reduced C∗-algebra of π1(M) does not contain non-trivial idempotents(the Kadison-Kaplansky conjecture). For details and more information, we refer to[8, Section 7].

The first author was partially supported by NSF grant DMS-1200475. The second author

was partially supported by a grant from the Simons Foundation (#245398). The third author waspartially supported by NSF grant DMS-1401126.

c©2016 American Mathematical Society

1

2 PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

Proofs of the Baum-Connes conjecture, or of its variants, generally involvesome sort of large-scale or coarse geometric hypothesis on the universal cover ofthe manifold M . A sample result, important in the context of this piece, is thatif the universal cover of M is coarsely embeddable in a Hilbert space, then theBaum-Connes assembly map is injective [64, 77], so that the Novikov conjectureholds for M .

For some time, it was thought possible that every bounded geometry metricspace was coarsely embeddable in Hilbert space. At least there was no counterexam-ple to this statement until Gromov made the following assertions [52]: an expanderdoes not coarsely embed in Hilbert space, and there exists a countable discretegroup that ‘contains’ an expander in an appropriate sense. With the appearance ofthis influential paper, there began a period of rapid progress on counterexamplesto the Baum-Connes conjecture [37], and other conjectures. In particular, theseso-called Gromov monster groups were found to be counterexamples to the Baum-Connes conjecture with coefficients ; they were also found to be the first examplesof non-exact groups in the sense of C∗-algebra theory; and expander graphs werefound to be counterexamples to the coarse Baum-Connes conjecture. We mentionthat, while counterexamples to most variants of the Baum-Connes conjecture havebeen found, there is still no known counterexample to the conjecture as we havestated it in (1.1).

The point of view we shall take in this survey is that the failure of exactness,and the failure of the Baum-Connes conjecture (with coefficients) are intimatelyrelated. This point of view is not particularly novel—the counterexamples givenby Higson, Lafforgue and Skandalis all have the failure of exactness as their rootcause [37]. More recent work has moreover suggested that at least some of thecounterexamples can be obviated by forcing exactness [21,25,57,74,75].

We shall exploit this point of view to reformulate the conjecture by replacing thereduced C∗-algebra of the fundamental group, and the associated reduced crossedproduct that is used when coefficients are allowed, on the right hand side by anew group C∗-algebra and crossed product; the new crossed product will by itsdefinition be exact . By doing so, we shall undercut the arguments that have lead inthe past to the counterexamples, and indeed, we shall see that some of the formercounterexamples are confirming examples for the new, reformulated conjecture.

To close this introduction, we give a brief outline of the paper. The firstseveral sections are essentially historical. In Sections 2 and 3 we provide backgroundinformation on exact groups, crossed products, and the group theoretic and coarsegeometric properties relevant for the theory surrounding the Baum-Connes andNovikov conjectures. We discuss the relationships among these properties, andtheir connection to other areas of C∗-algebra theory.

Section 4 contains a detailed discussion of expanders, focusing on the aspectsof the theory necessary to produce the counterexamples that will appear in latersections. Here, we follow an approach outlined by Higson in a talk at the 2000Mount Holyoke conference, but updated to a slightly more modern perspective.For the Baum-Connes conjecture itself, expanders provide counterexamples throughthe theory of Gromov monster groups. In Section 5 we describe the history andrecent progress on the existence of these groups, beginning with the original paperof Gromov and ending with the recent work of Arzhantseva and Osajda.

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE 3

The remainder of the paper is dedicated to discussion of the implications forthe Baum-Connes conjecture itself. We begin in Section 6 by recalling the nec-essary machinery to define the conjecture, focusing on the aspects needed for thesubsequent discussion. In Section 7 we describe how Gromov monster groups givecounterexamples to the conjecture, by essentially reducing to the discussion of ex-panders given in Section 4. In Section 8 we describe how to adjust the right handside of the Baum-Connes conjecture by replacing the reduced C∗-algebraic crossedproduct with a new crossed product functor having better functorial properties andin Section 9 we explain how and why this reformulated conjecture outperforms theoriginal by verifying it in the setting of the counterexamples from Section 7. InSection 10 we give an application of the reformulated conjecture to the Kadison-Kaplansky conjecture for the �1-algebra of a group.

2. Exact groups and crossed products

Throughout, G will be a countable discrete group. Much of what follows makessense for arbitrary (second countable) locally compact groups, and indeed this isthe level of generality we worked at in our original paper [10]. Here, we restrict tothe discrete case because it is the most relevant for non-exact groups, and becauseit simplifies some details.

A G-C∗-algebra is a C∗-algebra equipped with an action α of G by ∗-automor-phisms. The natural representations for G-C∗-algebras are the covariant repre-sentations : these consist of a C∗-algebra representation of A as bounded linearoperators on a Hilbert space H, together with a unitary representation of G on thesame Hilbert space,

π : A → B(H) and u : G → B(H),

satisfying the covariance condition π(αg(a)) = ugπ(a)ug−1 . Essentially, a covariantrepresentation spatially implements the action of G on A.

Crossed products of a G-C∗-algebra A encode both the algebra A and the G-action into a single, larger C∗-algebra. We introduce the notation A�alg G for thealgebra of finitely supported A-valued functions on G equipped with the followingproduct and involution:

f1 � f2(g) =∑h∈G

f1(h)αg(f2(h−1g)) and f∗(t) = αg(f(g

−1)∗).

The algebra A �alg G is functorial for G-equivariant ∗-homomorphisms in the ob-vious way. We shall refer to A �alg G as the algebraic crossed product of A byG. Finally, a covariant representation integrates to a ∗-representation of A�alg Gaccording to the formula

π � u(f) =∑g∈G

π(f(g))ug .

Two completions of the algebraic crossed product to a C∗-algebra are classicallystudied: the maximal and reduced crossed products. The maximal crossed productis the completion of A�alg G for the maximal norm, defined by

‖f‖max = sup{ ‖π � u(f)‖ : (π, u) a covariant pair }.Thus, the maximal crossed product has the universal property that every covariantrepresentation integrates (uniquely) to a representation of A �max G; indeed, it


is characterized by this property. The reduced crossed product is defined to bethe image of the maximal crossed product in the integrated form of a particularcovariant representation. Precisely, fix a faithful and non-degenerate representationπ of A on a Hilbert space H and define a covariant representation

π : A → B(H⊗ �2(G)) and λ : G → B(H⊗ �2(G))

by

π(a)(v ⊗ δg) = π(αg−1(a))v ⊗ δg and λh(v ⊗ δg) = v ⊗ δhg.

The reduced crossed product A �red G is the image of A �max G under theintegrated form of this covariant representation. In other words, A �red G is thecompletion of A�alg G for the norm

‖f‖red = ‖π � λ(f)‖.

The reduced crossed product (and its norm) are independent of the choice of faithfuland non-degenerate representation of A. Incidentally, one may check that π � λ isinjective on A �alg G. It follows that the maximal norm is in fact a norm on thealgebraic crossed product—no non-zero element has maximal norm equal to zero.In particular, we may view the algebraic crossed product as contained in each ofthe maximal and reduced crossed products as a dense ∗-subalgebra.

Kirchberg and Wassermann introduced, in their work on continuous fields ofC∗-algebras, the notion of an exact group [42]. They define a group G to be exactif, for every short exact sequence of G-C∗-algebras

(2.1) 0 �� I �� A �� B �� 0

the corresponding sequence of reduced crossed products

0 �� I �red G �� A�red G �� B �red G �� 0

is itself short exact. Several remarks are in order here. First, the map to B�redG isalways surjective, the map from I�redG is always injective, and the composition ofthe two non-trivial maps is always zero. In other words, exactness of the sequencecan only fail in that the image of the map into A�redG may be properly containedin the kernel of the following map. Second, the sequence obtained by using themaximal crossed product (instead of the reduced) is always exact; this followsessentially from the universal property of the maximal crossed product.

There is a parallel theory of exact C∗-algebras in which one replaces the reducedcrossed products by the spatial tensor products . In particular, a C∗-algebra D isexact if for every short exact sequence of C∗-algebras (2.1)—now without groupaction—the corresponding sequence

0 �� I ⊗D �� A⊗D �� B ⊗D �� 0

is exact. Here, we use the spatial tensor product; the analogous sequence definedusing the maximal tensor products is always exact, for any D. In the presentcontext, the theories of exact (discrete) groups and exact C∗-algebras are relatedby the following result of Kirchberg and Wassermann [42, Theorem 5.2].

2.1. Theorem. A discrete group is exact (as a group) precisely when its reducedC∗-algebra is exact (as a C∗-algebra).


A central notion for us will be that of a crossed product functor . By this weshall mean, for each G-C∗-algebra A a completion A�τ G of the algebraic crossedproduct fitting into a sequence

A�max G → A�τ G → A�red G,

in which each map is the identity when restricted to A�algG (a dense ∗-subalgebra ofeach of the three crossed product C∗-algebras). This is equivalent to requiring thatthe τ -norm dominates the reduced norm on the algebraic crossed product. Further,we require that A�τ G be functorial, in the sense that if A → B is a G-equivariant∗-homomorphism then the associated map on algebraic crossed products extends(uniquely) to a ∗-homomorphism A�τ G → B �τ G.

We shall call a crossed product functor τ exact if, for every short exact sequenceof G-C∗-algebras (2.1) the associated sequence

0 �� I �τ G �� A�τ G �� B �τ G �� 0

is itself short exact. For example, the maximal crossed product is exact for everygroup, but the reduced crossed product is exact only for exact groups. We shall seeother examples of exact (and non-exact) crossed products later.

3. Some properties of groups, spaces, and actions

In this section, we shall discuss some properties that are important for the studyof the Baum-Connes conjecture, and for issues related to exactness: a-T-menabilityof groups and coarse embeddability of metric spaces, and their relation to variousforms of amenability.

The following definition—due to Gromov [26, Section 7.E]—is fundamental forwork on the Baum-Connes conjecture.

3.1. Definition. A countable discrete group G is a-T-menable if it admits anaffine isometric action on a Hilbert space H such that the orbit of every v ∈ Htends to infinity; precisely,

‖g · v‖ → ∞ ⇐⇒ g → ∞Note here that the forward implication is always satisfied; the essential part of

the definition is the reverse implication which asserts that as g leaves every finitesubset of G the orbit g · v must leave every bounded subset of H.

To discuss the coarse geometric properties relevant for the Novikov conjecture,we must view the countable discrete group G as a metric space. Let us for themoment imagine that G is finitely generated. We fix a finite generating set S,so that every element of G is a finite product, or word, of elements of S and theirinverses. We define the associated word length by declaring the length of an elementg to be the minimal length of such a word; we denote this by |g|. This word lengthfunction is a proper length function, meaning that it is a non-negative real valuedfunction with the following properties:

|g| = 0 iff g = identity, |g−1| = |g|, |gh| ≤ |g|+ |h|;and infinite subsets of G have unbounded image in [0,∞). Returning to the generalsetting, it is well kown (and not difficult to prove) that a countable discrete groupadmits a proper length function.

We now equip G with a proper length function, for example a word length, anddefine the associated metric on G by d(g, h) = |g−1h|. This metric has bounded


geometry , meaning that there is a uniform bound on the number of elements ina ball of fixed radius. It is also left-invariant , meaning that the the left actionof G on itself is by isometries. This metric is not intrinsic to G, but depends onthe particular length function chosen. Nevertheless, the identity map is a coarseequivalence between any two bounded geometry, left invariant metrics on G. Weshall expand on this fact below. Thus, it makes sense to attribute metric propertiesto G as long as these properties are insentive to coarse equivalence.

Having equipped G with a metric, we are ready to state the following definition,also due to Gromov [26, Section 7.E].1

3.2. Definition. A countable discrete group G is coarsely embeddable (inHilbert space) if it admits a map f : G → H to a Hilbert space such

‖f(g)− f(h)‖ → ∞ ⇐⇒ d(g, h) → ∞.

In this case, f is a coarse embedding.

3.3. Remark. To relate coarse embeddability and a-T-menability, suppose thatG acts on a Hilbert space H as in Definition 3.1. Fix v ∈ H and notice that

‖g · v − h · v‖ = ‖g−1h · v − v‖ ∼ ‖g−1h · v‖(where∼means differing at most by a universal additive constant). Thus, forgettingthe action and recalling that the metric on G has bounded geometry, we see thatthe orbit map f(g) = g · v is a coarse embedding as in Definition 3.2.

3.4.Remark. Only the metric structure ofH enters into the definition of coarseembeddability; the same definition applies equally well to maps from one metricspace to another. In this more general setting, a coarse embedding f : X → Y ofmetric spaces is a coarse equivalence if for some universal constant C every elementof Y is a distance at most C from f(X). It is in this sense that the identity mapon G is a coarse equivalence between any two bounded geometry, left invariantmetrics. The key point here is that the balls centered at the identity in each metricare finite, so that the length function defining each metric is bounded on the ballsfor the other.

To motivate the relevance of these properties for the Baum-Connes and Novikovconjectures suppose, for example, that G acts on a finite dimensional Hilbert spaceas in Definition 3.1. It is then a discrete subgroup of some Euclidean isometrygroup Rn�O(n) (at least up to taking a quotient by a finite subgroup). That suchgroups satisfy the Baum-Connes conjecture follows already from Kasparov’s 1981conspectus [41, Section 5, Lemma 4], which predates the conjecture itself!

The relevance of the general, infinite dimensional version of a-T-menability, andof coarse embeddability, was apparent to some authors more than twenty years ago.See for example [51, Problems 3 and 4] of Gromov and [68, Problem 3] of Connes.The key technical advance that allowed progress is the infinite dimensional Bottperiodicity theorem of Higson, Kasparov, and Trout [36]. One has the followingtheorem: the part dealing with a-T-menability is due to Higson and Kasparov[34, 35], while the part dealing with coarse embeddability is due in main to Yu[77], although with subsequent improvements of Higson [33] and of Skandalis, Tu,and Yu [64].

1Gromov originally used the terminology uniformly embeddable and uniform embedding ; thisusage has fallen out of favor since it conflicts with terminology from Banach space theory.


3.5. Theorem. Let G be a countable discrete group. The Baum-Connes as-sembly map with coefficients in any G-C∗-algebra A

Ktop∗ (G;A) → K∗(A�red G)

is an isomorphism if G is a-T-menable, and is split injective if G is coarsely em-beddable.

The class of a-T-menable groups is large: it contains all amenable groups aswell as free groups, and classical hyperbolic groups; see [22] for a survey. A-T-menability admits many equivalent formulations: the key results are those ofAkemann and Walter studying positive definite and negative type functions [1],and of Bekka, Cherix and Valette relating a-T-menabilty as defined above to theproperties studied by Akemann and Walter [12] (the latter are usually called theHaagerup property due to their appearance in important work of Haagerup [31] onC∗-algebraic approximation results). There are, however, many non a-T-menablegroups: the most important examples are those with Kazhdan’s property (T ) suchas SL(3,Z): see the monograph [13].

The class of coarsely embeddable groups is huge: as well as all a-T-menablegroups, it contains for example all linear groups (over any field) [29] and all Gromovhyperbolic groups [60]. Indeed, for a long time it was unknown whether thereexisted any group that did not coarsely embed: see for example [26, Page 218, point(b)]. Thanks to expander based techniques which we will explore in later sections,it is now known that non coarsely embeddable groups exist; it is enormously usefulhere that coarse emeddability makes sense for arbitrary metric spaces, and not justgroups.

Before we turn to a discussion of expanders in the next section, we shall de-scribe the close relationship of coarse embeddability to exactness and some otherproperties of metric spaces, groups and group actions. The key additional idea isthat of Property A, which was introduced by Yu to be a relatively easily verifiedcriterion for coarse embeddability [77, Section 2]. Property A was quickly realizedto be relevant to exactness: see for example [43, Added note, page 556].

All the properties we have discussed so far can be characterized in terms ofpositive definite kernels , and doing so brings the distinctions among them intosharp focus. Recall that a (normalized) positive definite kernel on a set X is afunction f : X ×X → C satisfying the following properties:

(i) k(x, x) = 1 and k(x, y) = k(y, x), for all x, y ∈ X;(ii) for all finite subsets {x1, . . . , xn} of X and {a1, . . . , an} of C,

n∑i,j=1

aiajk(xi, xj) ≥ 0.

If we are working with a countable discrete group G we may additionally requirethe kernel to be left invariant, in the sense that k(g1g, g1h) = k(g, h) for every g1,g and h ∈ G.

3.6. Theorem. Let X be a bounded geometry, uniformly discrete metric space.Then X has Property A if and only if for every R (large) and ε (small), there existsa positive definite kernel k : X ×X → C such that

(i) |1− k(x, y)| < ε whenever d(x, y) < R;(ii) the set { d(x, y) ∈ [0,∞) : k(x, y) �= 0 } is bounded;


X is coarsely embeddable if and only if for every R and ε there exists a positivedefinite kernel k satisfying (i) above, but instead of (ii) the following weaker con-dition:

(ii)’ for every δ > 0 the set { d(x, y) ∈ [0,∞) : |k(x, y)| ≥ δ } is bounded.

A countable discrete group G is amenable if and only if for every R and ε thereexists a left invariant positive definite kernel satisfying (i) and (ii) above; it isa-T-menable if and only if for every R and ε there exists a left invariant positivedefinite kernel satisfying (i) and (ii)’ above.

The characterization of Property A for bounded geometry, uniformly discretemetric spaces in this theorem is due to Tu [66]. It is particularly useful for studyingC∗-algebraic approximation properties, in particular exactness, as it can be used toconstruct so-called Schur multipliers. The characterization of coarse embeddabilitycan be found in [59, Theorem 11.16]; that of a-T-menability comes from combining[1] and [12]; that of amenability is well known.

The following diagram, in which all the implications are clear from the previoustheorem, summarizes the properties we have discussed:

(3.1) amenability

��

�� Property A≡ exactness

��a-T-menability �� coarse embeddability .

The class of groups with Property A covers all the examples of coarsely embed-dable groups mentioned earlier. Indeed, proving the existence of groups withoutProperty A is as difficult as proving the existence groups that do not coarsely em-bed. Nonetheless, Osajda has recently shown the existence of coarsely embeddable(and even a-T-menable) groups without Property A [56]. In particular, there areno further implications between any of the properties in the diagram.

The following theorem summarizes some of the most important implicationsrelating Property A to C∗-algebra theory.

3.7. Theorem. Let G be a countable discrete group. The following are equiv-alent:

(i) G has Property A;(ii) G admits an amenable action on a compact space;(iii) G is an exact group;(iv) the group C∗-algebra C∗

red(G) is an exact C∗-algebra.

The reader can see the survey [72] or [17, Chapter 5] for proofs of most ofthese results, as well as the definitions that we have not repeated here. The originalreferences are: [38] for the equivalence of (i) and (ii); [30] (partially) and [58] forthe equivalence of (i) and (iv); [43, Theorem 5.2] for (iv) implies (iii) as we alreadydiscussed in Section 2; and (iii) implies (iv) is easy.

Almost all these implications extend to second countable, locally compactgroups with appropriately modified versions of Property A [2,16,24]; the exceptionis (iv) implies (iii), which is an open question in general. Finally, note that Theo-rem 3.7 has a natural analog in the setting of discrete metric spaces: see Theorem4.5 below.


3.8. Remark. There is an analog of the equivalence of (i) and (ii) for coarselyembeddable groups appearing in [64, Theorem 5.4]: a group is coarsely embeddableif and only if it admits an a-T-menable action on a compact space in the sense ofDefinition 9.2 below.

4. Expanders

In this section, we study expanders : highly connected, sparse graphs. Ex-panders are the easiest examples of metric spaces that do not coarsely embed.They are also connected to K-theory through the construction of Kazhdan projec-tions ; this construction is at the root of the counterexamples to the Baum-Connesconjecture.

For our purposes, a graph is a simplicial graph, meaning that we allow neitherloops nor multiple edges. More precisely, a graph Y comprises a (finite) set ofvertices, which we also denote Y , and a set of 2-element subsets of the vertex set,which are the edges. Two vertices x and y are incident if there is an edge containingthem, and we write x ∼ y in this case. The number of vertices incident to a givenvertex x is its degree, denoted deg(x).

A central object for us is the Laplacian of a graph Y , the linear operator�2(Y ) → �2(Y ) defined by

Δf(x) = deg(x)f(x)−∑

y : y∼x

f(y) =∑

y : y∼x

f(x)− f(y).

A straightforward calculation shows that Δ is a positive operator; in fact

(4.1) 〈Δf, f 〉 =∑

(x,y) : x∼y

|f(x)− f(y)|2 ≥ 0.

The kernel of the Laplacian on a connected graph is precisely the space of constantfunctions. Indeed, it follows directly from the definition that constant functionsbelong to the kernel; conversely, it follows from (4.1) that if Δf = 0 and x ∼ y thenf(x) = f(y), so that f is a constant function (using the connectedness). Thus, thesecond smallest eigenvalue (including multiplicity) of the Laplacian on a connectedgraph is strictly positive. We shall denote this eigenvalue by λ1(Y ).

An expander is a sequence (Yn) of finite connected graphs with the followingproperties:

(i) the number of vertices in Yn tends to infinity, as n → ∞;(ii) there exists d such that deg(x) ≤ d, for every vertex in every Yn;(iii) there exists c > 0 such that λ1(Yn) ≥ c, for every n.

From the discussion above, we see immediately that the property of being an ex-pander is about having both the degree bounded above, and the first eigenvaluebounded away from 0, independent of n. The existence of expanders can be provenwith routine counting arguments which in fact show that in an appropriate sensemost graphs are expanders: see [47, Section 1.2]. Nevertheless, the explicit con-struction of expanders was elusive. The first construction was given by Margulis[49]. Shortly thereafter the close connection with Kazhdan’s Property (T) wasunderstood—the collection of finite quotients of a residually finite discrete groupwith Property (T) are expanders, when equipped with the (Cayley) graph structurecoming from a fixed finite generating set of the parent group, and ordered so that


their cardinalities tend to infinity [47, Section 3.3]. In particular, the congruencequotients of SL(3,Z) are an expander.

In the present context, the first counterexamples provided by expanders areto questions in coarse geometry. Given a sequence Yn of graphs comprising anexpander, we consider the associated box space, which is a metric space Y with thefollowing properties:

(i) as a set, Y is the disjoint union of the Yn;(ii) the restriction of the metric to each Yn is the graph metric;(iii) d(Yn, Ym) → ∞ for n �= m and n+m → ∞.

Here, the distance between two vertices in the graph metric is the smallest possiblenumber of edges on a path connecting them. It is not difficult to construct a boxspace; one simply declares that the distance from a vertex in Ym to a vertex in theunion of the Yn for n < m is sufficiently large. Further, the identity map providesa coarse equivalence (see Remark 3.4) between any two box spaces, so that theircoarse geometry is well defined.

4.1. Proposition. A box space associated to an expander sequence is notcoarsely embeddable, and hence does not have property A.

This proposition was originally stated by Gromov, and proofs were later sup-plied by several authors including Higson and Dranishnikov: see for example [59,Proposition 11.29]. More recently, many results of this type have been proven,primarily, negative results about the impossibility of coarsely embedding varioustypes of expanders in various types of Banach space, and other non-positively curvedspaces. See for example [44,46,50].

On the analytic side, expanders have also proven useful for counterexamples,essentially because of the presence of Kazhdan type projections . On a single, con-nected, finite graph Y , we have the projection p onto the constant functions, whichis to say, onto the kernel of the Laplacian. This projection can be obtained as aspectral function of the Laplacian; precisely, p = f(Δ) provided that f(0) = 1 andthat f ≡ 0 on the remaining eigenvalues of Δ.

Now suppose that Y is the box space of a sequence Yn of finite graphs withuniformly bounded vertex degrees. We can then consider the operators

(4.2) p =

⎛⎜⎜⎜⎝p1 0 0 . . .0 p2 0 . . .0 0 p3 . . ....

......

. . .

⎞⎟⎟⎟⎠ and Δ =

⎛⎜⎜⎜⎝Δ1 0 0 . . .0 Δ2 0 . . .0 0 Δ3 . . ....

......

. . .

⎞⎟⎟⎟⎠acting on �2(Y ), identified with the direct sum of the spaces �2(Yn). While p will notgenerally be a spectral function of Δ, it will be when Yn is an expander sequence.Indeed, in this case if f is a continuous function on [0,∞) satisfying f(0) = 1 andf ≡ 0 on [c,∞) then p = f(Δ). We shall refer to p as the Kazhdan projection ofthe expander.

The importance of the Kazhdan projection is difficult to overstate: one canoften show that Kazhdan projections are not in the range of Baum-Connes typeassembly maps, and are therefore fundamental for counterexamples. This is bestunderstood in the context of metric spaces, and to proceed we need to introduce thecoarse geometric analog of the group C∗-algebra. For convenience, we consider here


only the uniform Roe algebra of a (discrete) metric space X. A bounded operatorT on �2(X) has finite propagation if there exists R > 0 such that T cannot propa-gate signals over a distance greater than R: precisely, for every finitely supportedfunction f on X, the support of T (f) is contained in the R-neighborhood of thesupport of f . The collection of all bounded operators having finite propagation is a∗-algebra, and its closure is the uniform Roe algebra of X. We denote the uniformRoe algebra of X by C∗(X), and remark that it contains the compact operators on�2(X) as an ideal.

4.2. Proposition. Let Y be the box space of an expander sequence. The Kazh-dan projection p is not compact, and belongs to C∗(Y ).

Proof. The Kazhdan projection has infinite rank; it projects onto the space offunctions that are constant on each Yn. Further, the Laplacian propagates signalsa distance at most 1, so that both Δ and its spectral function p = f(Δ) belong tothe C∗-algebra C∗(Y ). �

The Kazhdan projection of an expander Y has another significant property:it is a ghost . Here, returning to a discrete metric space X, a ghost is an elementT ∈ C∗(X) whose ‘matrix entries tend to 0 at infinity’; precisely, the suprema

supz∈X

|Txz| and supz∈X

|Tzx|

of matrix entries over the ‘xth row’ and ‘xth column’ tend to zero as x tends toinfinity. With this definition it is immediate that compact operators are ghosts,and easy to see that a finite propagation operator is a ghost precisely when it iscompact. The Kazhdan projection in C∗(Y ) is a (non-compact!) ghost because theelements pn in its matrix representation (4.2) are

pn =1

card(Yn)

⎛⎜⎜⎜⎝1 1 . . . 11 1 . . . 1...

.... . .

...1 1 . . . 1

⎞⎟⎟⎟⎠ .

To understand the importance of ghostliness we recast the definition slightly.We shall denote the Stone-Cech compactification of X by βX, and shall identifyits elements with ultrafilters on X. Each element of X gives rise to an ultrafilter,so that X ⊂ βX. We shall be primarily concerned with the free ultrafilters, thatis, the elements of the Stone-Cech corona β∞X = βX \X. A bounded function φon X has a limit against each ultrafilter. If an ultrafilter corresponds to a point ofX this limit is simply the evaluation of φ at that point; if ω is a free ultrafilter, weshall denote the limit by ω-lim(φ).

Suppose now we are given a (free) ultrafilter ω ∈ β∞X. We define a linearfunctional on C∗(X) by the formula

Ω(T ) = ω-lim(x �→ Txx).

Here, the Txx are the diagonal entries of the matrix representing the operator T inthe standard basis of �2(X). We check that Ω is a state on C∗(X). Indeed, onechecks immediately that Ω(1) = 1, and a simple calculation shows that the diagonal


entries of T ∗T are given by

(4.3) (T ∗T )xx =∑z

|Tzx|2 ;

thus they are non-negative so their limit is as well. Finally, we define C∗∞(X) to be

the image of C∗(X) in the direct sum of the Gelfand-Naimark-Segal representationsof the states defined in this way from free ultrafilters ω; this is a quotient of C∗(X).While the following proposition is well known, not being able to locate a proof inthe literature we provide one here.

4.3. Proposition. The kernel of the ∗-homomorphism C∗(X) → C∗∞(X) is

the set of all ghosts.

Proof. Suppose T is a ghost. Since the ghosts form an ideal in C∗(X) we havethat for every R and S ∈ C∗(X) the product RTS is also a ghost. In particular,its on-diagonal matrix entries (RTS)xx tend to zero as x → ∞, so that their limitagainst every free ultrafilter is also zero. This mean that the norm of T in the GNSrepresentation associated to every free ultrafilter is zero, so that T maps to zero inC∗

∞(X).Conversely, suppose that T maps to zero in C∗

∞(X), so that T ∗T does as well.Hence the limit of the on-diagonal matrix entries (T ∗T )xx is zero against every freeultrafilter, so that they converge to zero in the ordinary sense as x → ∞. Now,according to (4.3) we have

(T ∗T )xx =∑z

|Tzx|2 ≥ supz

|Tzx|2,

so that supz |Tzx|2 → 0 as x → ∞ as well. Applying the same argument to TT ∗

shows that supz |Txz|2 tends to 0 as x → ∞, and thus T is a ghost. �

Putting everything together, we have for the box space Y of an expander ashort sequence

(4.4) 0 �� K �� C∗(Y ) �� C∗∞(Y ) �� 0.

The sequence is not exact because the Kazhdan projection belongs to the kernel ofthe quotient map, although it is not compact. As we shall now show, it is possibleto detect the K-theory class of the Kazhdan projection and to see that the sequenceis not exact even at the level of K-theory. We shall see in Section 7 that this is thephenomenon underlying the counterexamples to the Baum-Connes conjecture.

4.4. Proposition. The K-theory class of the Kazhdan projection is not in theimage of the map K0(K) → K0(C

∗(Y )).

Proof. We have, for each ‘block’ Yn a contractive linear map C∗(Y ) →B(�2(Yn)) defined by cutting down by the appropriate projection. These are asymp-totically multiplicative on the algebra of finite propagation operators, and we obtaina ∗-homomorphism

(4.5) C∗(Y ) →∏

n B(�2(Yn))

⊕nB(�2(Yn)).


Taking the rank in each block (equivalently, taking the map on K-theory in-duced by the canonical matrix trace on B(�2(Yn))) gives a homomorphism

(4.6) K0

(∏n

B(�2(Yn))

)→

∏n

Z.

As K1(⊕nB(�2(Yn)) = 0, the six term exact series in K-theory specializes to anexact sequence

K0(⊕nB(�2(Yn))) �� K0(∏

n B(�2(Yn))) �� K0

(∏n B(�2(Yn))

⊕nB(�2(Yn))

)�� 0 .

Composing the ‘rank homomorphism’ in line (4.6) with the quotient map∏

n Z →∏n Z/ ⊕n Z clearly annihilates the image of K0(⊕nB(�2(Yn))), and thus from the

sequence above gives rise to a homomorphism

K0

(∏n B(�2(Yn))

⊕nB(�2(Yn))

)→

∏n Z

⊕nZ.

Finally, combining with the K-theory map induced by the ∗-homomorphism in line(4.5) gives a group homomorphism

K0(C∗(Y )) →

∏n Z

⊕nZ.

AnyK-theory class in the image of the mapK0(K) → K0(C∗(Y )) goes to zero under

the map in the line above. On the other hand, the Kazhdan projection restricts toa rank one projection on each �2(Yn) and therefore its image is [1, 1, 1, 1, . . . ], andso is non-zero. �

The failure of the above sequence (4.4) to be exact is at the base of manyfailures of exactness and other approximation properties in operator theory andoperator algebras. See for example the results of Voiculescu [70] and Wassermann[71]. More recently, the results in the following theorem (an analog of Theorem 3.7for metric spaces) have been filled in, clarifying the relationship between PropertyA, ghosts, amenability and exactness. Note that the box space of an expander, or acountable discrete group with proper, left invariant metric, satisfy the hypotheses.

4.5. Theorem. Let X be a bounded geometry uniformly discrete metric space.Then the following are equivalent:

(i) X has Property A;(ii) the coarse groupoid associated to X is amenable;(iii) the uniform Roe algebra C∗(X) is an exact C∗-algebra;(iv) all ghost operators are compact.

These results can be found in the following references: [64, Theorem 5.3] for theequivalence of (i) and (ii), and the definition of the coarse groupoid; [62] for theequivalence of (i) and (iii); [59, Proposition 11.4.3] for (i) implies (iv); and [61] for(iv) implies (i).

5. Gromov monster groups

As mentioned in the introduction, the search for counterexamples to the Baum-Connes conjecture began in earnest with the provocative remarks found in thelast section of [52]. There, Gromov describes a model for a random presentation


of a group and asserts that under certain conditions such a random group willalmost surely not be coarsely embeddable in Hilbert space, or in any �p-space forfinite p. The non-embeddable groups arise by randomly labeling the edges of asuitable expander family with labels that correspond to the generators of a given,for example, free group. For the method to work, it is necessary that the expanderhave large girth: the length of the shortest cycle in the nth constituent graph tendsto infinity with n. Thus labeled, cycles in the expander graphs give words in thegenerators which are viewed as relators in an (infinite) presentation of a randomgroup. Gromov then goes on to state that a further refinement of the method wouldreveal that certain of these random and non-embeddable groups are themselvessubgroups of finitely presented groups, which are therefore also non-embeddable.More details appeared in the subsequent paper of Gromov [27], and in the furtherwork of Arzhantseva and Delzant [3].

From the above sketch given by Gromov, it is immediately clear that the originalexpander graphs Yn would in some sense be ‘contained’ in the Cayley graph of therandom group G. And groups ‘containing expanders’ became known as Gromovmonsters . As is clearly explained in a recent paper of Osajda [56], it is an inherentlimitation of Gromov’s method that the expanders will not themselves be coarselyembedded in the random group. Rather, they will be ‘contained’ in the followingweaker sense: there exist constants a, b and cn such that cn is much smaller thanthe diameter of Yn, and such that for each n there exists a map fn : Yn → Gsatisfying

bd(x, y)− cn ≤ d(f(x), f(y)) ≤ ad(x, y).

In other words, each Yn is quasi-isometrically embedded in the Cayley graph of G,but the additive constant involved in the lower bound decays as n → ∞. This isnevertheless sufficient for the non-embeddability of G, and for the counterexamplesof Higson, Lafforgue and Skandalis [37] (who in fact use a still weaker form of‘containment’).

In part as a matter of convenience, and in part out of necessity, we shall adoptthe following more restricted notion of Gromov monster group.

5.1. Definition. A Gromov monster (or simply monster) group is a discretegroup G, equipped with a fixed finite generating set and which has the followingproperty: there exists a subset Y of G which is isometric to a box space of a largegirth, constant degree, expander.

Here, it is equivalent to require that each of the individual graphs Yn comprisingthe expander are isometrically embedded in G; using the isometric action of G onitself, it is straightforward to arrange the Yn (rather, their images in G) into a boxspace.

Building on earlier work with Arzhantseva [5], groups as in this definition wereshown to exist by Osajda: see [56, Theorem 3.2]. We recall in rough outline themethod. The basic data is a sequence of finite, connected graphs Yn of uniformlyfinite degree satisfying the following conditions:

(i) diam(Yn) → ∞;(ii) diam(Yn) ≤ A girth(Yn), for some constant A independent of n;(iii) girth(Yn) ≤ girth(Yn+1), and girth(Y1) > 24.

Here, recall that the girth of a graph is the length of the shortest simple cycle.While the method is more general, in order to construct monster groups the Yn


will, of course, be taken to be a suitable family of expanders. These conditions areless restrictive than those originally proposed by Gromov, and in his paper Osajdadescribes an explicit set of expanders that satisfy them.

Using a combination of combinatorial and probabilistic arguments, Osajda pro-duces two labelings of the edges in the individual Yn with letters from a finite alpha-bet: one satisfies a small cancellation condition for pieces from different blocks andthe other for pieces from a common block. He then combines these in a straightfor-ward way to obtain a labeling that globally satisfies the C ′(1/24) small cancellationcondition. The monster group G is the quotient of the free group on the lettersused in the final labeling by the normal subgroup generated by the relations readalong the cycles of the graphs Yn. It was known from previous work that theC ′(1/24) condition implies that the individual Yn will be isometrically embeddedin the Cayley graph of G [5,55].

The infinitely presented Gromov monsters described here may seem artificial.After all, in the introduction we formulated the Baum-Connes conjecture for fun-damental groups of closed aspherical manifolds, and one may prefer to confineattention to finitely presented groups. Fully realizing Gromov’s original statement,Osajda remarks that a general method developed earlier by Sapir [63] leads tothe existence of closed, aspherical manifolds whose universal covers exhibit similarpathologies. Summarizing, we have the following result.

5.2. Theorem. Gromov monster groups (in the sense of Definition 5.1) exist.Further, there exist closed aspherical manifolds whose fundamental groups containquasi-isometrically embedded expanders.

While groups as in the second statement of this theorem would not qualify asGromov monster groups under our restricted definition above, their existence isvery satisfying.

We shall close this section with a more detailed discussion of the relationshipbetween the properties introduced in Section 3. As we mentioned previously, noneof the implications in diagram (3.1) is reversible. The most difficult point concernsthe existence of discrete groups (or even bounded geometry metric spaces) thatare coarsely embeddable but do not have Property A. The first example of such aspace was given by Arzhantseva, Guentner and Spakula [4] (non-bounded geometryexamples were given earlier by Nowak [54]); their space is the box space in whichthe blocks are the iterated Z/2-homology covers of the figure-8 space, i.e. the wedgeof two circles.

In the case of groups, a much more ambitious problem is the existence of adiscrete group which is a-T-menable, but does not have Property A. Building onearlier work with Arzhantseva, this problem was recently solved by Osajda [5,56].The strategy is similar to the construction of Gromov monsters: use a graphicalsmall cancellation technique to embed large girth graphs with uniformly boundeddegree (at least 3) into the Cayley graph of a finitely generated group. Again, underthe C ′(1/24) hypothesis the graphs will be isometrically embedded. The large girthhypothesis and the assumption that each vertex has degree at least 3 ensure, by aresult of Willett [73], that the group will not have Property A.

The remaining difficulty is to show that the group constructed is a-T-menableunder appropriate hypotheses on the graphs and the labelling. The key idea isdue to Wise, who showed that certain finitely presented classical small cancellation


groups are a-T-menable, by endowing their Cayley graphs with the structure of aspace with walls for which the wall pseudometric is proper [76].

6. The Baum-Connes conjecture with coefficients

When discussing the Baum-Connes conjecture in the introduction, we consid-ered it as a higher index map

(6.1) K∗(M) → K∗(C∗red(G)),

which takes the K-homology class defined by an elliptic differential operator D ona closed aspherical manifold M with fundamental group G to the higher index ofD in the K-theory of the reduced group C∗-algebra of G. This point of view isperhaps the most intuitive way to view the conjecture and also leads to some of itsmost important applications. Here however, we need to get ‘under the hood’ of theBaum-Connes machinery, and give enough definitions so that we can explain ourconstructions.

To formulate the conjecture more generally, and in particular to allow coeffi-cients in a G-C∗-algebra, it is usual to use bivariant K-theory and the notion ofdescent. Even if one is only interested in the classical conjecture of (6.1), the extragenerality is useful as it grants access to many powerful tools, and has much betternaturality and permanence properties under standard operations on groups. Thereare two standard bivariant K-theories available: the KK-theory of Kasparov, andthe E-theory of Connes and Higson. These two theories have similar formal prop-erties, and for our purposes, it would not make much difference which theory weuse (see Remark 8.2 below). However, at the time we wrote our paper [10] it wasonly clear how to make our constructions work in E-theory, and for the sake ofconsistency we use E-theory here as well.

We continue to work with a countable discrete group G. We shall denote thecategory whose objects are G-C∗-algebras and whose morphisms are equivariant

∗-homomorphisms by GC*; similarly, C* denotes the category whose objects areC∗-algebras and whose morphisms are ∗-homomorphisms. Further, we shall assumethat all C∗-algebras are separable.

The equivariant E-theory category, defined in [28] and which we shall denote

EG, is obtained from the category GC* by appropriately enlarging the morphismsets. More precisely, the objects of EG are the G-C∗-algebras. An equivariant∗-homomorphism A → B gives a morphism in EG and further, there is a covariant

functor from GC* to EG that is the identity on objects. We shall denote themorphisms sets in EG by EG(A,B). These are abelian groups, and it follows thatfor a fixed G-C∗-algebra B, the assignments

A �→ EG(A,B) and A �→ EG(B,A)

are, respectively, a contravariant and a covariant functor from GC* to the categoryof abelian groups.

Let now EG denote a universal space for proper actions of G; this means thatEG is a metrizable space equipped with a proper G-action such that the quotientspace is also metrizable, and moreover that any metrizable proper G-space admits acontinuous equivariant map into EG, which is unique up to equivariant homotopy.Such spaces always exist [8]. Suppose X ⊆ EG is a G-invariant and cocompactsubset; this means thatX is closed and that there is a compact subsetK ⊆ EG such


that X ⊆ G ·K. Such an X is locally compact (and Hausdorff), and if X ⊆ Y aretwo such subsets of EG there is an equivariant ∗-homomorphism C0(Y ) → C0(X)defined by restriction. In this way the various C0(X), with X ranging over the G-invariant and cocompact subsets of EG, becomes a directed set of G-C∗-algebrasand equivariant ∗-homomorphisms.

It follows from this discussion that for any G-C∗-algebra A we may form thedirect limit

Ktop0 (G;A) := lim

X⊆EG

X cocompact

EG(C0(X);A),

and similarly for K1 using suspensions. The universal property of EG togetherwith homotopy invariance of the E-theory groups implies that Ktop

∗ (G;A) does notdepend on the choice of EG up to unique isomorphism. It is called the topologicalK-theory of G. This group will be the domain of the Baum-Connes assembly map.

To define the assembly map, we need to discuss descent. Specializing the con-struction of the equivariant E-theory category to the trivial group gives the E-theory category, which we shall denote by E. The objects in this category arethe C∗-algebras, and the morphisms from A to B are an abelian group denotedE(A,B). A ∗-homomorphism A → B gives a morphism in this category, and thereis a covariant functor from the category of C∗-algebras and ∗-homomorphisms to Ethat is the identity on objects. Moreover for any C∗-algebra B, the group E(C, B)identifies naturally with the K-theory group K0(B).

Recall from Section 2 that the maximal crossed product defines a functor from

the category GC* to the category C*. The following theorem asserts that it ispossible to extend this functor to the category EG, so that it becomes defined on

the generalized morphisms belonging to EG but not toGC*; see [28, Theorem 6.22]for a proof.

6.1. Theorem. There is a (maximal) descent functor �max : EG → E whichagrees with the usual maximal crossed product functor both on objects and on mor-phisms in EG coming from equivariant ∗-homomorphisms.

To complete the definition of the Baum-Connes assembly map, we need to knowthat if X is a locally compact, proper and cocompact G-space, then C0(X)�maxGcontains a basic projection, denoted pX , with properties as in the next result: see[28, Chapter 10] for more details.

6.2. Proposition. Let X be a locally compact, proper, cocompact G-space. TheK-theory class of the basic projection [pX ] ∈ K0(C0(X)�maxG) = E(C, C0(X)�max

G) has the following properties:

(i) [pX ] depends only on X (and not on choices made in the definition of pX);(ii) [pX ] is functorial for equivariant maps.

Here, functoriality means that if X → Y is an equivariant map of spaces as inthe statement of the proposition, then the classes [pX ] and [pY ] correspond underthe functorially induced map on K-theory.


Now, let X be a proper, locally compact G-space and let A be a G-C∗-algebra.The assembly map for X with coefficients in A is defined as the composition

EG(C0(X), A) → E(C0(X)�max G,A�max G)

→ E(C, A�max G)

→ E(C, A�red G),

in which the first arrow is the descent functor, the second is composition in E withthe basic projection, and the third is induced by the quotient map A �max G →A �red G. It follows now from property (ii) in Proposition 6.2 that if X → Y isan equivariant inclusion of locally compact, proper, cocompact G-spaces, then thediagram

E(C0(X)�max G,A�max G) ��

��

E(C, A�max G)

E(C0(Y )�max G,A�max G) �� E(C, A�max G)

commutes. Here, the horizontal arrows are given by composition with the appro-priate basic projections, and the left hand vertical arrow is composition with the∗-homomorphism C0(Y )�maxG → C0(X)�maxG induced by the inclusion X → Y .

Hence the assembly maps are compatible with the direct limit defining Ktop0 (G;A),

and give a well-defined homomorphism

Ktop0 (G;A) → E(C, A�red G) = K0(A�red G).

Everything works similarly on the level of K1 using suspensions, and thus we get ahomomorphism

μ : Ktop∗ (G;A) → K∗(A�red G),

which is, by definition, the Baum-Connes assembly map. The Baum-Connes con-jecture states that this map is an isomorphism.

6.3.Remark. Following through the construction above without passing throughthe quotient to the reduced crossed product gives the maximal Baum-Connes as-sembly map

μ : Ktop∗ (G;A) → K∗(A�max G).

It plays an important role in the theory, but is known not to be an isomorphism ingeneral thanks to obstructions that exist whenever G has Kazhdan’s property (T )[13]; we will come back to this point later.

7. Counterexamples to the Baum-Connes conjecture

In this section, we discuss a class of counterexamples to the Baum-Connesconjecture with coefficients. These are based on [37, Section 7] and [74, Section 8],but are a little simpler and more concrete than others appearing in the literature.The possibility of a simpler construction comes down to the straightforward waythe monster groups constructed by Osajda contain expanders.

The existence of counterexamples depends on the following key fact: the leftand right hand sides of the Baum-Connes conjecture see short exact sequences ofG-C∗-algebras differently. To see this, note that the properties of E-theory asdiscussed in Section 6 imply that the Baum-Connes assembly map is functorial in


the coefficient algebra: precisely, an equivariant ∗-homomorphism A → B inducesa commutative diagram

Ktop∗ (G;A)

��

�� K∗(A�red G)

��

Ktop∗ (G;B) �� K∗(B �red G),

in which the horizontal maps are the Baum-Connes assembly maps, and the verticalmaps are induced from the associated morphism A → B in the equivariant E-theorycategory. The following lemma gives a little more information when the maps comefrom a short exact sequence.

7.1. Lemma. Let

0 �� I �� A �� B �� 0

be a short exact sequence of separable G-C∗-algebras. There is a commutative dia-gram of Baum-Connes assembly maps

Ktop0 (G; I) ��

��

Ktop0 (G;A) ��

��

Ktop0 (G;B)

��

K0(I �red G) �� K0(A�red G) �� K0(B �red G),

in which the horizontal arrows are the functorially induced ones. Moreover, the toprow is exact in the middle.

Proof. The existence and commutativity of the diagram follows from ourdiscussion of E-theory. Exactness of the top row follows from exactness propertiesof E-theory (see [28, Theorem 6.20]) and the fact that exactness is preserved underdirect limits. �

The following consequence of the Baum-Connes conjecture with coefficients isimmediate from the lemma.

7.2. Corollary. Let

0 �� I �� A �� B �� 0

be a short exact sequence of separable G-C∗-algebras. If the Baum-Connes conjec-ture for G with coefficients in all of I, A, and B is true then the correspondingsequence of K-groups

K0(I �red G) �� K0(A�red G) �� K0(B �red G)

is exact in the middle.

We will now use Gromov monster groups to give a concrete family of exampleswhere this fails, thus contradicting the Baum-Connes conjecture with coefficients.Assume that G is a monster as in Definition 5.1. In particular, there is assumed tobe a subset Y ⊆ G which is (isometric to) a large girth, constant degree expander.The essential idea is to relate Proposition 4.4 to appropriate crossed products.

To do this, equip �∞(G) with the action induced by the right translation actionof G on itself. Consider the (non-unital) G-invariant C∗-subalgebra of �∞(G) gen-erated by the functions supported in Y ; as this C∗-algebra is commutative, we may


as well write it as C0(W ) where W is its spectrum, a locally compact G-space. Notethat as G acts on itself by isometries on the left, the right action of g ∈ G moveselements by distance exactly the length |g|: in symbols, d(x, xg) = |g| for all x ∈ G.Hence C0(W ) is the closure of the ∗-subalgebra of �∞(G) consisting of all functionssupported within a finite distance from Y . It follows that C0(W ) contains C0(G) asan essential ideal, whence G is an open dense subset of W . Defining ∂W := W \G,it follows that C0(∂W ) = C0(W )/C0(G).

Let ρ denote the right regular representation of G on �2(G) and M the multipli-cation action of �∞(G) on �2(G). Then the pair (M,ρ) is a covariant representationof �∞(G) for the right G-action. Moreover, it is well-known (compare for exam-ple [17, Proposition 5.1.3]) that this pair integrates to a faithful representation of�∞(G)�redG on �2(G) that takes C0(G)�redG onto the compact operators. As thereduced crossed product preserves inclusions, it makes sense to restrict this repre-sentation to C0(W )�red G, thus giving a faithful representation of C0(W )�red Gon �2(G).

The key facts we need to build our counterexamples are contained in the fol-lowing lemma. To state it, let C∗(Y ) denote the uniform Roe algebra of Y andC∗

∞(Y ) the quotient as in Section 4. Represent C∗(Y ) on �2(G) by extending byzero on the orthogonal complement �2(G \ Y ) of �2(Y ).

7.3. Lemma. The faithful representations of C0(W )�redG and C∗(Y ) on �2(G)defined above give rise to a commutative diagram

K(�2(Y ))

��

�� C∗(Y )

��

�� C∗∞(Y )

��

C0(G)�red G �� C0(W )�red G �� C0(∂W )�red G

where the vertical arrows are all inclusions of subsets of the bounded operators on�2(G). Moreover, the vertical arrows are all inclusions of full corners.

Proof. Let χ denote the characteristic function of Y , considered as an elementof C0(W ). Our first goal is to identify the C∗-algebras in the top row of the diagramwith the corners of those in the botom row corresponding to the projection χ. Webegin with the C∗-algebra C0(W ) �red G, which is generated by operators of theform fρg, where f ∈ C0(W ) and g ∈ G. The compression of such an operator

(7.1) χ(fρg)χ : �2(Y ) → �2(Y )

has matrix coefficients

(7.2) 〈 δx, χfρgχ(δy) 〉 = 〈 δx, fρg(δy) 〉 = (fδyg−1)(x) =

{f(x), y = xg

0, else,

for x, y ∈ Y . As discussed above d(x, xg) = |g|, so that the operator in line (7.1) hasfinite propagation (at most |g|). Hence the corner χ(C0(W )�red G)χ is containedin C∗(Y ).

Conversely, suppose T is a finite propagation operator on �2(Y ). For each g ∈ Gdefine a complex valued function fg on G by

fg(x) =

{〈 δx, T δxg 〉, x, xg ∈ Y

0, else.


Now, fg is identically = 0 if |g| is greater than the propagation of T , and anelementary check of matrix coefficients using line (7.2) shows that T is given by the(finite) sum

T =∑g

χ(fgρg)χ.

In particular, T belongs to the corner χ(C0(W )�redG)χ. Since this corner containsall the finite propagation operators on Y , we see that it contains C∗(Y ) as well.

The C∗-algebra C0(∂W )�redG is handled by analogous computations, regard-ing χ as an element of C0(∂W ). Finally, under the identification of C0(G)�red Gwith K(�2(G)), it is clear that K(�2(Y )) = χ(C0(G)�red G)χ.

Having identified the C∗-algebras in the top row of the diagram with cornersof those in the bottom row corresponding to the projection χ it remains to see thatthese corners are full. Again, we begin with the C∗-algebra C0(W ) �red G. Thiscrossed product is generated by operators of the form fρg where f is a boundedfunction with support in the set Y h, for some h ∈ G. Thus, it suffices to show thateach such operator belongs to the ideal of C0(W )�red G generated by χ. Now, thecharacteristic function of Y h, viewed as an operator on �2(G), is ρ∗hχρh. It followsthat

fρg = f(ρ∗hχρh)ρg = (fρh−1)χρhg

belongs to the ideal generated by χ, and we are through.In a similar way, the image of χ is a full projection in C0(∂W ) �red G. Fi-

nally, any non-zero projection on �2(G), and in particular χ, is a full multiplier ofC0(G)�red G = K(�2(G)). �

Now, consider the diagram

(7.3) K0(K)

��

�� K0(C∗(Y ))

��

�� K0(C∗∞(Y ))

��

K0(C0(G)�red G) �� K0(C0(W )�red G) �� K0(C0(∂W )�red G)

functorially induced by the diagram in the above lemma. We showed in Proposition4.4 that the top line is not exact: the class of the Kazhdan projection in K0(C

∗(Y ))is not the image of a class from K0(K), but gets sent to zero in K0(C

∗∞(Y )). As the

vertical maps are induced by inclusions of full corners, they are isomorphisms onK-theory, and so the bottom line is also not exact in the middle: again, the failureof exactness is detected by the class of the Kazhdan projection.

Unfortunately, we cannot appeal directly to Corollary 7.2 to show that Baum-Connes with coefficients fails for G, as the C∗-algebras C0(W ) and C0(∂W ) arenot separable. To get separable C∗-algebras with similar properties, let C0(Z) beany G-invariant C∗-subalgebra of C0(W ) that contains C0(G); it follows that Zcontains G as a dense open subset, and writing ∂Z = Z \ G gives a short exactsequence of G-C∗-algebras.

0 �� C0(G) �� C0(Z) �� C0(∂Z) �� 0

We want to guarantee that the crossed product C0(Z)�redG contains the Kazhdanprojection. There is a straightforward way to do this: our efforts in this sectionculminate in the following theorem.


7.4. Theorem. With notation as above, let C0(Z) denote any separable G-invariant C∗-subalgebra of C0(W ) that contains C0(G) and the characteristic func-tion χ of the expander Y . Then

(i) the crossed product C0(Z)�red G contains the Kazhdan projection associ-ated to Y ;(ii) the sequence

K0(C0(G)�red G) �� K0(C0(Z)�red G) �� K0(C0(∂Z)�red G)

is not exact in the middle;(iii) the Baum-Connes conjecture with coefficients is false for G.

Proof. For (i), let d ∈ N be the degree of all the vertices in Y . The Laplacianon Y (compare line (4.2) above) is then given by

Δ = dχ−∑

g∈G, |g|=1

χρgχ

and is thus in C0(Z)�red G. As both Δ and χ are elements of C0(Z) �red G, theKazhdan projection p is as well, by the functional calculus.

Part (ii) follows from part (i), our discussion of C0(W ) above, and the commu-tative diagram

K0(C0(G)�red G) �� K0(C0(Z)�red G) ��

��

K0(C0(∂Z)�red G)

��

K0(C0(G)�red G) �� K0(C0(W )�red G) �� K0(C0(∂W )�red G) ,

where the vertical arrows are all induced by the canonical inclusions. Part (iii) isimmediate from part (ii) and Corollary 7.2. �

At this point, we do not know exactly for which of the coefficients C0(Z)or C0(∂Z) Baum-Connes fails. Indeed, the fact that Baum-Connes is true withcoefficients in C0(G) and a chase of the diagram from Lemma 7.1 shows that eithersurjectivity fails for C0(Z), or injectivity fails for C0(∂Z). A more detailed analysisin Theorem 9.7 below shows that in fact the assembly map is an isomorphism withcoefficients in C0(∂Z), so that surjectivity fails for G with coefficients in C0(Z).

8. Reformulating the conjecture: exotic crossed products

In this section, we discuss how to adapt the Baum-Connes conjecture to takethe counterexamples from Section 7 into account. The counterexamples to theconjecture stem from analytic properties of the reduced crossed product: a naturalway to adapt the conjecture is then to change the crossed product to one with‘better’ properties.

Indeed, it is quite simple to define a ‘conjecture of Baum-Connes type’ for anarbitrary crossed product functor �τ . Define the τ -Baum-Connes assembly mapto be the composition

Ktop∗ (G;A) → K∗(A�max G) → K∗(A�τ G)

of the maximal assembly map, and the map induced on K-theory by the quotientmap A�maxG → A�τ G; it follows from the discussion in Section 6 that this is theusual Baum-Connes assembly map when τ is the reduced crossed product. And


one may hope that the τ -Baum-Connes assembly map is an isomorphism underfavorable conditions for well behaved τ .

One certainly cannot expect all of these ‘τ -Baum-Connes assembly maps’ to beisomorphisms, however: indeed, we have already observed that isomorphism fails forsome groups when τ is the reduced crossed product. There are even examples of a-T-menable groups and associated crossed products τ for which the τ -Baum-Connesassembly map (with trivial coefficients) is not an isomorphism: see [10, AppendixA]. Considering these examples as well as naturality issues, one is led to the follow-ing desirable properties of a crossed product functor τ that might be used to ‘fix’the Baum-Connes conjecture.

Exactness. It should fix the exactness problems: that is, for any short exact se-quence

0 �� I �� A �� B �� 0

of G-C∗-algebras, the induced sequence of C∗-algebras

0 �� I �τ G �� A�τ G �� B �τ G �� 0

should be exact.

Compatibility with Morita equivalences. Two G-C∗-algebras are equivariantly sta-bly isomorphic if A⊗KG is equivariantly ∗-isomorphic to B⊗KG. Here, KG denotesthe compact operators on the direct sum ⊕∞

1 �2(G), equipped with the conjugationaction arising from the direct sum of copies of the regular representation. It followsdirectly from the definition of E-theory that the domain of the Baum-Connes as-sembly map cannot detect the difference between equivariantly stably isomorphiccoefficient algebras. Therefore we would like our crossed product to have the sameproperty: see [10, Definition 3.2] for the precise condition we use.

This is a manifestation of Morita invariance. Indeed, separable G-C∗-algebrasare equivariantly stably isomorphic if and only if they are equivariantly Moritaequivalent, as follows from results in [23] and [53], which leads to a general Moritainvariance result in E-theory [28, Theorem 6.12]. See also [18, Sections 4 and 7]for the relationship to other versions of Morita invariance.

Existence of descent. There should be a descent functor�τ : EG → E, which agrees

with �τ : GC* → C* on G-C∗-algebras and ∗-homomorphisms. This is importantfor proving the conjecture: indeed, following the paradigm established by Kasparov[40], the most powerful known approaches to the Baum-Connes conjecture proceedby proving that certain identities hold in EG (or in the KKG-theory category, orsome related more versatile setting as in Lafforgue’s work [45]), and then usingdescent to deduce consequences for crossed products.

Consistency with property (T ). The three properties above hold for the maximalcrossed product. However, it is well-known that the maximal crossed product is notthe right thing to use for the Baum-Connes conjecture: the Kazhdan projections(see [67] or [39, Section 3.7]) in C∗

max(G) = C �max G are not in the image of the

maximal assembly map Ktop∗ (G;C) → K∗(C �max G) (see [32, Discussion below

5.1]). We would thus like that all Kazhdan projections get sent to zero under thequotient map C�max G → C�τ G.


Summarizing the above discussion, any crossed product that ‘fixes’ the Baum-Connes conjecture should have the following properties:

(i) it is exact ;(ii) it is Morita compatible;(iii) it has a descent functor in E-theory;(iv) it annihilates Kazhdan projections.

Such crossed products do indeed exist! In order to prove this, we introduced in[10, Section 3] a partial order on crossed product functors by saying that �τ ≥ �σ

if the norm on A �alg G coming from A �τ G is at least as large as that comingfrom A �σ G. The following theorem is one of the main results of [10]; the partdealing with exactness is due to Kirchberg.

8.1. Theorem. With respect to the partial order above, there is a (unique)minimal crossed product �E with properties (i) and (ii). This crossed productautomatically also has properties (iii) and (iv).

Summarizing, our reformulation of the Baum-Connes conjecture is that theE-Baum-Connes assembly map

Ktop∗ (G;A) → K∗(A�max G) → K∗(A�E G)

is an isomorphism; we shall refer to this assertion as the E-Baum-Connes conjecture.It is quite natural to consider the minimal crossed product satisfying (i) and (ii)

above: indeed, it is in some sense the ‘closest’ to the reduced crossed product amongall the crossed products with properties (i) to (iv) above and, for exact groups itis the reduced crossed product. Consequently, for exact groups the reformulatedconjecture is nothing other than the original Baum-Connes conjecture.

8.2. Remark. As mentioned above, we chose to work with E-theory, insteadof the more common KK-theory to formulate the Baum-Connes conjecture. It isnatural to ask whether the development above can be carried out in KK-theory,and in particular whether �E admits a descent functor �E : KKG → KK. Theanswer is yes, as long as we restrict as usual to countable groups and separableG-C∗-algebras [18].

9. The counterexamples and the reformulated conjecture

In this section we shall revisit the counterexamples presented in Section 7 tothe original Baum-Connes conjecture, and study them from the point of view ofthe reformulated conjecture of Section 8. In particular, we shall continue with thenotation of Section 7: G is a Gromov monster group, containing an expander Y ;C0(W ) is the minimal G-invariant C∗-subalgebra of �∞(G) that contains �∞(Y );and ∂W = W \G.

In Theorem 7.4 we saw that if C0(Z) is any separable G-invariant C∗-subalgebraof C0(W ) containing both C0(G) and the characteristic function of Y , then theoriginal Baum-Connes conjecture fails for G with coefficients in at least one ofC0(Z) and C0(∂Z), where again ∂Z = Z \ G. The key point was the failure ofexactness of the sequence

K0(C0(G)�red G) �� K0(C0(Z)�red G) �� K0(C0(∂Z)�red G)

in the middle, as evidenced by the Kazhdan projection of Y .


In the case of the reformulated conjecture, however, we would be working withthe analogous sequence involving the E-crossed product, which is exact (even atthe level of C∗-algebras). Thus, at this point we know that the proof of Theorem7.4 will not apply to show the existence of counterexamples to the E-Baum-Connesconjecture. However, something much more interesting happens: in this section,we shall show that for any Z as above, the E-Baum-Connes conjecture is true forG with coefficients in both C0(Z) and C0(∂Z)! This is a stronger result than inour original paper [10], where we just showed that there exists some Z with theproperty above.

There are two key-ingredients. First, we need a ‘two out of three’ lemma, whichwill allow us to deduce the E-Baum-Connes conjecture for C0(Z) from the E-Baum-Connes conjecture for C0(G) and C0(∂Z). Second, we need to show that the actionof G on ∂Z is always a-T-menable: this implies via work of Tu [65] that a strongform of the Baum-Connes conjecture holds in the equivariant E-theory category,and allows us to deduce the E-Baum-Connes conjecture for G with coefficients inC0(∂Z). The crucial geometric assumption needed for the second step is that theexpander Y has large girth, and therefore looks ‘locally like a tree’.

The first ingredient is summarized in the following lemma, which is a moreprecise version of Lemma 7.1. See [10, Proposition 4.6] for a proof. See also [19,Section 4] for a proof for the original formulation of the Baum-Connes conjectureusing the reduced crossed product, and the additional assumption that G is exacton the level of K-theory.

9.1. Lemma. Let

0 �� I �� A �� B �� 0

be a short exact sequence of separable G-C∗-algebras. There is a commutative dia-gram of six term sequences

Ktop0 (G; I)

��

��

�� Ktop0 (G;A) ��

��

��

Ktop0 (G;B)

��

��

��

K0(I �E G) �� K0(A �E G) �� K0(B �E G)

��

Ktop1 (G;B)

��

��

��

Ktop1 (G;A)��

��

��

Ktop1 (G; I)��

��

��

K1(B �E G)

��

K1(A �E G)�� K1(I �E G),��

in which the front and back rectangular six term sequences are exact, and the mapsfrom the back sequence to the front are E-Baum-Connes assembly maps. In partic-ular, if the Baum-Connes conjecture holds with coefficients in two out of three ofI, A, and B, then it holds with coefficients in the third.

We now move on to the second key ingredient, the a-T-menability of the actionof a Gromov monster group G on any of the spaces ∂Z.

9.2. Definition. Let G be a discrete group acting on the right on a locallycompact space X by homeomorphisms. The action is a-T-menable if there is acontinuous function

h : X ×G → [0,∞)


with the following properties.

(i) The restriction of h to X × {e} is 0.(ii) For all x ∈ X and g ∈ G, h(x, g) = h(xg, g−1).(iii) For any finite subset {g1, . . . , gn} of G, any finite subset {t1, . . . , tn} of Rsuch that

∑ni=1 ti = 0, and any x ∈ X, we have that

n∑i,j=1

titjh(xgi, g−1i gj) ≤ 0.

(iv) For any compact subset K of X, the restriction of h to the set

{(x, g) ∈ X ×G | x ∈ K,xg ∈ K}is proper.

The following result is essentially [10, Theorem 7.9].

9.3. Theorem. Let G be a Gromov monster group with isometrically embeddedexpander Y , and let W and ∂W be as in Section 7 (and as explained at the beginningof this section). Let π : G → Y be any function such that d(x, π(x)) = d(x, Y ) forall x ∈ G.

Define a function h : G × G → [0,∞) by h(x, g) = d(π(x), π(xg)). Thenh extends by continuity to a function h : W × G → [0,∞), and the restrictionh : ∂W × G → [0,∞) has all the properties in Definition 9.2. In particular, theaction of G on ∂W is a-T-menable.

The crucial geometric input into the proof is the fact that Y has large girth.This means that as one moves out to infinity in Y , then Y ‘looks like a tree’ onlarger and larger sets. One can then use the negative type property of the distancefunction on a tree to prove that h has the right properties.

Now, let C0(Z) be any separable G-invariant subalgebra of C0(W ) containingC0(G) and the characteristic function χ of Y . Set ∂Z = Z \ G. We would like toshow that the action of G on ∂Z is also a-T-menable; as ∂Z is a quotient of ∂W ,it suffices from Theorem 9.3 to show that the function h : G×G → [0,∞) extendto h : Z × G → [0,∞) (at least for some choice of function π : G → Y with theproperties in the statement). We will do this via a series of lemmas.

9.4. Lemma. For each r > 0, let Nr(Y ) = {g ∈ G | d(g, Y ) ≤ r} denote

the r-neighborhood of Y in G. If Nr(Y ) denotes the closure of Nr(Y ) in Z, then

{Nr(Y )}r∈N is a cover of Z by an increasing sequence of compact, open subsets.

Proof. For g ∈ G, let χY g denote the characteristic function of the right trans-late of Y by g, which is in C0(Z) by definition of this algebra. Hence f =

∑|g|≤r χY g

is in C0(Z). The closure of Nr(Y ) is equal to f−1(0,∞) and to f−1[1,∞), and isthus compact and open as f is an element of C0(Z). Finally, note that finitely sup-ported elements of �∞(G) and translates of χ by the right action of G are supported

in Nr(Y ) for some r > 0; as such elements generate Z, it follows that {Nr(Y )}r∈N

is a cover of Z. �

Choose now an order g1, g2, . . . on the elements of G such that g1 = e and sothat the function N → R defined by n �→ |gn| is non-decreasing. For each x ∈ G, letn(x) be the smallest integer such that xgn(x) is in Y , and define a map π : G → Yby setting π(x) = xgn(x). Note that d(π(x), x) = d(x, Y ) for all x ∈ G.


9.5. Lemma. Fix g ∈ G and r ∈ N, and define a function

hr,g : Nr(Y ) → [0,∞), x �→ d(π(x), π(xg)).

Then hr,g extends continuously to the closure Nr(Y ) of Nr(Y ) in Z.

Proof. Write the elements of {x ∈ G | |x| ≤ r} as g1, . . . , gn with respect tothe order used to define π. For each m ∈ {1, . . . , n}, let

Em = {x ∈ Nr(Y ) | xgm ∈ Y }and note that the characteristic function χEm

of Em is equal to χY gm · χNr(Y ) andis thus in C0(Z) by Lemma 9.4. On the other hand, if we let

Fm = {x ∈ Nr(Y ) | π(x) = xgm}then the characteristic function of Fm equals χEm

(1−

∑m−1i=1 χEi

)and is thus also

in C0(Z). Similarly, if we write the elements of {h ∈ G | |h| ≤ r+ |g|} as g1, . . . , gn′

and for each m ∈ {1, . . . , n′} let

F ′m = {x ∈ Nr(Y ) | π(xg) = xgm},

then the characteristic function of F ′m is in C0(Z).

For each (k, l) ∈ {1, . . . , n} × {1, . . . , n′}, let χk,l denote the characteristicfunction of Fk ∩ F ′

l , which is in C0(Z) by the above discussion. Note that therestriction of hr,g to Fk ∩ F ′

l sends x ∈ Nr(Y ) to

d(π(x), π(xg)) = d(xgk, xggl) = |g−1k ggl|.

Hence

hr,g =n∑

k=1

n′∑l=1

|g−1k ggl|χk,l,

and thus hr,g is in C0(Z) as claimed. �

9.6. Corollary. The function

h : G×G → [0,∞), x �→ d(π(x), π(xg))

extends by continuity to h : Z ×G → [0,∞). In particular, the action of G on ∂Zis a-T-menable.

Proof. For each fixed g ∈ G, Lemma 9.5 implies that the restriction of thefunction

hg : G → [0,∞), x �→ d(π(x), π(xg))

to Nr(Y ) extends continuously to Nr(Y ); as {Nr(Y )}r∈N is a compact, open coverof Z, it follows that hg extends to a continuous function on all of Z. Hence thefunction

h : G×G → [0,∞), (x, g) �→ d(π(x), π(xg))

extends to a continuous function on all of Z × G. The result now follows fromTheorem 9.3 as ∂Z is a quotient of ∂W . �

The following corollary is the culmination of our efforts in this section. First, itgives us more information about what goes wrong with the Baum-Connes conjecturethan Section 7 does. More importantly for our current work, it shows that the E-Baum-Connes conjecture is true for this counterexample: we thus have a concreteclass of example where our reformulated conjecture ‘out-performs’ the original one.


9.7. Theorem. Let G be a Gromov monster group with isometrically embeddedexpander Y . Equip �∞(G) with the action induced by the right translation action ofG on itself, and let C0(W ) denote the G-invariant C∗-subalgebra of �∞(G) generatedby �∞(Y ). Let C0(Z) be any separable G-invariant C∗-subalgebra of C0(W ) thatcontains C0(G) and the characteristic function χ of Y . Then:

(i) the usual Baum-Connes assembly map for G with coefficients in C0(Z) isinjective;(ii) the usual Baum-Connes assembly map for G with coefficients in C0(Z)fails to be surjective;

(iii) the E-Baum-Connes assembly map for G with coefficients in C0(Z) is anisomorphism.

Proof. The essential point is that work of Tu [65] shows that a-T-menabilityof the action of G on ∂Z implies that a strong version of the Baum-Connes conjec-ture for G with coefficients in C0(∂Z) holds in the equivariant E-theory categoryEG. This in turn implies the τ -Baum-Connes conjecture for G with coefficients inC0(∂Z) for any crossed product τ that admits a descent functor. See [10, Theorem6.2] for more details.

The result follows from this, Lemma 9.1, and the fact that the Baum-Connesconjecture is true for any crossed product with coefficients in a proper G-algebralike C0(G). �

10. The Kadison-Kaplansky conjecture for �1(G)

The Kadison-Kaplansky conjecture states that for a torsion free discrete groupG, there are no idempotents in C∗

red(G) other than the ‘trivial’ examples givenby 0 and 1. It is well-known that the usual Baum-Connes conjecture implies theKadison-Kaplansky conjecture. As �1(G) is a subalgebra of C∗

red(G), the Kadison-Kaplansky conjecture implies that �1(G) contains no idempotents other than 0 or1. In this section, we show that the E-Baum-Connes conjecture, and in fact any ‘ex-otic’ Baum-Connes conjecture, implies that �1(G) has no non-trivial idempotents.Thus the E-Baum-Connes conjecture implies a weak form of the Kadison-Kaplanskyconjecture. Compare [14, Corollary 1.6] for a similar result in the context of theBost conjecture.

10.1. Theorem. Let G be a countable torsion free group and let σ be a crossedproduct functor for G. If the σ-Baum-Connes conjecture holds for G with trivialcoefficients then the only idempotents in the Banach algebra �1(G) are zero and theidentity.

Recall that there is a canonical tracial state

τ : C∗red(G) → C, τ (a) = 〈 δe, aδe 〉.

The trace τ is well known to be faithful in the sense that a non-zero positive elementof C∗

red(G) has strictly positive trace: see, for example [17, Proposition 2.5.3]. Onehas the following standard C∗-algebraic lemma.

10.2. Lemma. Let e ∈ C∗red(G) be an idempotent. If τ (e) is an integer then

e = 0 or e = 1.

Proof. The idempotent e is similar in C∗red(G) to a projection p [15, Propo-

sition 4.6.2] so that τ (p) = τ (e) ∈ Z. Positivity of τ and the operator inequality


0 ≤ p ≤ 1 imply that 0 ≤ τ (p) ≤ 1 and so τ (p) = 0 or τ (p) = 1. Since τ is faithful,we conclude p = 0 or p = 1, and the same for e. �

Recall that the trace τ defines a map τ∗ : K0(C∗red(G)) → R which sends the

K-theory class of an idempotent e ∈ C∗red(G) to τ (e) [15, Section 6.9]. The key

point in the proof of Theorem 10.1 is the following result concerning the imageunder τ∗ of elements in the range of the Baum-Connes assembly map.

10.3. Proposition. Let G be a countable torsion free group. If x∈K0(C∗red(G))

is in the image of the Baum-Connes assembly map with trivial coefficients then τ∗(x)is an integer.

Proof. This is a corollary of Atiyah’s covering index theorem [6] (see also[20]). The most straight-forward way to connect Atiyah’s covering index theoremto the Baum-Connes conjecture, and thus to prove the proposition, is via the Baum-Douglas geometric model for K-homology [9,11]: this is explained in [69, Section6.3] or [7, Proposition 6.1]. See also [48], particularly Theorem 0.3, for a slightlydifferent approach (and a more general statement that takes into account the casewhen G has torsion). �

Proof of Theorem 10.1. Let e be an idempotent in �1(G). Since �1(G) is asubalgebra of both C∗

red(G) and C∗σ(G) := C �σ G, we may consider the K-theory

classes [e]red and [e]σ defined by e for each of these C∗-algebras. The usual (reduced)Baum-Connes assembly map factors through the quotient map C∗

σ(G) → C∗red(G),

and this quotient map is the identity on �1(G), so takes [e]σ to [e]red. Thus, since[e]σ is in the range of the σ-Baum-Connes assembly map, [e]red is in the range ofthe reduced Baum-Connes assembly map. Proposition 10.3 implies now that τ (e)is an integer, and Lemma 10.2 implies that e is equal to either 0 or 1. �

11. Concluding remarks

In our reformulated version of the Baum-Connes conjecture, the left side isunchanged, that is, is the same as in the original conjecture as stated by Baumand Connes [8]. At first glance, it may seem surprising that in the reformulatedconjecture only the right hand side is changed. In this section we shall motivate,via the Bost conjecture, precisely why the left side should remain unchanged.

Recall that the original Baum-Connes assembly map for the group G withcoefficients in a G-C∗-algebra A factors as

Ktop∗ (G,A) → K∗(�

1(G,A)) → K∗(A�red G),

where �1(G,A) is the Banach algebra crossed product. According to the Bostconjecture, the first arrow in this display is an isomorphism; the second arrow isinduced by the inclusion �1(G,A) → A�red G.

The Bost conjecture is known to hold in a great many cases, in particular, forfundamental groups of Riemannian locally symmetric spaces; see [45]. In thesecases, the Baum-Connes conjecture is equivalent to the assertion that the K-theoryof the Banach algebra �1(G,A) is isomorphic to the K-theory of the C∗-algebraA�redG, and in fact that the inclusion �1(G,A) → A�redG induces an isomorphism.

While the Bost conjecture may seem more natural because it has the appro-priate functoriality in the group G, it does not have the important implications forgeometry and topology that the Baum-Connes conjecture does. In particular, it isnot known to imply either the Novikov higher signature conjecture or the stable


Gromov-Lawson-Rosenberg conjecture about existence of positive scalar curvaturemetrics on spin manifolds. As described here, our reformulation, which involves anappropriate C∗-algebra completion of �1(G,A) retains these implications.

From this point of view, an attempt to reformulate the Baum-Connes conjec-ture should involve finding a pre-C∗-norm on �1(G,A) with the property that theK-theory of the Banach algebra �1(G,A) equals the K-theory of its C∗-algebracompletion. This problem involves in a fundamental way the harmonic analysis ofthe group, and this paper can be viewed as indicating a possible solution.

References

[1] C. A. Akemann and M. E. Walter, Unbounded negative definite functions, Canad. J. Math.33 (1981), no. 4, 862–871, DOI 10.4153/CJM-1981-067-9. MR634144 (83b:43009)

[2] C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and theirC∗-algebras, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4153–4178 (electronic), DOI10.1090/S0002-9947-02-02978-1. MR1926869 (2004e:46082)

[3] G. Arzhantseva and T. Delzant, Examples of random groups, Available on the authors’ web-

sites, 2008.[4] G. Arzhantseva, E. Guentner, and J. Spakula, Coarse non-amenability and coarse em-

beddings, Geom. Funct. Anal. 22 (2012), no. 1, 22–36, DOI 10.1007/s00039-012-0145-z.MR2899681

[5] G. Arzhantseva and D. Osajda, Graphical small cancellation groups with the Haagerup prop-erty, arXiv:1404.6807, 2014.

[6] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Anal-yse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Soc. Math. France, Paris, 1976,pp. 43–72. Asterisque, No. 32-33. MR0420729 (54 #8741)

[7] P. Baum, On the index of equivariant elliptic operators, Operator algebras, quantization,and noncommutative geometry, Contemp. Math., vol. 365, Amer. Math. Soc., Providence,RI, 2004, pp. 41–49, DOI 10.1090/conm/365/06699. MR2106816 (2005h:58034)

[8] P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory of group C∗-algebras, C∗-algebras: 1943–1993 (San Antonio, TX, 1993), Con-temp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291, DOI10.1090/conm/167/1292018. MR1292018 (96c:46070)

[9] P. Baum and R. G. Douglas, K homology and index theory, Operator algebras and applica-tions, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc.,Providence, R.I., 1982, pp. 117–173. MR679698 (84d:58075)

[10] P. Baum, E. Guentner, and R. Willett. Expanders, exact crossed products, and the Baum-Connes conjecture. To appear, Annals of K-theory, 2013.

[11] P. Baum, N. Higson, and T. Schick, A geometric description of equivariant K-homology forproper actions, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence,

RI, 2010, pp. 1–22. MR2732043 (2012g:19017)[12] M. E. B. Bekka, P.-A. Cherix, and A. Valette, Proper affine isometric actions of amenable

groups, Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), LondonMath. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 1–4,DOI 10.1017/CBO9780511629365.003. MR1388307 (97e:43001)

[13] B. Bekka, P. de la Harpe, and A. Valette. Kazhdan’s Property (T). Cambridge UniversityPress, 2008.

[14] A. J. Berrick, I. Chatterji, and G. Mislin, From acyclic groups to the Bass conjecture foramenable groups, Math. Ann. 329 (2004), no. 4, 597–621, DOI 10.1007/s00208-004-0521-6.MR2076678 (2005g:20085)

[15] B. Blackadar, K-theory for operator algebras, 2nd ed., Mathematical Sciences Research In-stitute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR1656031(99g:46104)

[16] J. Brodzki, C. Cave, and K. Li, Exactness of locally compact second countable groups,Preprint, 2015.


[17] N. P. Brown and N. Ozawa, C∗-algebras and finite-dimensional approximations, Gradu-ate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008.MR2391387 (2009h:46101)

[18] A. Buss, S. Echterhoff, and R. Willett, Exotic crossed produts and the Baum-Connes conjec-ture, arXiv:1409.4332, to appear in J. Reine Angew. Math., 2014.

[19] J. Chabert and S. Echterhoff, Permanence properties of the Baum-Connes conjecture, Doc.Math. 6 (2001), 127–183 (electronic). MR1836047 (2002h:46117)

[20] I. Chatterji and G. Mislin, Atiyah’s L2-index theorem, Enseign. Math. (2) 49 (2003), no. 1-2,85–93. MR1998884 (2004g:58027)

[21] X. Chen, Q. Wang, and G. Yu, The maximal coarse Baum-Connes conjecture for spaceswhich admit a fibred coarse embedding into Hilbert space, Adv. Math. 249 (2013), 88–130,DOI 10.1016/j.aim.2013.09.003. MR3116568

[22] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette. Groups with the HaagerupProperty (Gromov’s a-T-menability), volume 197 of Progress in mathematics. Birkhauser,2001.

[23] R. E. Curto, P. S. Muhly, and D. P. Williams, Cross products of strongly Morita equiva-lent C∗-algebras, Proc. Amer. Math. Soc. 90 (1984), no. 4, 528–530, DOI 10.2307/2045024.MR733400 (85i:46083)

[24] S. Deprez and K. Li, Property A and uniform embedding for locally compact groups,arXiv:1309.7290v2, to appear in J. Noncommut. Geom., 2013.

[25] M. Finn-Sell and N. Wright, Spaces of graphs, boundary groupoids and the coarse Baum-Connes conjecture, Adv. Math. 259 (2014), 306–338, DOI 10.1016/j.aim.2014.02.029.MR3197659

[26] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex,1991), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge,1993, pp. 1–295. MR1253544 (95m:20041)

[27] M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146,DOI 10.1007/s000390300002. MR1978492 (2004j:20088a)

[28] E. Guentner, N. Higson, and J. Trout, Equivariant E-theory for C∗-algebras, Mem.Amer. Math. Soc. 148 (2000), no. 703, viii+86, DOI 10.1090/memo/0703. MR1711324

(2001c:46124)[29] E. Guentner, N. Higson, and S. Weinberger, The Novikov conjecture for linear groups,

Publ. Math. Inst. Hautes Etudes Sci. 101 (2005), 243–268, DOI 10.1007/s10240-005-0030-5.MR2217050 (2007c:19007)

[30] E. Guentner and J. Kaminker, Exactness and the Novikov conjecture, Topology 41 (2002),no. 2, 411–418, DOI 10.1016/S0040-9383(00)00036-7. MR1876896 (2003e:46097a)

[31] U. Haagerup, An example of a nonnuclear C∗-algebra, which has the metric approximationproperty, Invent. Math. 50 (1978/79), no. 3, 279–293, DOI 10.1007/BF01410082. MR520930(80j:46094)

[32] N. Higson, The Baum-Connes conjecture, Proceedings of the International Congress of Math-ematicians, Vol. II (Berlin, 1998), Doc. Math. Extra Vol. II (1998), 637–646 (electronic).MR1648112 (2000e:46088)

[33] N. Higson, Bivariant K-theory and the Novikov conjecture, Geom. Funct. Anal. 10 (2000),no. 3, 563–581, DOI 10.1007/PL00001630. MR1779613 (2001k:19009)

[34] N. Higson and G. Kasparov, Operator K-theory for groups which act properly and isometri-cally on Hilbert space, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 131–142 (elec-tronic), DOI 10.1090/S1079-6762-97-00038-3. MR1487204 (99e:46090)

[35] N. Higson and G. Kasparov, E-theory and KK-theory for groups which act prop-erly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74, DOI10.1007/s002220000118. MR1821144 (2002k:19005)

[36] N. Higson, G. Kasparov, and J. Trout, A Bott periodicity theorem for infinite-dimensional Eu-clidean space, Adv. Math. 135 (1998), no. 1, 1–40, DOI 10.1006/aima.1997.1706. MR1617411(99j:19005)

[37] N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum-Connes conjecture,Geom. Funct. Anal. 12 (2002), no. 2, 330–354, DOI 10.1007/s00039-002-8249-5. MR1911663(2003g:19007)

[38] N. Higson and J. Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew.Math. 519 (2000), 143–153, DOI 10.1515/crll.2000.009. MR1739727 (2001h:57043)


[39] N. Higson and J. Roe, Analytic K-homology, Oxford Mathematical Monographs, OxfordUniversity Press, Oxford, 2000. Oxford Science Publications. MR1817560 (2002c:58036)

[40] G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91(1988), no. 1, 147–201, DOI 10.1007/BF01404917. MR918241 (88j:58123)

[41] G. G. Kasparov, K-theory, group C∗-algebras, and higher signatures (conspectus), Novikovconjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc.Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 101–146, DOI

10.1017/CBO9780511662676.007. MR1388299 (97j:58153)[42] E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of C∗-algebras, Math.

Ann. 315 (1999), no. 2, 169–203, DOI 10.1007/s002080050364. MR1721796 (2000i:46050)[43] E. Kirchberg and S. Wassermann, Permanence properties of C∗-exact groups, Doc. Math. 4

(1999), 513–558 (electronic). MR1725812 (2001i:46089)[44] T. Kondo, CAT(0) spaces and expanders, Math. Z. 271 (2012), no. 1-2, 343–355, DOI

10.1007/s00209-011-0866-y. MR2917147[45] V. Lafforgue,K-theorie bivariante pour les algebres de Banach et conjecture de Baum-Connes

(French), Invent. Math. 149 (2002), no. 1, 1–95, DOI 10.1007/s002220200213. MR1914617(2003d:19008)

[46] V. Lafforgue, Un renforcement de la propriete (T) (French, with English and Frenchsummaries), Duke Math. J. 143 (2008), no. 3, 559–602, DOI 10.1215/00127094-2008-029.MR2423763 (2009f:22004)

[47] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathe-matics, vol. 125, Birkhauser Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski.MR1308046 (96g:22018)

[48] W. Luck, The relation between the Baum-Connes conjecture and the trace conjecture, Invent.Math. 149 (2002), no. 1, 123–152, DOI 10.1007/s002220200215. MR1914619 (2003f:19005)

[49] G. A. Margulis, Explicit constructions of expanders (Russian), Problemy Peredaci Informacii9 (1973), no. 4, 71–80. MR0484767 (58 #4643)

[50] M. Mendel and A. Naor, Nonlinear spectral calculus and super-expanders, Publ. Math. Inst.

Hautes Etudes Sci. 119 (2014), 1–95, DOI 10.1007/s10240-013-0053-2. MR3210176[51] Mikhael Gromov. Annotated problem list. In S. Ferry, A. Ranicki, and J. Rosenberg, editors,

Novikov conjectures, index theory theorems and rigidity, page 67. London Mathematical So-ciety, 1993.

[52] M. Gromov, Spaces and questions, Geom. Funct. Anal. Special Volume (2000), 118–161.GAFA 2000 (Tel Aviv, 1999). MR1826251 (2002e:53056)

[53] J. A. Mingo and W. J. Phillips, Equivariant triviality theorems for Hilbert C∗-modules, Proc.Amer. Math. Soc. 91 (1984), no. 2, 225–230, DOI 10.2307/2044632. MR740176 (85f:46111)

[54] P. W. Nowak, Coarsely embeddable metric spaces without Property A, J. Funct. Anal. 252(2007), no. 1, 126–136, DOI 10.1016/j.jfa.2007.06.014. MR2357352 (2008i:54026)

[55] Y. Ollivier, On a small cancellation theorem of Gromov, Bull. Belg. Math. Soc. Simon Stevin

13 (2006), no. 1, 75–89. MR2245980 (2007e:20066)[56] D. Osajda, Small cancellation labellings of some infinite graphs and applications,

arXiv:1406.5015, 2014.[57] H. Oyono-Oyono and G. Yu, K-theory for the maximal Roe algebra of certain expanders,

J. Funct. Anal. 257 (2009), no. 10, 3239–3292, DOI 10.1016/j.jfa.2009.04.017. MR2568691(2010h:46117)

[58] N. Ozawa, Amenable actions and exactness for discrete groups (English, with English andFrench summaries), C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), no. 8, 691–695, DOI10.1016/S0764-4442(00)00248-2. MR1763912 (2001g:22007)

[59] J. Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathemat-ical Society, Providence, RI, 2003. MR2007488 (2004g:53050)

[60] J. Roe, Hyperbolic groups have finite asymptotic dimension, Proc. Amer. Math. Soc. 133(2005), no. 9, 2489–2490 (electronic), DOI 10.1090/S0002-9939-05-08138-4. MR2146189(2005m:20102)

[61] J. Roe and R. Willett, Ghostbusting and property A, J. Funct. Anal. 266 (2014), no. 3,1674–1684, DOI 10.1016/j.jfa.2013.07.004. MR3146831

[62] H. Sako, Translation C∗-algebras and property A for uniformly locally finite spaces,arXiv:1213.5900v1, 2012.


[63] M. Sapir, A Higman embedding preserving asphericity, J. Amer. Math. Soc. 27 (2014), no. 1,1–42, DOI 10.1090/S0894-0347-2013-00776-6. MR3110794

[64] G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids,Topology 41 (2002), no. 4, 807–834, DOI 10.1016/S0040-9383(01)00004-0. MR1905840(2003c:58020)

[65] J.-L. Tu, La conjecture de Baum-Connes pour les feuilletages moyennables (French,with English and French summaries), K-Theory 17 (1999), no. 3, 215–264, DOI

10.1023/A:1007744304422. MR1703305 (2000g:19004)[66] J.-L. Tu, Remarks on Yu’s “property A” for discrete metric spaces and groups (English,

with English and French summaries), Bull. Soc. Math. France 129 (2001), no. 1, 115–139.MR1871980 (2002j:58038)

[67] A. Valette, Minimal projections, integrable representations and property (T), Arch. Math.(Basel) 43 (1984), no. 5, 397–406, DOI 10.1007/BF01193846. MR773186 (86j:22006)

[68] A. Valette. Annotated problem list. In S. Ferry, A. Ranicki, and J. Rosenberg, editors, Novikovconjectures, index theory theorems and rigidity, pages 74–75. London Mathematical Society,1993.

[69] A. Valette, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETHZurich, Birkhauser Verlag, Basel, 2002. From notes taken by Indira Chatterji; With an ap-pendix by Guido Mislin. MR1907596 (2003f:58047)

[70] D. Voiculescu, Property T and approximation of operators, Bull. London Math. Soc. 22(1990), no. 1, 25–30, DOI 10.1112/blms/22.1.25. MR1026768 (90m:46101)

[71] S. Wassermann, C∗-algebras associated with groups with Kazhdan’s property T , Ann. of Math.(2) 134 (1991), no. 2, 423–431, DOI 10.2307/2944351. MR1127480 (92g:46085)

[72] R. Willett, Some notes on property A, Limits of graphs in group theory and computer science,EPFL Press, Lausanne, 2009, pp. 191–281. MR2562146 (2010i:22005)

[73] R. Willett, Property A and graphs with large girth, J. Topol. Anal. 3 (2011), no. 3, 377–384,DOI 10.1142/S179352531100057X. MR2831267 (2012j:53047)

[74] R. Willett and G. Yu, Higher index theory for certain expanders and Gromov monster groups,I, Adv. Math. 229 (2012), no. 3, 1380–1416, DOI 10.1016/j.aim.2011.10.024. MR2871145

[75] R. Willett and G. Yu, Higher index theory for certain expanders and Gromov monster groups,

II, Adv. Math. 229 (2012), no. 3, 1762–1803, DOI 10.1016/j.aim.2011.12.016. MR2871156[76] D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1,

150–214, DOI 10.1007/s00039-004-0454-y. MR2053602 (2005c:20069)[77] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding

into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240, DOI 10.1007/s002229900032.MR1728880 (2000j:19005)

Department of Mathematics, Pennsylvania State University, University Park, Penn-

sylvania 16802

E-mail address: [email protected]

University of Hawai‘i at Manoa, Department of Mathematics, 2565 McCarthy Mall,

Honolulu, Hawaii 96822-2273


University of Hawai‘i at Manoa, Department of Mathematics, 2565 McCarthy Mall,

Honolulu, HI 96822-2273



Generalization of C∗-algebra methods via real positivityfor operator and Banach algebras

David P. Blecher

Dedicated with affection and gratitude to Richard V. Kadison.

Abstract. With Charles Read we have introduced and studied a new notionof (real) positivity in operator algebras, with an eye to extending certain C∗-algebraic results and theories to more general algebras. As motivation notethat the ‘completely’ real positive maps on C∗-algebras or operator systemsare precisely the completely positive maps in the usual sense; however withreal positivity one may develop a useful order theory for more general spacesand algebras. This is intimately connected to new relationships between anoperator algebra and the C∗-algebra it generates. We have continued this worktogether with Read, and also with Matthew Neal. Recently with NarutakaOzawa we have investigated the parts of the theory that generalize further toBanach algebras. In the present paper we describe some of this work, and alsogive some new updates and complementary results, which are connected withgeneralizing various C∗-algebraic techniques initiated by Richard V. Kadison.In particular Section 2 is in part a tribute to him in keeping with the occasion ofthis volume, and also discusses some of the origins of the theory of positivityin our sense in the setting of algebras, which the later parts of our paperdevelopes further. The most recent work will be emphasized.

1. Introduction

This is a much expanded version of our talk given at the AMS Special Ses-sion “Tribute to Richard V. Kadison” in January 2015. We survey some of ourwork on a new notion of (real) positivity in operator algebras (by which we meanclosed subalgebras of B(H) for a Hilbert space H), unital operator spaces, andBanach algebras, focusing on generalizing various C∗-algebraic techniques initiatedby Richard V. Kadison. In particular Section 2 is in part a tribute to Kadison inkeeping with the occasion of this volume, and we will describe a small part of hisopus relevant to our setting. This section also discusses some of the origins of thetheory of positivity in our sense in the setting of algebras, which the later partsof our paper developes further. In the remainder of the paper we illustrate our

2010 Mathematics Subject Classification. Primary 46B40, 46L05, 47L30; Secondary 46H10,46L07, 46L30, 47L10.

Key words and phrases. Nonselfadjoint operator algebras, ordered linear spaces, approximateidentity, accretive operators, state space, quasi-state, hereditary subalgebra, Banach algebra, idealstructure.

Supported by NSF grant DMS 1201506.


35

36 DAVID P. BLECHER

real-positivity theory by showing how it relates to these results of Kadison, andalso give some small extensions and additional details for our recent paper withOzawa [21], and for [20] with Neal.

With Charles Read we have introduced and studied a new notion of (real) pos-itivity in operator algebras, with an eye to extending certain C∗-algebraic resultsand theories to more general algebras. As motivation note that the ‘completely’real positive maps on C∗-algebras or operator systems are precisely the completelypositive maps in the usual sense (see Theorem 3.2 below); however with real posi-tivity one may develop an order theory for more general spaces and algebras thatis useful at least for some purposes. We have continued this work together withRead, and also with Matthew Neal; giving many applications. (See papers withthese coauthors referenced in the bibliography below.) Recently with NarutakaOzawa we have investigated the parts of the theory that generalize further to Ba-nach algebras. In all of this, our main goal is to generalize certain nice C∗-algebraicresults, and certain function space or function algebra results, which use positivityor positive approximate identities, but using our real positivity. As we said above,in the present paper we survey some of this work which is connected with workof Kadison. The most recent work will be emphasized, particularly parts of theBanach-algebraic paper [21]. One reason for this emphasis is that less backgroundis needed here (for example, we shall avoid discussion of noncommutative topology,and our work on noncommutative peak sets and peak interpolation, which we havesurveyed recently in [12] although we have since made more progress in [25]). An-other reason is that we welcome this opportunity to add some additional details andcomplements to [21] (and to [20]). In particular we will prove some facts that werestated there without proof. A subsidiary goal of Sections 6 and 7 is to go throughversions for general Banach algebras of results in Sections 3, 4, and 7 of [21] statedfor Banach algebras with approximate identities. We will also pose several openquestions. The drawback of course with this focus is that the Banach algebra caseis sometimes less impressive and clean than the operator algebra case, there usuallybeing a price to be paid for generality.

Of course an operator algebra or function algebra A may have no positive el-ements in the usual sense. However we see e.g. in Theorem 5.2 below that anoperator algebra A has a contractive approximate identity iff there is a great abun-dance of real-positive elements; for example, iff A is spanned by its real-positiveelements. Below Theorem 5.2 we will point out that this is also true for certainclasses of Banach algebras. Of course in the theory of C∗-algebras, positivity andthe existence of positive approximate identities are crucial. Some form of our ‘pos-itive cone’ already appeared in papers of Kadison and Kelley and Vaught in theearly 1950’s, and in retrospect it is a natural idea to attempt to use such a cone togeneralize various parts of C∗-algebra theory involving positivity and the existenceof positive approximate identities. However nobody seems to have pursued this un-til now. In practice, some things are much harder than the C∗-algebra case. Andmany things simply do not generalize beyond the C∗-theory; that is, our approachis effective at generalizing some parts of C∗-algebra theory, but not others. Theworst problem is that although we have a functional calculus, it is not as good.Indeed often at first sight in a given C∗-subtheory, nothing seems to work. But inmany cases if one looks a little closer something works, or an interesting conjec-ture is raised. Successful applications so far include for example noncommutative

C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 37

topology (eg. noncommutative Urysohn and Tietze theorems for general operatoralgebras, and the theory of open, closed and compact projections in the bidual),lifting problems, the structure of completely contractive idempotent maps on anoperator algebra (described in Section 3 below), noncommutative peak sets, peakinterpolation, and some other noncommutative function theory, comparison theory,the structure of operator algebras, new relationships between an operator algebraand the C∗-algebra it generates, approximate identities, etc. We refer the readerto our recent papers in the bibliography for these.

2. Richard Kadison and the beginnings of positivity

The first published words of Richard V. Kadison appear to be the following:

“It is the purpose of the present note to investigate the order prop-erties of self-adjoint operators individually and with respect to con-taining operator algebras”.

This was from the paper [49], which appeared in 1950. In the early 1950s the warwas over, John von Neumann was editor of the Annals of Mathematics and wastalking to anybody who was interested about ‘rings of operators’, Kadison was inChicago and the IAS, and all was well with the world. In 1950, von Neumann wrotea letter to Kaplansky (IAS Archives, reproduced in [65]) which begins as follows:

“Dear Dr. Kaplansky,

Very many thanks for your letter of February 11th and yourmanuscript on ”Projections in Banach Algebras”. I am very gladthat you are submitting it for THE ANNALS, and I will immedi-ately recommend it for publication.

Your results are very interesting. You are, of course, very right:I am and I have been for a long time strongly interested in a “purelyalgebraical” rather than “vectorial-spatial” foundation for theoriesof operator-algebras or operatorlike-algebras. To be more precise:It always seemed to me that there were three successive levels ofabstraction - first, and lowest, the vectorial-spatial, in which theHilbert space and its elements are actually used; second, the purelyalgebraical, where only the operators or their abstract equivalentsare used; third, the highest, the approach when only linear spacesor their abstract equivalents (i.e. operatorially speaking, the pro-jections) are used. [. . . ] After Murray and I had reached somewhatrounded results on the first level, I neglected to make a real efforton the second one, because I was tempted to try immediately thethird one. This led to the theory of continuous geometries. Instudying this, the third level, I realized that one is led there to thetheory of “finite” dimensions only. The discrepancy between whatmight be considered the “natural” ranges for the first and the thirdlevel led me to doubt whether I could guess the correct degree ofgenerality for the second one. . . ”.

It is remarkable here to recall that von Neumann invented the abstract def-inition of Hilbert spaces, the theory of unbounded operators (as well as much ofthe bounded theory), ergodic theory, the mathematical formulation of quantummechanics, many fundamental concepts associated with groups (like amenability),

38 DAVID P. BLECHER

and several other fields of analysis. Even today, teaching a course in functionalanalysis can sometimes feel like one is mainly expositing the work of this one man.However von Neumann is saying above that he had unfortunately neglected whathe calls the ‘second level’ of ‘operator algebra’, and at the time of this letter thiswas ripe and timely for exploration.

Happily, about the time the above letter was written, Richard Vincent Kadi-son entered the world with a bang: a spate of amazing papers at von Neumann’s‘second level’. Indeed Kadison soon took leadership of the American school of op-erator algebras. Some part of his early work was concerned with positive cones andtheir properties. We will now briefly describe a few of these and spend much ofthe remainder of our article showing how they can be generalized to nonselfadjointoperator algebras and Banach algebras. The following comprises just a tiny part ofKadison’s opus; but nonetheless is still foundational and seminal. Indeed much ofC∗-algebra theory would disappear without this work. At the start of this sectionwe already mentioned his first paper, devoted to ‘order properties of self-adjoint op-erators individually and with respect to containing operator algebras’. His memoir“A representation theory for commutative topological algebra” [51] soon followed,one small aspect of which was the introduction and study of positive cones, states,and square roots in Banach algebras. In the 1951 Annals paper [50], Kadison gen-eralized the “Banach-Stone” theorem, characterizing surjective isometries betweenC∗-algebras. This result has inspired very many functional analysts and innumer-able papers. See for example [38] for a collection of such results, together withtheir history, although this reference is a bit dated since the list grows all the time.See also e.g. [11, Section 6]. In a 1952 Annals paper [52] he proved the Kadison–Schwarz inequality, a fundamental inequality satisfied by positive linear maps onC∗-algebras. His student Størmer continued this in a very long (and still contin-uing) series of deep papers. Later this Kadison–Schwarz work was connected tocompletely positive maps, Stinespring’s theorem and Arveson’s extension theorem(see the next paragraph and e.g. [68]), conditional expectations, operator systemsand operator spaces, quantum information theory, etc. A related enduring interestof Kadison’s is projections and conditional expectations on C∗-algebras and vonNeumann algebras. A search of his collected works finds very many contributionsto this topic (e.g. [53]).

In 1960, Kadison together with I. M. Singer [57] initiated the study of non-selfadjoint operator algebras on a Hilbert space (henceforth simply called operatoralgebras). Five years later or so, the late Bill Arveson in his thesis continued thestudy of nonselfadjoint operator algebras, using heavily the Kadison-Fuglede deter-minant of [54] and positivity properties of conditional expectations. This work waspublished in [4]; it developes a von Neumann algebraic theory of noncommutativeHardy spaces. We mention in passing that we continued Arveson’s work from [4] ina series of papers with Labuschagne, again using the Kadison-Fuglede determinantof [54] as a main tool (see e.g. the survey [14]), as well as positive conditionalexpectations and the Kadison–Schwarz inequality. This is another example of us-ing C∗-algebraic methods, and in particular tools originating in seminal work ofKadison, in a more general (noncommutative function theoretic) setting. Howeversince this lies in a different direction to the rest of the present article we will sayno more about this. In the decade after [4], Arveson went on to write many otherseminal papers on nonselfadjoint operator algebras, perhaps most notably [5], in


which completely positive maps and the Kadison–Schwarz inequality play a decisiverole, and which may be considered a source of the later theory of operator spacesand operator systems.

Another example: in 1968 Kadison and Aarnes, his first student at Penn, in-troduced strictly positive elements in a C∗-algebra A, namely x ∈ A which satisfyf(x) > 0 for every state f of A. They proved the fundamental basic result:

Theorem 2.1 (Aarnes–Kadison). For a C∗-algebra A the following are equiv-alent:

(1) A has a strictly positive element.(2) A has a countable increasing contractive approximate identity.(3) A = zAz for some positive z ∈ A.(4) The positive cone A+ has an element z of full support (that is, the support

projection s(z) is 1).

The approximate identity in (2) may be taken to be commuting, indeed it may be

taken to be (z1n ) for z as in (3). If A is a separable C∗-algebra then these all hold.

Aarnes and Kadison did not prove (4). However (4) is immediate from the rest

since s(z) is the weak* limit of z1n , and the converse is easy. This result is related

to the theory of hereditary subalgebras, comparison theory in C∗-algebras, etc. Infact much of modern C∗-algebra theory would collapse without basic results likethis. For example, the Aarnes–Kadison theorem implies the beautiful characteri-zation due to Prosser [71] of closed one-sided ideals in a separable C∗-algebra Aas the ‘topologically principal (one-sided) ideals’ (we are indebted to the refereefor pointing out that Prosser was a student of Kelley). The latter is equivalentto the characterization of hereditary subalgebras of such A as the subalgebras ofform zAz. (We recall that a hereditary subalgebra, or HSA for short, is a closedselfadjoint subalgebra D satisfying DAD ⊂ D.) These results are used in manymodern theories such as that of the Cuntz semigroup. Or, as another example, theAarnes–Kadison theorem is used in the important stable isomorphism theorem forMorita equivalence of C∗-algebras (see e.g. [10,28]).

Indeed in some sense the Aarnes–Kadison theorem is equivalent to the firstassertion of the following:

Theorem 2.2. A HSA (resp. closed right ideal) in a C∗-algebra A is (topo-logically) principal, that is of the form zAz (resp. zA) for some z ∈ A iff it hasa countable (resp. countable left) contractive approximate identity. Every closedright ideal (resp. HSA) is the closure of an increasing union of such (topologically)principal right ideals (resp. HSA’s).

Indeed separable HSA’s (resp. closed right ideals) in C∗-algebras have countable(resp. countable left) approximate identities.

One final work of Kadison which we will mention here is his first paper with GertPedersen [55], which amongst other things initiates the development of a compari-son theory for elements in C∗-algebras generalizing the von Neumann equivalenceof projections. Again positivity and properties of the positive cone are key to thatwork. This paper is often cited in recent papers on the Cuntz semigroup.

The big question we wish to address in this article is how to generalize suchresults and theories, in which positivity is the common theme, to not necessarily

40 DAVID P. BLECHER

selfadjoint operator algebras (or perhaps even Banach algebras). In fact one of-ten can, as we have shown in joint work with Charles Read, Matt Neal, NarutakaOzawa, and others. In the Banach algebra literature of course there are many gen-eralizations of C∗-algebra results, but as far as we are aware there is no ‘positivity’approach like ours (although there is a trace of it in [37]). In particular we mentionSinclair’s generalization from [74] of part of the Aarnes–Kadison theorem:

Theorem 2.3 (Sinclair). A separable Banach algebra A with a bounded ap-proximate identity has a commuting bounded approximate identity.

If A has a countable bounded approximate identity then Sinclair and othersshow results like A = xA = Ay for some x, y ∈ A. In part of our work we followSinclair in using variants of the proof of the Cohen factorization method to achievesuch results but with ‘positivity’.

We now explain one of the main ideas. Returning to the early 1950s: it wasonly then becoming perfectly clear what a C∗-algebra was; a few fundamental factsabout the positive cone were still being proved. We recall that an unpublishedresult of Kaplansky removed the final superfluous abstract axiom for a C∗-algebra,and this used a result in a 1952 paper of Fukamiya, and in a 1953 paper of JohnKelley and Vaught [58] based on a 1950 ICM talk by those authors. These sourcesare referenced in almost every C∗-algebra book. The paper of Kelley and Vaughtwas titled “The positive cone in Banach algebras”, and in the first section of thepaper they discuss precisely that. The following is not an important part of theirpaper, but as in Kadison’s paper a year earlier they have a small discussion on howto make sense of the notion of a positive cone in a Banach algebra, and they provesome basic results here. Both Kadison and Kelley and Vaught have some use forthe set

FA = {x ∈ A : ‖1− x‖ ≤ 1}.In their case A is unital (that is has an identity of norm 1), but if not one may take1 to be the identity of a unitization of A. In [22], Charles Read and the authorbegan a study of not necessarily selfadjoint operator algebras on a Hilbert spaceH; henceforth operator algebras. In this work, FA above plays a pivotal role, andalso the cone R+

FA. In [23] we looked at the slightly larger cone rA of so calledaccretive elements (this is a non-proper cone or ‘wedge’). In an operator algebrathese are the elements with positive real part; in a general Banach algebra they arethe elements x with Re ϕ(x) ≥ 0 for every state ϕ on a unitization of A. We recallthat a state on a unital Banach algebra is, as usual in the theory of numerical range[27], a norm one functional ϕ such that ϕ(1) = 1. That is, accretive elements arethe elements with numerical range in the closed right half-plane. We sometimesalso call these the real positive elements. We will see later in Proposition 6.6 that

R+FA = rA. That is, the one cone above is the closure of the other. We write CA

for either of these cones.The following lemma is known, some of it attributable to Lumer and Phillips, or

implicit in the theory of contraction semigroups, or can be found in e.g. [63, Lemma2.1]. The latter paper was no doubt influential on our real-positive theory in [21].

Lemma 2.4. Let A be a unital Banach algebra. If x ∈ A the following areequivalent:

(1) x ∈ rA, that is, x has numerical range in the closed right half-plane.(2) ‖1− tx‖ ≤ 1 + t2‖x‖2 for all t > 0.


(3) ‖ exp(−tx)‖ ≤ 1 for all t > 0.(4) ‖(t+ x)−1‖ ≤ 1

t for all t > 0.

(5) ‖1− tx‖ ≤ ‖1− t2x2‖ for all t > 0.

Proof. For the equivalence of (1) and (3), see [27, p. 17]. Clearly (5) implies(2). That (2) implies (1) follows by applying a state ϕ to see |1− tϕ(x)| ≤ 1+Kt2,which forces Reϕ(x) ≥ 0) (see [63, Lemma 2.1]). Given (4) with t replaced by 1

t ,we have

‖1− tx‖ = ‖(1 + tx)−1(1 + tx)(1− tx)‖ ≤ ‖1− t2x2‖.This gives (5). Finally (1) implies (4) by e.g. Stampfli and Williams result [76,Lemma 1] that the norm in (4) is dominated by the reciprocal of the distance from−t to the numerical range of x. �

(We mention another equivalent condition: given ε > 0 there exists a t > 0with ‖1− tx‖ < 1 + εt. See e.g. [27, p. 30].)

Real positive elements, and the smaller set FA above, will play the role for usof positive elements in a C∗-algebra. While they are not the same, real positivityis very compatible with the usual definition of positivity in a C∗-algebra, as will beseen very clearly in the sequel, and in particular in the next section.

3. Real completely positive maps and projections

Recall that a linear map T : A → B between C∗-algebras (or operator systems)is completely positive if T (A+) ⊂ B+, and similarly at the matrix levels. By a unitaloperator space below we mean a subspace of B(H) or a unital C∗-algebra containingthe identity. We gave abstract characterizations of these objects with Matthew Nealin [16,19], and have studied them elsewhere.

Definition 3.1. A linear map T : A → B between operator algebras or unitaloperator spaces is real positive if T (rA) ⊂ rB. It is real completely positive, or RCPfor short, if Tn is real positive on Mn(A) for all n ∈ N.

(This and the following two results are later variants from [9] of matchingmaterial from [22] for FA.)

Theorem 3.2. A (not necessarily unital) linear map T : A → B between C∗-algebras or operator systems is completely positive in the usual sense iff it is RCP.

We say that an algebra is approximately unital if it has a contractive approxi-mate identity (cai).

Theorem 3.3 (Extension and Stinespring-type Theorem). A linear map T :A → B(H) on an approximately unital operator algebra or unital operator space is

RCP iff T has a completely positive (in the usual sense) extension T : C∗(A) →B(H). Here C∗(A) is a C∗-algebra generated by A. This is equivalent to being ableto write T as the restriction to A of V ∗π(·)V for a ∗-representation π : C∗(A) →B(K), and an operator V : H → K.

Of course this result is closely related to Kadison’s Schwarz inequality. Inparticular, if one is trying to generalize results where completely positive maps andthe Kadison’s Schwarz inequality are used in the C∗-theory, to operator algebras,one can see how Theorem 3.2 would play a key role. And indeed it does, for examplein the remaining results in this section.

42 DAVID P. BLECHER

We will not say more about unital operator spaces in the present article, exceptto say that it is easy to see that completely contractive unital maps on a unitaloperator space are RCP.

We give two or three applications from [20] of Theorem 3.3. The first is relatedto Kadison’s Banach–Stone theorem for C∗-algebras [50], and uses our Banach–Stone type theorem [15, Theorem 4.5.13].

Theorem 3.4. (Banach–Stone type theorem) Suppose that T : A → B isa completely isometric surjection between approximately unital operator algebras.Then T is real (completely) positive if and only if T is an algebra homomorphism.

In the following discussion, by a projection P on an operator algebra A, wemean an idempotent linear map P : A → A. We say that P is a conditionalexpectation if P (P (a)bP (c)) = P (a)P (b)P (c) for a, b, c ∈ A.

Proposition 3.5. A real completely positive completely contractive map (resp.projection) on an approximately unital operator algebra A, extends to a unital com-pletely contractive map (resp. projection) on the unitization A1.

Much earlier, we studied completely contractive projections P and conditionalexpectations on unital operator algebras. Assuming that P is also unital (that is,P (1) = 1) and that Ran(P ) is a subalgebra, we showed (see e.g. [15, Corollary4.2.9]) that P is a conditional expectation. This is the operator algebra variant ofTomiyama’s theorem for C∗-algebras. A well known result of Choi and Effros statesthat the range of a completely positive projection P : B → B on a C∗-algebra B, isagain a C∗-algebra with product P (xy). The analogous result for unital completelycontractive projections on unital operator algebras is true too, and is implicit in theproof of our generalization of Tomiyama’s theorem above. Unfortunately, there isno analogous result for (nonunital) completely contractive projections on possiblynonunital operator algebras without adding extra hypotheses on P . However if weadd the condition that P is also ‘real completely positive’, then the question doesmake good sense and one can easily deduce from the unital case and Proposition3.5 one direction of the following:

Theorem 3.6. [20] The range of a completely contractive projection P : A →A on an approximately unital operator algebra is again an operator algebra withproduct P (xy) and cai (P (et)) for some cai (et) of A, iff P is real completelypositive.

Proof. For the ‘forward direction’ note that P ∗∗ is a unital complete contrac-tion, and hence is real completely positive as we said in above Theorem 3.4. For the‘backward direction’ the following proof, due to the author and Neal, was originallya remark in [20]. By passing to the bidual we may assume that A is unital. IfP (P (1)x) = P (xP (1)) = x for all x ∈ Ran(P ) then we are done by the abstractcharacterization of operator algebras from [15, Section 2.3], since then P (xy) de-fines a bilinear completely contractive product on Ran(P ) with ‘unit’ P (1). LetI(A) be the injective envelope of A. We may extend P to a completely positive

completely contractive map P : I(A) → I(A), by [9, Theorem 2.6] and injectivity

of I(A). We will abusively sometimes write P for P , and also for its second adjointon I(A)∗∗. The latter is also completely positive and completely contractive. Then

P (P (1)1n ) ≥ P (P (1)) = P (1) ≥ P (P (1)

1n ).


Hence these quantities are equal. In the limit, P (s(P (1))) = P (1), if s(P (1)) isthe support projection of P (1). Hence P (z) = 0 where z = s(P (1)) − P (1). Ify ∈ I(A)+ with ‖y‖ ≤ 1, then P (y) ≤ P (1) ≤ s(P (1)), and so s(P (1))P (y) =

P (y) = P (y)s(P (1)). It follows that s(P (1))x = xs(P (1)) = x for all x ∈ Ran(P ).If also ‖x‖ ≤ 1, then

P (P (1)x) = P (s(P (1))x)− P (zx) = P (s(P (1))x) = P (x)

by the Kadison-Schwarz inequality, since

P (zx)P (zx)∗ ≤ P (zxx∗z) ≤ P (z2) ≤ P (z) = 0.

Thus P (P (1)x) = x if x ∈ P (A). Similarly, P (xP (1)) = x as desired. Thus P (xy)defines a bilinear completely contractive product on Ran(P ) with ‘unit’ P (1). �

The main thrust of [20] is the investigation of the completely contractive pro-jections and conditional expectations, and in particular the ‘symmetric projectionproblem’ and the ‘bicontractive projection problem’, in the category of operator al-gebras, attempting to find operator algebra generalizations of certain deep results ofStørmer, Friedman and Russo, Effros and Størmer, Robertson and Youngson, andothers (see papers of these authors referenced in the bibliography below), concern-ing projections and their ranges, assuming in addition that our projections are realcompletely positive. We say that an idempotent linear P : X → X is completelysymmetric (resp. completely bicontractive) if I−2P is completely contractive (resp.if P and I − P are completely contractive). ‘Completely symmetric’ implies ‘com-pletely bicontractive’. The two problems mentioned at the start of this paragraphconcern 1) Characterizing such projections P ; or 2) characterizing the range of suchprojections. On a unital C∗-algebra B the work of some of the authors mentionedat the start of this paragraph establish that unital positive bicontractive projec-tions are also symmetric, and are precisely 1

2 (I+θ), for a period 2 ∗-automorphismθ : B → B. The possibly nonunital positive bicontractive projections P are of asimilar form, and then q = P (1) is a central projection in M(B) with respect towhich P decomposes into a direct sum of 0 and a projection of the above form12 (I+ θ), for a period 2 ∗-automorphism θ of qB. Conversely, a map P of the latterform is automatically completely bicontractive, and the range of P , which is theset of fixed points of θ, is a C∗-subalgebra, and P is a conditional expectation.

One may ask what from the last paragraph is true for general (approximatelyunital) operator algebras A? The first thing to note is that now ‘completely bi-contractive’ is no longer the same as ‘completely symmetric’. The following is oursolution to the symmetric projection problem, and it uses Kadison’s Banach–Stonetheorem for C∗-algebras [50], and our variant of the latter for approximately unitaloperator algebras (see e.g. [15, Theorem 4.5.13]):

Theorem 3.7. [20] Let A be an approximately unital operator algebra, andP : A → A a completely symmetric real completely positive projection. Then therange of P is an approximately unital subalgebra of A. Moreover, P ∗∗(1) = q is aprojection in the multiplier algebra M(A) (so is both open and closed).

Set D = qAq, a hereditary subalgebra of A containing P (A). There exists aperiod 2 surjective completely isometric homomorphism θ : A → A such that θ(q) =q, so that θ restricts to a period 2 surjective completely isometric homomorphism

44 DAVID P. BLECHER

D → D. Also, P is the zero map on q⊥A+Aq⊥ + q⊥Aq⊥, and

P =1

2(I + θ) on D.

In fact

P (a) =1

2(a+ θ(a)(2q − 1)) , a ∈ A.

The range of P is the set of fixed points of θ.Conversely, any map of the form in the last equation is a completely symmetric

real completely positive projection.

Remark. In the case that A is unital but q is not central in the last theorem,if one solves the last equation for θ, and then examines what it means that θ is ahomomorphism, one obtains some interesting algebraic formulae involving q, q⊥, Aand θ|qAq.

For the more general class of completely bicontractive projections, a first lookis disappointing–most of the last paragraph no longer works in general. Onedoes not always get an associated completely isometric automorphism θ such thatP = 1

2 (I + θ), and q = P (1) need not be a central projection. However, as alsoseems to be sometimes the case when attempting to generalize a given C∗-algebrafact to more general algebras, a closer look at the result, and at examples, doesuncover an interesting question. Namely, given an approximately unital operatoralgebra A and a real completely positive projection P : A → A which is com-pletely bicontractive, when is the range of P a subalgebra of A and P a conditionalexpectation? This seems to be the right version of the ‘bicontractive projectionproblem’ in the operator algebra category. We give in [20] a sequence of threereductions that reduce the question. The first reduction is that by passing to thebidual we may assume that the algebra A is unital. The second reduction is thatby cutting down to qAq, where q = P (1) (which one can show is a projection), wemay further assume that P (1) = 1 (one can show P is zero on q⊥A + Aq⊥). Thethird reduction is by restricting attention to the closed algebra generated by P , wemay further assume that P (A) generates A as an operator algebra. We call thisthe ‘standard position’ for the bicontractive projection problem. It turns out thatwhen in standard position, Ker(P ) is forced to be an ideal with square zero.

In the second reduction above, that is if A and P are unital, then one may showthat A decomposes as A = C ⊕ B, where 1A ∈ B = P (A), C = (I − P )(A), andwe have the relations C2 ⊂ B,CB +BC ⊂ C (see [20, Lemma 4.1] and its proof).The period 2 map θ : x + y �→ x− y for x ∈ B, y ∈ C is a homomorphism (indeedan automorphism) on A iff P (A) is a subalgebra of A, and we have, similarly toTheorem 3.7:

Corollary 3.8. If P : A → A is a unital idempotent on a unital operatoralgebra then P is completely bicontractive iff there is a period 2 linear surjectionθ : A → A such that ‖I ± θ‖cb ≤ 2 and P = 1

2 (I + θ). The range of P is asubalgebra iff θ is also a homomorphism, and then the range of P is the set of fixedpoints of this automorphism θ. Also, P is completely symmetric iff θ is completelycontractive.

We remark that for the subcategory of uniform algebras (that is, closed unital(or approximately unital) subalgebras of C(K), for compact K), there is a completesolution to the bicontractive projection problem.


Theorem 3.9. Let P : A → A be a real positive bicontractive projection ona (unital or approximately unital) uniform algebra. Then P is symmetric, and soof course by Theorem 3.7 we have that P (A) is a subalgebra of A, and P is aconditional expectation.

Proof. We sketch the idea, found in a conversation with Joel Feinstein. By thefirst two reductions described above we can assume that A and P are unital. We alsoknow that B = P (A) is a subalgebra, since if it were not then the third reductiondescribed above would yield nonzero nilpotents, which cannot exist in a functionalgebra. Thus by the discussion above the theorem, the map θ(x+y) = x−y there isan algebra automorphism of A, hence an isometric isomorphism (since norm equalsspectral radius). So P = 1

2 (I + θ) is symmetric. �

The same three step reduction shows that we can also solve the problem in theaffirmative for real completely positive completely bicontractive projections P on aunital operator algebra A such that the closed algebra generated by A is semiprime(that is, it has no nontrivial square-zero ideals). We have found counterexamples tothe general question, but we have also have found conditions that make all known(at this point) counterexamples go away. See [20] for details.

4. More notation, and existence of ‘positive’ approximate identities

We have already defined the cone rA of accretive or ‘real positive’ elements,and its dense subcone R+

FA. Another subcone which is occasionally of interestis the cone consisting of elements of A which are ‘sectorial’ of angle θ < π

2 . Forthe purposes of this paper being sectorial of angle θ will mean that the numericalrange in A (or in a unitization of A if A is nonunital) is contained in the sectorSθ consisting of complex numbers reiρ with r ≥ 0 and |ρ| ≤ θ. This third cone isa dense subset of the second cone R+

FA if A is an operator algebra [25, Lemma2.15]. We remark that there exists a well established functional calculus for sectorialoperators (see e.g. [43]). Indeed the advantages of this cone and the last one seemsto be mainly that these have better functional calculi. For the cone R+

FA, ifA is an operator algebra, one could use the functional calculus coming from vonNeumann’s inequality. Indeed if ‖I − x‖ ≤ 1 then f �→ f(I − x) is a contractivehomomorphism on the disk algebra. If x is real positive in an operator algebra, onecould also use Crouzeix’s remarkable functional calculus on the numerical range ofx (see e.g. [31]). If x is sectorial in a Banach algebra, one may use the functionalcalculus for sectorial operators [43].

A final notion of positivity which we introduced in the work with Read, which isslightly more esoteric, but which is a close approximation to the usual C∗-algebraicnotion of positivity: In the theorems below we will sometimes say that an elementx is nearly positive; this means that in the statement of that result, given ε > 0one can also choose the element in that statement to be real positive and withinε of its real part (which is positive in the usual sense). In fact whenever we say‘x is nearly positive’ below, we are in fact able, for any given ε > 0, to choosex to also be a contraction with numerical range within a thin ‘cigar’ centered onthe line segment [0, 1] of height < ε. That is, x has sectorial angle < arcsin ε. Inan operator algebra any contraction x with such a sectorial angle is accretive andsatisfies ‖x−Rex‖ ≤ ε, so x is within ε of an operator which is positive in the usualsense. Indeed if a is an accretive element in an operator algebra then (principal)

46 DAVID P. BLECHER

n-th roots of a have spectrum and numerical radius within a sector S π2n, and hence

are as close as we like (for n sufficiently large) to an operator which is positivein the usual sense (see Section 6). Thus one obtains ‘nearly positive elements’ bytaking n-th roots of accretive elements. A nearly positive approximate identity (et)means that it is real positive and the sectorial angle of et converges to 0 with t.(We remark that at the time of writing we do not know for general Banach algebrasif roots (or rth powers for 0 < r < 1) of accretive elements are in R+

FA, or if thethird cone in the last paragraph is contained in the second cone. We note thatthe roots in the last sentence need not be in this third cone, as may be seen using[21, Example 3.13].)

In the last paragraphs we have described several variants of ‘positivity’, whichat least in an operator algebra are each successively stronger than the last. It isconvenient to mentally picture each of these notions by sketching the region con-taining the numerical range of x. Thus for the first notion, the accretive elements,one simply pictures the right half plane in C. One pictures the second, the coneR+

FA, as a dense cone in the right half plane composed of closed disks center aand radius a, for all a > 0. The third cone is pictured as increasing sectors Sθ inC, for increasing θ < π

2 . And the ‘nearly positive’ elements are pictured by thethin ‘cigar’ mentioned a paragraph or so back, centered on the line segment [0, 1]of height < ε, and contained in the closed disk center 1

2 of radius 12 .

We now list some more of our notation and general facts: We write Ball(X) forthe set {x ∈ X : ‖x‖ ≤ 1}. For us Banach algebras satisfy ‖xy‖ ≤ ‖x‖‖y‖. If x ∈ Afor a Banach algebra A, then ba(x) denotes the closed subalgebra generated by x.If A is a Banach algebra which is not Arens regular, then the multiplication weusually use on A∗∗ is the ‘second Arens product’ (� in the notation of [32]). This isweak* continuous in the second variable. If A is a nonunital, not necessarily Arensregular, Banach algebra with a bounded approximate identity (bai), then A∗∗ hasa so-called ‘mixed identity’ [32,34,67], which we will again write as e. This is aright identity for the first Arens product, and a left identity for the second Arensproduct. A mixed identity need not be unique, indeed mixed identities are just theweak* limit points of bai’s for A.

See the book of Doran and Wichmann [34] for a compendium of results aboutapproximate identities and related topics. If A is an approximately unital Banachalgebra, then the left regular representation embeds A isometrically in B(A). Wewill always write A1 for the multiplier unitization of A, that is, we identify A1

isometrically with A+C I in B(A). Below 1 will almost always denote the identityof A1, if A is not already unital. If A is a nonunital, approximately unital Banachalgebra then the multiplier unitization A1 may also be identified isometrically withthe subalgebra A+ C e of A∗∗ for a fixed mixed identity e of norm 1 for A∗∗.

We recall that a subspace E of a Banach space X is an M -ideal in X if E⊥⊥ iscomplemented inX∗∗ via a contractive projection P so thatX∗∗ = E⊥⊥⊕∞Ker(P ).In this case there is a unique contractive projection onto E⊥⊥. This concept wasinvented by Alfsen and Effros, and [44] is the basic text for their beautiful andpowerful theory. By an M -approximately unital Banach algebra we mean a Banachalgebra which is an M -ideal in its multiplier unitization A1. This is equivalent(see [21, Lemma 2.4] to saying that ‖1 − x‖(A1)∗∗ = ‖e − x‖A∗∗ for all x ∈ A∗∗,


unless the last quantity is < 1 in which case ‖1 − x‖(A1)∗∗ = 1. Here e is theidentity for A∗∗ if it has one, otherwise it is a mixed identity of norm 1. A resultof Effros and Ruan implies that approximately unital operator algebras are M -approximately unital (see e.g. [15, Theorem 4.8.5 (1)]). Also, all unital Banachalgebras are M -approximately unital.

We use states a lot in our work. However for an approximately unital Banachalgebra A with cai (et), the definition of ‘state’ is problematic. Although we havenot noticed this discussed in the literature, there are several natural notions, andwhich is best seems to depend on the situation. For example: (i) a contractivefunctional ϕ on A with ϕ(et) → 1 for some fixed cai (et) for A, (ii) a contractivefunctional ϕ on A with ϕ(et) → 1 for all cai (et) for A, and (iii) a norm 1 functionalon A that extends to a state on A1, where A1 is the ‘multiplier unitization’ above.If A satisfies a smoothness hypothesis then all these notions coincide [21, Lemma2.2], but this is not true in general. The M -approximately unital Banach algebrasin the last paragraph are smooth in this sense. Also, if e is a mixed identity for A∗∗

then the statement ϕ(e) = 1 may depend on which mixed identity one considers.In this paper though for simplicity, and because of its connections with the usualtheory of numerical range and accretive operators, we will take (iii) above as thedefinition of a state of A. In [21] we also consider some of the other variants above,and these will appear below from time to time. We define the state space S(A) tobe the set of states in the sense of (iii) above. The quasistate space Q(A) is {tϕ :t ∈ [0, 1], ϕ ∈ S(A)}. The numerical range of x ∈ A is WA(x) = {ϕ(x) : ϕ ∈ S(A)}.As in [21] we define rA∗∗ = A∗∗ ∩ r(A1)∗∗ . There is an unfortunate ambiguity withthe latter notation here and in [21] in the (generally rare) case that A∗∗ is unital. Itshould be stressed that in these papers rA∗∗ should not, if A∗∗ is unital, be confusedwith the real positive (i.e. accretive) elements in A∗∗. It is shown in [21, Section 2]that these are the same if A is an M -approximately unital Banach algebra, and inparticular if A is an approximately unital operator algebra. It is easy to see thatA∗∗ ∩ r(A1)∗∗ is contained in the accretive elements in A∗∗ if A∗∗ is unital, but theother direction seems unclear in general.

Of course in the theory of C∗-algebras, positivity and the existence of positiveapproximate identities are crucial. How does one get a ‘positive cai’ in an algebrawith cai? We have several ways to do this. First, for approximately unital operatoralgebras and for a large class of approximately unital Banach algebras (eg. the scaledBanach algebras defined in the next section; and we do not possess an example ofa Banach algebra that is not scaled yet) we have a ‘Kaplansky density’ result:

Ball(A) ∩ rAw∗

= Ball(A∗∗) ∩ rA∗∗ . See Theorem 5.8 below. (We remark thatalthough it seems not to be well known, the most common variants of the usualKaplansky density theorem for a C∗-algebra A do follow quickly from the weak*density of Ball(A) in Ball(A∗∗), if one constructs A∗∗ carefully.) If A∗∗ has a realpositive mixed identity e of norm 1, then one can then get a real positive cai byapproximating e by elements of Ball(A)∩rA. See Corollary 5.9. A similar argumentallows one to deduce the second assertion in the following result from the first (onealso uses the fact that in an M -approximately unital Banach algebra ‖1− 2e‖ ≤ 1for a mixed identity of norm 1 for A∗∗):

Theorem 4.1. [21,22] Let A be an M -approximately unital Banach algebra,for example any operator algebra. Then FA is weak* dense in FA∗∗ . Hence A hasa cai in 1

2FA.

48 DAVID P. BLECHER

Applied to approximately unital operator algebras (which as we said are all M -approximately unital) the last assertion of Theorem 4.1 becomes Read’s theoremfrom [72]. See also [12,25] for other proofs of the latter result.

Remark 4.2. For the conclusion that FA is weak* dense in FA∗∗ one may relaxthe M -approximately unital hypothesis to the following much milder condition: Ais approximately unital and given ε > 0 there exists a δ > 0 such that if y ∈ Awith ‖1 − y‖ < 1 + δ then there is a z ∈ A with ‖1 − z‖ = 1 and ‖y − z‖ < ε.Here 1 denotes the identity of any unitization of A. This follows from the proof of[21, Theorem 5.2]. For example, L1(R) satisfies this condition with δ = ε.

Another approach to finding a ‘real positive cai’ under a countability conditionfrom [21, Section 2] uses a slight variant of the ‘real positive’ definition. Namely fora fixed cai e = (et) for A define Se(A) = {ϕ ∈ Ball(A∗) : limt ϕ(et) = 1} (a subsetof S(A)). Define reA = {x ∈ A : Reϕ(x) ≥ 0 for all ϕ ∈ Se(A)}. If we multiply thesestates by numbers in [0, 1], we get the associated quasistate space Qe(A). Note thatreA contains rA. On the other hand, [21, Theorem 6.5] (or a minor variant of theproof of it) shows that if A∗∗ is unital then reA is never contained in rA∗∗ (or in theaccretive elements in A∗∗) unless rA = reA.

Theorem 4.3. [21] A Banach algebra A with a sequential cai e and with Qe(A)weak* closed, has a sequential cai in reA.

Proof. We give the main idea of the proof in [21], and a few more details forthe first step. Suppose that K is a compact space and (fn) is a bounded sequencein C(K,R), such that limn fn(x) exists for every x ∈ K and is non-negative. Claim:for every ε > 0, there is a function f ∈ conv{fn} such that f ≥ −ε on K. Indeed ifthis were not true, then there exists an ε > 0 such that for all f ∈ conv{fn} thereis a point x in K with f(x) < −ε. Moreover, for all g ∈ conv{fn}, if f ∈ conv{fn}with ‖f − g‖ < ε

4 , there is a point x in K with g(x) < − 3ε4 . So A = conv{fn} and

C = C(K)+ are clearly disjoint. Moreover, it is well known that convex sets E,Cin an LCTVS can be strictly separated iff 0 /∈ E − C, and this is clearly the casefor us here. So there is a continuous functional ψ on C(K,R) and scalars M,Nwith ψ(g) ≤ M < N ≤ ψ(h) for all g ∈ A and h ∈ C. Since C is a cone we maytake N = 0. By the Riesz–Markov theorem there is a Borel probability measure msuch that supn

∫Kfn dm < 0. This is a contradiction and proves the Claim, since

limn

∫K

fn dm ≥ 0 by Lebesgue’s dominated convergence theorem.Now letK = Qe(A) and fn(ψ) = Reψ(en) where e = (en). We have limn fn ≥ 0

pointwise on K, so by the last paragraph for any ε > 0 a convex combination ofthe fn is always ≥ −ε on K. By a standard geometric series type argument we canreplace ε with 0 here, so that we have a real positive element, and with more carethis convex combination may be taken to be a generic element in a cai. �

Finally, we state a ‘new’ result, which will be proved in Corollary 5.10 below(this result was referred to incorrectly in the published version of [21] as ‘Corollary3.4’ of the present paper).

Corollary 4.4. If A is an approximately unital Banach algebra with a cai esuch that S(A) = Se(A), and such that the quasistate space Q(A) is weak* closed,then A has a cai in rA.


We remark that we have no example of an approximately unital Banach algebrawhere Q(A) is not weak* closed. In particular, we have found that commonlyencountered algebras have this property.

5. Order theory in the unit ball

In the spirit of the quotation starting Section 2 we now discuss variants of wellknown order-theoretic properties of the unit ball of a C∗-algebra and its dual. Someof these results also may be viewed as new relations between an operator algebraand a C∗-algebra that it generates. There are interesting connections to the classicaltheory of ordered linear spaces (due to Krein, Ando, Alfsen, etc) as found e.g. inthe first chapters of [6]. In addition to striking parallels, some of this classicaltheory can be applied directly. Indeed several results from [21] (some of whichare mentioned below, see e.g. the proof of Theorem 5.4) are proved by appealingto results in that theory. See also [25] for more connections if the algebras are inaddition operator algebras.

The ordering induced by rA is obviously b � a iff a−b is accretive (i.e. numericalrange in right half plane). If A is an operator algebra this happens when Re(a−b) ≥0.

Theorem 5.1. [25] If an approximately unital operator algebra A generates aC∗-algebra B, then A is order cofinal in B. That is, given b ∈ B+ there existsa ∈ A with b � a. Indeed one can do this with b � a � ‖b‖ + ε. Indeed one cando this with b � Cet � ‖b‖ + ε, for a nearly positive cai (et) for A and a constantC > 0.

This and the next result are trivial if A unital.

Theorem 5.2. [25] Let A be an operator algebra which generates a C∗-algebraB, and let UA = {a ∈ A : ‖a‖ < 1}. The following are equivalent:

(1) A is approximately unital.(2) For any positive b ∈ UB there exists a ∈ cA with b � a.(2’) Same as (2), but also a ∈ 1

2FA and nearly positive.

(3) For any pair x, y ∈ UA there exist nearly positive a ∈ 12FA with x � a and

y � a.(4) For any b ∈ UA there exist nearly positive a ∈ 1

2FA with −a � b � a.

(5) For any b ∈ UA there exist x, y ∈ 12FA with b = x− y.

(6) CA is a generating cone (that is, A = CA − CA).

In any operator algebra A it is true that CA − CA is a closed subalgebra of A.It is the biggest approximately unital subalgebra of A, and it happens to also be aHSA in A [23]. We do not know if this is true for Banach algebras.

For ‘nice’ Banach algebras A the cone CA has some of the pleasant order prop-erties in items (3)–(6) in Theorem 5.2. See [21, Section 6] for various variants onthis theme. The following is a particularly clean case:

Theorem 5.3. [21, Section 6] Let A be an M -approximately unital Banachalgebra. Then

(1) For any pair x, y ∈ UA there exist a ∈ 12FA with x � a and y � a.

(2) For any b ∈ UA there exist a ∈ 12FA with −a � b � a.

(3) For any b ∈ UA there exist x, y ∈ UA ∩ 12FA with b = x− y.

(4) CA is a generating cone (that is, A = CA − CA).

50 DAVID P. BLECHER

During the writing of the present paper we saw the following improvement ofpart of Corollaries 6.7 and 6.8, and on some of 6.10 in the submitted version of thepaper [21]. At the galleys stage of that paper we incorporated those advances, butunfortunately slipped up in one proof. The correct version is as below.

Theorem 5.4. [21, Section 6] If a Banach algebra A has a cai e and satisfiesthat Qe(A) is weak* closed, then (1)–(4) in the last theorem hold, with 1

2FA replacedby Ball(A) ∩ reA, and � replaced by the linear ordering defined by the cone reA, andCA replaced by reA. One may drop the three superscript e’s in the last line if inaddition S(A) = Se(A).

Proof. Lemma 2.7 (1) in [21] implies that if Qe(A) is weak* closed, then the‘dual cone’ in A∗ of reA is R+ Se(A). By the remark before [21, Proposition 6.2]a similar fact holds for the real dual cone. Since ‖ϕ‖ = 1 for states and for theirreal parts, the norm on the real dual cone is additive. This is known to imply, bythe theory of ordered linear spaces [6, Corollary 3.6, Chapter 2], that the open ballof A is a directed set. So for any pair x, y ∈ UA there exist z ∈ UA with x �e zand y �e z. Applying this again to z,−z there exists w ∈ UA with ±z �e w. Thisimplies that w±z

2 ∈ reA, and z �e a where a = w+z2 . This proves (1). Applying (1)

to b,−b we get (2). Setting x = a+b2 , y = a−b

2 for a, b as in (2), we get (3) and hence(4). The final assertion is then obvious since if S(A) = Se(A) then rA = reA and �e

is just �. �

Recall that the positive part of the open unit ball UB of a C∗-algebra B is adirected set, and indeed is a net which is a positive cai for B. The first part of thisstatement is generalized by Theorems 5.2 (3), 5.3 (1), and 5.4 (1). The followinggeneralizes the second part of the statement to operator algebras:

Corollary 5.5. [25] If A is an approximately unital operator algebra, thenUA ∩ 1

2FA is a directed set in the � ordering, and with this ordering UA ∩ 12FA is

an increasing cai for A.

We do not know if the second part of the last result is true for any other classesof Banach algebras.

We say a Banach algebra A is scaled if every real positive linear map into thescalars is a nonnegative multiple of a state. Of course it is well known that C∗-algebras are scaled. Somewhat surprisingly, we do not know of an approximatelyunital Banach algebra that is not scaled, and certainly all commonly encounteredBanach algebras seem to be scaled. Unital Banach algebras are scaled by e.g. anargument in the proof of [63, Theorem 2.2].

Theorem 5.6. [21, 25] If A is an approximately unital operator algebra, ormore generally an M -approximately unital Banach algebra, then A is scaled.

For operator algebras, the last result implies Read’s theorem mentioned earlier.

Proposition 5.7. [21] If A is a nonunital approximately unital Banach alge-bra, then the following are equivalent:

(i) A is scaled.(ii) S(A1) is the convex hull of the trivial character χ0 and the set of states

on A1 extending states of A.


(iii) The quasistate space Q(A) = {ϕ|A : ϕ ∈ S(A1)}.(iv) Q(A) is convex and weak* compact.

If these hold then Q(A) = S(A)w∗

, and the numerical range satisfies

WA(a) = conv{0,WA(a)} = WA1(a), a ∈ A.

Theorem 5.8 (Kaplansky density type result). If A is a scaled approximatelyunital Banach algebra then Ball(A) ∩ rA is weak* dense in the unit ball of rA∗∗ .

The last result is from [21], although some operator algebra variant was doneearlier with Read.

Corollary 5.9. If A is a scaled approximately unital Banach algebra then Ahas a cai in rA iff A∗∗ has a mixed identity e of norm 1 in rA∗∗ , or equivalentlywith ‖1A1 − e‖ ≤ 1.

Proof. This is proved in [21, Proposition 6.4], relying on earlier results there,except for parts of the last assertion. For the remaining part, if A has a cai in rA

then a cluster point of this cai is a mixed identity of norm 1, and it is in r(A1)∗∗

since the latter is weak* closed and contains rA. However by a result from [21] (seeLemma 6.18 below), an idempotent is in r(A1)∗∗ iff it is in F(A1)∗∗ . �

Corollary 5.10. If A is a scaled approximately unital Banach algebra with acai e such that S(A) = Se(A) then A has a cai in rA.

Proof. Let e be any weak* limit point of e. Clearly ϕ(e) = 1 for all ϕ ∈Se(A) = S(A). If ϕ ∈ S((A1)∗∗) then its restriction to A1 is in S(A1), henceϕ(e) ≥ 0 by the last line and Proposition 5.7. So e ∈ A∗∗ ∩ r(A1)∗∗ = rA∗∗ , andso the result follows from our Kaplansky density type theorem in the form of itsCorollary 5.9. �

The class of algebras A in the last Corollary is the same as the class in the lastline of the statement of Theorem 5.4. Thus for such algebras, (1)–(4) in Theorem5.3 hold, with 1

2FA replaced by Ball(A)∩ rA, and CA replaced by rA. In particular,rA spans A.

6. Positivity and roots in Banach algebras

As we said in the Introduction, this section and the next have several purposes:We will describe results from our other papers (particularly [21], which generalizessome parts of the earlier work) connected to the work of Kadison summarized inSection 2, but we will also restate the results from several sections of [21] in themore general setting of Banach algebras with no kind of approximate identity. Alsowe will give a detailed discussion of roots (fractional powers) in relation to ourpositivity (see also [9,22,23,25] for results not covered here).

Thus let A be a Banach algebra without a cai, or without any kind of bai.If B is any unital Banach algebra isometrically containing A as a subalgebra, forexample any unitization of A, we define

FBA = {a ∈ A : ‖1B − a‖ ≤ 1},

and write rBA for the set of a ∈ A whose numerical range in B is contained in theright half plane. These sets are closed and convex. Also we define

FA = ∪B FBA , rA = ∪B rBA ,

52 DAVID P. BLECHER

the unions taken over all unital Banach algebras B containing A. Unfortunately itis not clear to us that FA and rA are always convex, which is needed in most ofSection 7 below (indeed we often need them closed too there), and so we will haveto use FB

A and rBA there instead. Of course we could fix this problem by definingFA = ∩B FB

A and rA = ∩B rBA , the intersections taken over all B as above. If we didthis then we could remove the superscript B in all results in Section 7 below; thiswould look much cleaner but may be less useful in practice.

Obviously FA and rA are convex and closed if there is an unitization B0 of Asuch that F

B0

A = FA (resp. rB0

A = rA). This happens if A is an operator algebrabecause then there is a unique unitization by a theorem of Ralf Meyer (see [15,Section 2.1]). The following is another case when this happens.

Lemma 6.1. Let A be a nonunital Banach algebra.

(1) Suppose that there exists a ‘smallest’ unitization norm on A⊕C. That is,there exists a smallest norm on A ⊕ C making it a normed algebra withproduct (a, λ)(b, μ) = (ab + μa + λb, λμ), and satisfying ‖(a, 0)‖ = ‖a‖Afor a ∈ A. Let B0 be A⊕C with this smallest norm. Then F

B0

A = FA and

rB0

A = rA.(2) Suppose that the left regular representation embeds A isometrically in

B(A). (This is the case for example if A is approximately unital.) DefineB0 to be the span in B(A) of IA and the isometrically embedded copy of

A. This has the smallest norm of any unitization of A. Hence FB0

A = FA

and rB0

A = rA.

Proof. If B is any unital Banach algebra containing A, and a ∈ FBA then

a ∈ FB0

A . So FB0

A = FA. A similar argument shows that rB0

A = rA, using Lemma 2.4(2), namely that

rBA = {a ∈ A : ‖1B − ta‖ ≤ 1 + t2‖a‖2 for all t ≥ 0}.(2) The first assertion here is well known and simple: If B is any unital Banach

algebra containing A note that

‖a+ λ1B0‖B0

= sup{‖(a+ λ1)x‖A : x ∈ Ball(A)}= sup{‖(a+ λ1B)x‖B : x ∈ Ball(A)},

so that ‖a+ λ1B0‖B0

≤ ‖a+ λ1B‖B. So (1) holds. �

Proposition 6.2. If A is a nonunital subalgebra of a unital Banach algebraB, and if C is a subalgebra of A, then FB

C = C ∩ FBA and rBC = C ∩ rBA.

We now discuss roots (that is, r’th powers for r ∈ [0, 1]) in a subalgebra Aof a unital Banach algebra B. Actually, we only discuss the principal root (orpower); we recall that the principal rth power, for 0 < r < 1, is the one whosespectrum is contained in a sector Sθ of angle θ < 2rπ. There are several ways todefine these that we are aware of. We will review these and show that they are thesame. As far as we know, Kelley and Vaught [58] were the first to define the squareroot of elements of FA, but their argument works for r’th powers for r ∈ [0, 1]. If‖1− x‖ ≤ 1, define

xr =∞∑k=0

(r

k

)(−1)k(1− x)k , r > 0.


For k ≥ 1 the sign of(rk

)(−1)k is always negative, and

∑∞k=1

(rk

)(−1)k = −1. Thus

the series above converges absolutely, hence converges in A. Indeed it is now easy tosee that the series given for xr is a norm limit of polynomials in x with no constantterm. Using the Cauchy product formula in Banach algebras in a standard way,one deduces that (x

1n )n = x for any positive integer n.

Proposition 6.3 (Esterle). If A is a Banach algebra then FA is closed underr’th powers for any r ∈ [0, 1].

Proof. Let x ∈ A ∩ FB where B is a unital Banach algebra containing A.We have 1B − xr =

∑∞k=1

(rk

)(−1)k (1B − x)k, which is a convex combination in

Ball(B). So xr ∈ A ∩ FB ⊂ FA. �

From [37, Proposition 2.4] if x ∈ FA then we also have (xt)r = xtr for t ∈ [0, 1]and any real r.

One cannot use the usual Riesz functional calculus to define xr if 0 is in thespectrum of x, since such r’th powers are badly behaved at 0. However if 0 is inthe spectrum of x, and x ∈ rBA , one may define xr = limε→0+ (x+ ε1B)

r where thelatter is the r’th power according to the Riesz functional calculus. We will soonsee that this limit exists and lies in A, and then it follows that it is independent ofthe particular unital algebra B containing A as a subalgebra (since all unitizationnorms for A are equivalent). A second way to define r’th powers for r ∈ [0, 1])in Banach algebras is found in [61], following the ideas in Hilbert space operatorcase from the Russian literature from the 50’s [62]. Namely, suppose that B is aunital Banach algebra containing A as a subalgebra, and x ∈ A with numericalrange in B excluding all negative numbers. Since the numerical range is convex,it follows that this numerical range is in fact contained in a sector (i.e. a cone inthe complex plane with vertex at 0) of angle ≤ π. Since this is the case we areinterested in, we will assume that the numerical range of x is in the closed righthalf plane. (This is usually not really any loss of generality, since x and hence thejust mentioned cone can be ‘rotated’ to ensure this.) Thus the numerical range ofx is contained inside a semicircle, namely the one containing the right half of thecircle center 0 radius R > 0. We enlarge this semicircle to a slightly larger ‘slice’of this circle of radius R; thus let Γ be the positively oriented contour which issymmetric about the x-axis, and is composed of an arc of the circle slightly biggerthat the right half of the circle, and two line segments which connect zero with thearc. Let Γε be Γ but with points removed that are distance less than ε to the origin.One defines xt to be the limit as ε → 0 of 1

2πi

∫Γε

λt(λ1B − x)−1 dλ. The latterintegral lies in A+ C 1B, by the usual facts about such integrals. If A is nonunitaland χ0 is the character on A + C 1B annihilating A then χ0(x

t) is the limit of1

2πi

∫Γε

λt(λ1− χ0(x))−1 dλ, which is 1

2πi

∫Γλt dλ = 0. So xt ∈ A. Note that xt is

independent of the particular unitization B used, using the fact that all unitizationnorms are equivalent. If in addition x is invertible then 0 /∈ SpB(x), so that we canreplace Γ by a curve that stays to one side of 0, so that xr is the rth power of x asgiven by the Riesz functional calculus. In fact it is shown in [61, Proposition 3.1.9]that xr = limε→0+ (x + ε1B)

r for t > 0, giving the equivalence with the definitionat the start of this discussion. In addition we now see, as we discussed earlier, thatthe latter limit exists, lies in A, and is independent of B. By [61, Corollary 1.3],the rth power function is continuous on rBA , for any r ∈ (0, 1). Principal nth roots

54 DAVID P. BLECHER

of elements x whose numerical range avoids the negative real numbers are unique,for any positive integer n (see [61] and Theorem 2.5 in [26]).

Another way to define r’th powers xr for r ∈ [0, 1] and x ∈ rA, is via thefunctional calculus for sectorial operators [43] (see also e.g. [79, IX, Section 11] forsome of the origins of this approach). Namely, if B is a unitization of A (or a unitalBanach algebra containing A as a subalgebra) and x ∈ rBA , view x as an operatoron B by left multiplication. This is sectorial of angle ≤ π

2 , and so we can use thetheory of roots (fractional powers) from e.g. [43, Section 3.1] (see also [78]). Moregenerally, if x has no strictly negative numbers in its numerical range with respectto B, then the formula of Stampfli and Williams [76, Lemma 1] and some basictrigonometry shows that x is sectorial of some angle θ < π in the sense of e.g. [43],so that all the facts about fractional powers from that text apply. Basic propertiesof such powers include: xsxt = xs+t and (cx)t = ctxt, for positive scalars c, s, t, andt → xt is continuous. There are very many more in e.g. [43]. Also [43, Proposition3.1.9] shows that xr = limε→0+ (x + εI)r for r > 0, the latter power with respectto the usual Riesz functional calculus. It is easy to see from the last fact thatthe definitions of xr given in this paragraph and in the last paragraph coincide ifx ∈ rA and r > 0; so that again xr is in (the copy inside B(B) of) A. Another

formula we have occasionally found useful is xr = sin(tπ)π

∫∞0

sr−1 (s+x)−1x ds, theBalakrishnan formula (see e.g. [43,79]). Finally, we mention that there are somelovely iterative descriptions of the square root that we discuss in a forthcomingpaper [26] together with a few more facts about roots that are omitted here.

We now show that if x ∈ FA then the definitions of xr given in the last para-graphs and in Proposition 6.3 coincide, if r > 0. We may assume that 0 < r ≤ 1 andwork in a unital algebra B containing A. Let y = 1

1+ε (x+ε1B). Then ‖1B−y‖ < 1,

and so yr as defined in the last paragraphs equals∑∞

k=0

(rk

)(−1)k(1B − y)k since

both are easily seen to equal the rth power of y as given by the Riesz functional cal-culus. However

∑∞k=0

(rk

)(−1)k(1−y)k converges uniformly to

∑∞k=0

(rk

)(−1)k(1−

x)k, as ε → 0+, since the norm of the difference of these two series is dominated by

∞∑k=1

(r

k

)(−1)k (

1

1 + ε− 1) ‖(1− x)k‖ ≤ ε

1 + ε→ 0,

using the fact that for k ≥ 1 the sign of(rk

)(−1)k is always negative. Also, with the

definition of powers in the last paragraphs we have yr = ( 11+ε )

r = (x+ ε1B)r → xr

as ε → 0+. Thus the definitions of xr given in the last paragraphs and in Proposition6.3 coincide in this case.

If A is a subalgebra of a unital Banach algebra B then we define the F-transformon A to be F(x) = x(1B + x)−1 = 1B − (1B + x)−1 for x ∈ rA. This is a relativeof the well known Cayley transform in operator theory. Note that F(x) ∈ ba(x) bythe basic theory of Banach algebras, and it does not depend on B, again because allunitization norms forA are equivalent. The inverse transform takes y to y(1B−y)−1.For operator algebras we have ‖F(x)‖ ≤ ‖x‖ and ‖κ(x)‖ ≤ ‖x‖ for x ∈ rA. ForBanach algebras this is not true; for example on the group algebra of Z2.

Unless explicitly said to the contrary, the remaining results in this section aregeneralizations to general Banach algebras of results from [21]. The main resultshere in the operator algebra case were proved earlier by the author and Read (someare much sharper in that setting).


Lemma 6.4. If A is a subalgebra of a unital Banach algebra B then F(rBA) ⊂ FBA

and F(rA) ⊂ FA.

Proof. This is because by a result of Stampfli and Williams [76, Lemma 1],

‖1B − x(1B + x)−1‖ = ‖(1B + x)−1‖ ≤ d−1 ≤ 1,

where d is the distance from −1 to the numerical range in B of x. �

The following was stated in [21] without proof details.

Proposition 6.5. If A is a unital Banach algebra and x ∈ rA and ε > 0 then

x+ ε1 ∈ CFA where C = ε+ ‖x‖2

ε .

Proof. We have

‖1− C−1(x+ ε1)‖ = C−1 ‖(C − ε)1− x‖ = C−1 ‖x‖2ε

‖1− ε

‖x‖2x‖.

By Lemma 2.4 (2), this is dominated by C−1 ‖x‖2

ε (1 + ε2

‖x‖2 ) = 1. �

It follows easily from Proposition 6.5 that R+FA = rA if A is unital. For

nonunital algebras we use a different argument:

Proposition 6.6. If A is a subalgebra of a unital Banach algebra B then

R+FBA = rBA and R+

FA = rA.

Proof. If x ∈ rBA and t ≥ 0, then tx(1B + tx)−1 ∈ FBA by Lemma 6.4.

By elementary Banach algebra theory, (1B + tx)−1 → 1B as t ↘ 0. So x =limt→0+

1t tx(1B + tx)−1, from which the results are clear. �

Remark. There is a numerical range lifting result that works in quotientsof Banach spaces with ‘identity’ or of approximately unital Banach algebras, ifone takes the quotient by an M -ideal (see [30] and the end of Section 8 in [21]).This may be viewed as a noncommutative Tietze theorem, as explained in the lastparagraph of Section 8 in [21]. As a consequence one can lift a real positive elementin such a quotient A/J to a real positive in A. This again is a generalization ofa well known C∗-algebraic positivity results since as pointed out by Alfsen andEffros (and Effros and Ruan), M -ideals in a C∗-algebras (or, for that matter, in anapproximately unital operator algebra) are just the two-sided closed ideals (with acai). See e.g. [15, Theorem 4.8.5].

Lemma 6.7. Let A be a Banach algebra. If x ∈ rA, then ||xt|| ≤ 2 sin(tπ)πt(1−t) ‖x‖t

if 0 < t < 1. If A is an operator algebra one may remove the 2 in this estimate.

To prove this and the next corollary: by the above we may as well work in anyunital Banach algebra containing A, and this case was done in [21]. In the operatoralgebra case a recent paper of Drury [35] is a little more careful with the estimatesfor the integral in the Balakrishnan formula mentioned above for xt, and obtains

‖xt‖ ≤Γ( t2 ) Γ(

1−t2 )

2√πΓ(t)Γ(1− t)

if 0 < t < 1 and ‖x‖ ≤ 1. Drury states this for (strictly) positive semidefinitematrices x, and seems to have a typo in the proof, but the proof is easily seen towork for accretive operators on Hilbert space.

56 DAVID P. BLECHER

Lemma 6.8. There is a nonnegative sequence (cn) in c0 such that for any

Banach algebra A, and x ∈ FA or x ∈ Ball(A) ∩ rA, we have ‖x 1nx − x‖ ≤ cn for

all n ∈ N.

Remark 6.9. If A is a Banach algebra and x ∈ FA or or x ∈ Ball(A) ∩ rA is

nonzero then lim supn ‖x 1n ‖ ≤ 1 is the same as saying limn ‖x 1

n ‖ = 1. For

‖x‖ ≤ ‖x 1nx− x‖+ ‖x 1

nx‖ ≤ cn + ‖x 1n ‖‖x‖, n ∈ N,

where (cn) ∈ c0 as in Lemma 6.8. This property holds if A is an operator algebraby the last assertion of Lemma 6.7.

Corollary 6.10. A Banach algebra A with a left bai (resp. right bai, bai) inrA has a left bai (resp. right bai, bai) in FA. And a similar statement holds with rA

and FA replaced by rBA and FBA for any unital Banach algebra B containing A as a

subalgebra.

Proof. If (et) is a left bai (resp. right bai, bai) in rA, let bt = F(et) ∈ FA. By

the proof in [21, Corollary 3.9], (b1nt ) is a left bai (resp. right bai, bai) in FA. �

Remark 6.11. If the bai in the last result is sequential, then so is the oneconstructed in FA.

We imagine that if a Banach algebra has a cai in rA then under mild conditionsit has a cai in FA. We give a couple of results along these lines, that are not in [21].

Corollary 6.12. Suppose that A is a Banach algebra with the property thatthere is a sequence (dn) of scalars with limit 1 such that ‖x 1

n ‖ ≤ dn for all n ∈ Nand x ∈ FA (this is the case for operator algebras by Lemma 6.7). If A has a leftbai (resp. right bai, bai) in rA then A has a left cai (resp. right cai, cai) in FA. Anda similar statement holds with rA and FA replaced by rBA and FB

A for any unitalBanach algebra B containing A as a subalgebra.

Proof. For the first case, let (fs)s∈Λ = (b1nt ) be the left bai in FA from Corol-

lary 6.10. Note that ‖fs‖ ≤ dn and so it is easy to see that ‖fs‖ → 1 by the Remarkafter Lemma 6.8. If there is a contractive subnet of (fs) we are done, so assumethat there is no contractive subnet. So for every s ∈ Λ there is an s′ ≥ s with‖fs′‖ > 1. Let Λ0 = {s ∈ Λ : ‖fs‖ > 1}. A straightforward argument shows thatΛ0 is directed, and that (fs)s∈Λ0

is a subset of (fs)t∈Λ which is a left bai in FA.Then ( 1

‖fs‖ fs)s∈Λ0is in FA since ‖fs‖ > 1. So ( 1

‖fs‖ fs)s∈Λ0is a left cai in FA. The

other cases are similar. �

The hypothesis in the next result that A∗∗ is unital is, by [7, Theorem 1.6],equivalent to there being a unique mixed identity (we thank Matthias Neufang forthis reference).

Proposition 6.13. Let A be a Banach algebra such that A∗∗ is unital and Ahas a real positive cai, or more generally suppose that there exists a real positive caifor A and a bai for A in FA with the same weak* limit. Then A has a cai in FA.This latter cai may be chosen to be sequential if in addition A has a sequential bai.

Proof. That the second hypothesis is more general follows by Corollary 6.10since a subnet of the ensuing bai for A in FA has a weak* limit. Note that if(fs)s∈Λ is a bai in FA with ‖fs‖ → 1 then either there is a subnet of (fs) consisting


of contractions, in which case this subnet is a cai in FA, or Λ0 = {s ∈ Λ : ‖fs‖ ≥ 1}is a directed set and ( 1

‖fs‖ fs)s∈Λ0is a cai in FA.

Next, suppose that (et) is a cai in rA, and (fs) is a bai in FA and they havethe same weak* limit f . By a re-indexing argument, we can assume that they areindexed by the same directed set. Then et−ft → 0 weakly in A. If E = {x1, · · · , xn}is a finite subset of A define Fs,E to be the subset

{(et − ft, etx1 − x1, x1et − x1, ftx1 − x1, x1ft − x1,

etx2 − x2, · · · , x1ft − x1) : t ≥ s}

of A(4m+1). Since (A(4m+1))∗ is the 1 direct sum of 4m+1 copies of A∗, it is easy tosee that 0 is in the weak closure of Fs,E (since et − ft → 0 weakly and etxk → xk,etc). Thus by Mazur’s theorem 0 is in the norm closure of the convex hull of Fs,E .For each n ∈ N there are a finite subset t1, · · · , tK (whereK may depend on n, s, E),

and positive scalars (αn,s,Ek )Kk=1 with sum 1, such that if rn,s,E =

∑Kk=1 αn,s,E

k etkand wn,s,E =

∑Kk=1 αn,s,E

k ftk , then ‖rn,s,Exk − xk‖, ‖xkrn,s,E − xk‖, ‖xkwn,s,E −xk‖, ‖wn,s,Exk − xk‖, and ‖rn,s,E − wn,s,E‖, are each less than 2−n for all k =1, · · · ,m. Note that (rn,s,E) is then a cai in rA, and (wn,s,E) is a bai in FA. Sincern,s,E − wn,s,E → 0 with n, it follows that ‖wn,s,E‖ → 1 with (n,E). So as in thelast paragraph one may obtain from (wn,s,E) a cai in FA.

If we have a sequential cai in rA then it follows from e.g. Sinclair’s Aarnes-Kadison type theorem (see the lines after Theorem 2.3; alternatively one may useour Aarnes-Kadison type theorem 7.13 below) that A = xAx for some x ∈ A. Givena cai (ft) in FA, choose t1 < t2 < · · · with ‖ftkx− x‖+ ‖xftk − x‖ < 2−k. Then itis clear that (ftk) is a sequential cai in FA. �

Remark 6.14. It follows that under the conditions of the last result, one mayimprove [21, Corollary 6.10] in the way described after that result (using the factin the remark after [21, Corollary 2.10]).

Corollary 6.15. If A is a Banach algebra then rA is closed under rth powersfor any r ∈ [0, 1]. So is rBA for any unital Banach algebra B isometrically containingA as a subalgebra.

Proof. We saw in the proof of Proposition 6.6 that if x ∈ rBA then x =limt→0+

1t tx(1 + tx)−1, and tx(1 + tx)−1 ∈ FB

A . Thus by [61, Corollary 1.3] we

have that xr = limt→0+1tr (tx(1 + tx)−1)r for 0 < r < 1. By Proposition 6.3 and

its proof, the latter powers are in R+FBA , so that xr ∈ R+

FBA = rBA ⊂ rA. �

In an operator algebra, much is known about the numerical range of fractionalpowers (see e.g. Section 2 in [26]). In particular, if x is sectorial of angle θ ≤ π

2

then xt has sectorial angle ≤ tθ. Indeed this is what allows us to produce ‘nearlypositive elements’, as discussed in Section 4. The following fact, which we havenot seen in the literature, is the best one has in a general Banach algebra, and thisdisappointment means that some of the theory from [22,23,25] will not generalizeto Banach algebras. In particular, one can have accretive elements whose nth rootsall have the same numerical range, and also are not sectorial of any angle < π/2(see e.g. Example 3.14 in [21]).

Proposition 6.16. If x is sectorial of angle θ ≤ π2 in a unital Banach algebra

then xt has sectorial angle ≤ tθ + (1− t)π2 .

58 DAVID P. BLECHER

Proof. This is Corollary 6.15 if θ = π2 . Suppose that WB(x) ⊂ Sθ. Then

e±i(π2 −θ)x is accretive. Hence (wx)t is accretive where w = e±i(π

2 −θ). By [43,Lemma 3.1.4] with f(z) = wz we have (wx)t = wt xt. So wtxt is accretive. Revers-ing the argument above we see that

WB(x) ⊂ ei(π2 −θ)Sπ

2∩ e−i(π

2 −θ)Sπ2= Stθ+(1−t)π

2

as desired. �Proposition 6.17. If A is a Banach algebra and x ∈ rA then ba(x) = ba(F(x)),

and so xA = F(x)A.

Proof. We said earlier that F(x) is in ba(x) and is independent of the partic-ular unital Banach algebra containing A. Thus this result follows from the unitalcase considered in [21, Proposition 3.11]. �

Lemma 6.18. If p is an idempotent in a Banach algebra A then p ∈ FA iffp ∈ rA.

Proof. This is clear from the unital case considered in [21, Lemma 3.12]. �Proposition 6.19. If A is a Banach algebra and x ∈ rA, then ba(x) has a bai

in FA. Hence any weak* limit point of this bai is a mixed identity residing in FA∗∗ .Indeed (x

1n ) is a bai for ba(x) in rA, and (F(x)

1n ) is a bai for ba(x) in FA.

Proof. If x ∈ rBA then the proof of [21, Proposition 3.17] shows that ba(x)has a bai in FB

A , and hence any weak* limit point of this bai is a mixed identity

residing in FB∗∗

A∗∗ ⊂ FA∗∗ . Indeed (x1n ) is a bai for ba(x) in rBA , �

The following new observation is a simple consequence of the above which wewill need later.

Corollary 6.20. If A is a nonunital Banach algebra and if E and F aresubsets of rA then EA = EB, AF = BF , and EAF = EBF , where B is anyunitization of A.

Proof. The first follows from the following fact: if x ∈ rA then

x ∈ xA = ba(x)A = xB,

since by Cohen factorization x ∈ ba(x) = ba(x)2 ⊂ xA. The other two are similar.�

We now turn to the support projection of an element, encountered in theAarnes–Kadison theorem 2.1. In an operator algebra or Arens regular Banachalgebra things are cleaner (see [9, 22, 23]). For a Banach algebra A and x ∈ rA,

we write s(x) for the weak* Banach limit of (x1n ) in A∗∗. That is s(x)(f) =

LIMn f(x1n ) for f ∈ A∗, where LIM is a Banach limit. It is easy to see that

xs(x) = s(x)x = x, by applying these to f ∈ A∗. Hence s(x) is a mixed identityof ba(x)∗∗, and is idempotent. By the Hahn–Banach theorem it is easy to see that

s(x) ∈ conv({x 1n : n ∈ N})

w∗. In x ∈ rBA then by an argument after [21, Proposi-

tion 3.17] we have s(x) ∈ FB∗∗ ∩A∗∗ = FB∗∗

A∗∗ ⊂ FA∗∗ . If ba(x) is Arens regular thens(x) will be the identity of ba(x)∗∗.

We call s(x) above a support idempotent of x, or a (left) support idempotentof xA (or a (right) support idempotent of Ax). The reason for this name is thefollowing result.


Corollary 6.21. If A is a Banach algebra, and x ∈ rA then xA has a left baiin FA and x ∈ xA = s(x)A∗∗∩A and (xA)⊥⊥ = s(x)A∗∗. (These products are withrespect to the second Arens product.)

Proof. The proof of [21, Corollary 3.18] works, and gives that xA has a leftbai in FB

A if x ∈ rBA . �

As in [22, Lemma 2.10] and [21, Corollary 3.19] we have:

Corollary 6.22. If A is a Banach algebra, and x, y ∈ rA, then xA ⊂ yA iffs(y)s(x) = s(x). In this case xA = A iff s(x) is a left identity for A∗∗. (Theseproducts are with respect to the second Arens product.)

As in [22, Corollary 2.7] we have:

Corollary 6.23. Suppose that A is a subalgebra of a Banach algebra B. Ifx ∈ A ∩ rB, then the support projection of x computed in A∗∗ is the same, via thecanonical embedding A∗∗ ∼= A⊥⊥ ⊂ B∗∗, as the support projection of x computedin B∗∗.

In Section 2 we mentioned the paper of Kadison and Pedersen [55] initiatingthe development of a comparison theory for elements in C∗-algebras generalizingthe von Neumann equivalence of projections. Again positivity and properties ofthe positive cone are key to that work. Admittedly their algebras were monotonecomplete, but many later authors have taken up this theme, with various versions ofequivalence or subequivalence of elements in general C∗-algebras (see for example[10] or [3,66] and references therein). Indeed recently the study of Cuntz equiva-lence and subequivalence within the context of the Elliott program has become oneof the most important areas of C∗-algebra theory. In [18] Neal and the author be-gan a program of generalizing basic parts of the theory of comparison, equivalence,and subequivalence, to the setting of general operator algebras. In that paper wefocused on comparison of elements in R+

FA, but we proved some lemmas in [25]that show that everything should work for elements in rA. In particular, we followthe lead of Lin, Ortega, Rørdam, and Thiel [66] in studying these equivalences, etc.,in terms of the roots and support projections s(x) discussed in this section above,or in terms of module isomorphisms of (topologically) principal modules of the formxA studied below. There is a lot more work needed to be done here, our paper wassimply the first steps. Also, we have not tried to see if any of this generalizes tolarger classes of Banach algebras. Much of our theory in [18] depends on facts fornth roots of real positive elements. Thus we would expect that a certain portion ofthis theory generalizes to Banach algebras using the facts about roots summarizedin Section 6.

7. Structure of ideals and HSA’s

We recall that an element x in an algebra A is pseudo-invertible in A if thereexists y ∈ A with xyx = x. The following result (which is the non-approximatelyunital case of [21, Theorem 3.21]) should be compared with the C∗-algebraic versionof the result due to Harte and Mbekhta [45, 46], and to the earlier version ofthe result in the operator algebra case (see particularly [22, Section 3], and [25,Subsection 2.4] and [24]).

60 DAVID P. BLECHER

Theorem 7.1. Let A be a Banach algebra, and x ∈ rA. The following areequivalent:

(i) s(x) ∈ A,(ii) xA is closed,(iii) Ax is closed,(iv) x is pseudo-invertible in A,(v) x is invertible in ba(x).

Moreover, these conditions imply

(vi) 0 is isolated in, or absent from, SpA(x).

Finally, if ba(x) is semisimple then (i)–(vi) are equivalent.

Proof. The first five equivalences are just as in [21, Theorem 3.21]; as is theassertions regarding (vi), since there we may assume A is unital by definition ofspectrum and because of the form of (v). �

The next results (which are the non-approximately unital cases of results in[21, Section 3]) follow from Theorem 7.1 just as the approximately unital cases didin [21], which in turn often rely on earlier arguments from e.g. [22]:

Corollary 7.2. If A is a closed subalgebra of a unital Banach algebra B, andif x ∈ rBA, then x is invertible in B iff 1B ∈ A and x is invertible in A, and iff ba(x)contains 1B; and in this case s(x) = 1B.

Corollary 7.3. Let A be a Banach algebra. A closed right ideal J of A is ofthe form xA for some x ∈ rA iff J = qA for an idempotent q ∈ FA.

Corollary 7.4. If a nonunital Banach algebra A contains a nonzero x ∈ rA

with xA closed, then A contains a nontrivial idempotent in FA. If a Banach algebraA has no left identity, then xA �= A for all x ∈ rA.

In [13] we generalized the concept of hereditary subalgebra (HSA), an importanttool in C∗-algebra theory, to operator algebras, and established that the basics ofthe C∗-theory of HSA’s is still true. Now of course HSA’s need not be selfadjoint,but are still norm closed approximately unital inner ideals in A, where by thelatter term we mean a subalgebra D with DAD ⊂ D. Generalizing Theorem 2.2above, we showed in [22,23] that HSA’s and right ideals with left cais in operatoralgebras aremanifestations of our cone rA, or if preferred, FA or the ‘nearly positive’elements. We now discuss some aspects of this in the case of Banach algebras from[21], and mention some of what is still true in that setting. In particular we studythe relationship between HSA’s and one-sided ideals with one-sided approximateidentities. Some aspects of this relationship is problematic for general Banachalgebras (see [21, Section 4]), but it works much better in separable algebras. Aswe said around Theorem 2.3, our work is closely related to the results of Sinclairand others on the Cohen factorization method (see e.g. [37,74]), which does includesome similar sounding but different results.

We define a right F-ideal (resp. left F-ideal) in a Banach algebra A to be aclosed right (resp. left) ideal with a left (resp. right) bai in FA (or equivalently, byCorollary 6.10, in rA). Henceforth in this section, by a hereditary subalgebra (HSA)of A we will mean an inner ideal D with a two-sided bai in FA (or equivalently, byCorollary 6.10, in rA). Perhaps these should be called F-HSA’s to avoid confusionwith the notation of [13,22] where one uses cai’s instead of bai’s, but for brevity we


shall use the shorter term. And indeed for operator algebras (the setting of [13,22])the two notions coincide, and also right and left F-ideals are just the r-ideals and�-ideals of those papers (see Corollary 7.10). Note that a HSA D induces a pair ofright and left F-ideals J = DA and K = AD. Using the proof in [21, Lemma 4.2]we have:

Lemma 7.5. If A is a Banach algebra, and z ∈ FA, set J = zA, D = zAz,and K = Az. Then D is a HSA in A and J and K are the induced right and leftF-ideals mentioned above.

At this point we have to jettison FA and rA as defined at the start of Section6, if A is not approximately unital, because the remaining results are endangeredif FA and rA are not closed and convex. Indeed most results in Sections 4 and 7 of[21] would seem to need FA and rA to be replaced by FB

A and rBA for a fixed unitalBanach algebra B containing A as a subalgebra. That is, we need to fix a particularunitization of A, not consider all unitizations simultaneously. Of course if A is anoperator algebra then there is a unique unitization, hence all this is redundant. (Aswe said early in Section 6, we could also fix this problem by redefining FA = ∩B FB

A

and rA = ∩B rBA , the intersections taken over all B as above. Everything belowwould then look cleaner but may be less useful.) Thus we define a right FB-ideal(resp. left FB-ideal) in a A to be a closed right (resp. left) ideal with a left (resp.right) bai in FB

D (or equivalently, by Corollary 6.10, in rBD). Note that one-sided FB-ideals in A are exactly subalgebras of A which are one-sided FB-ideals in A+C 1Bin the sense of [21, Section 4].

We define a B-hereditary subalgebra (or B-HSA for short) of A to an inner idealD in A with a two-sided bai in FB

D (or equivalently, by Corollary 6.10, in rBD). Notethat B-HSA’s in A are exactly subalgebras of A which are HSA’s in A + C 1B inthe sense of [21, Section 4].

Again a B-HSA D induces a pair of right and left FB-ideals J = DA andK = AD. Lemma 7.5 becomes: If z ∈ FB

A , set J = zA, D = zAz, and K = Az.Then D is a B-HSA in A and J and K are the induced right and left FB-idealsmentioned above.

Because of the facts at the end of the second last paragraph, and because ofCorollary 6.20, in the following four results we can assume that A is unital, in whichcase the proofs are in [21]. These results are all stated for a Banach algebra withunitization B, but they could equally well be stated for a closed subalgebra of aunital Banach algebra B.

Lemma 7.6. Suppose that J is a right FB-ideal in a Banach algebra with unitiza-tion B. For every compact subset K ⊂ J , there exists z ∈ J∩FB

A with K ⊂ zJ ⊂ zA.

Applying this lemma gives the first assertion in the following result, takingK = { 1

nen}∪{0}, where (en) is the left bai. Taking K = { dn

n‖dn‖}∪{0} where {dn}is a countable dense set gives the second.

Corollary 7.7. Let A be a Banach algebra with unitization B. The closedright ideals of A with a countable left bai in rBA are precisely the (topologically)

‘principal right ideals’ zA for some z ∈ FBA which is also in the ideal. Every

separable right FB-ideal is of this form.

62 DAVID P. BLECHER

Corollary 7.8. Let A be a Banach algebra with unitization B. The rightFB-ideals in A are precisely the closures of increasing unions of right ideals in Aof the form zA for some z ∈ FB

A.

We say that a right module Z over A is algebraically countably generated (resp.algebraically finitely generated) over A if there exists a countable (resp. finite set){xk} in Z such that every z ∈ Z may be written as a finite sum

∑nk=1 xkak for some

ak ∈ A. Of course any algebraically finitely generated is algebraically countablygenerated.

Corollary 7.9. Let A be a Banach algebra with unitization B. A right FB-ideal J in A is algebraically countably generated as a right module over A iff J = qAfor an idempotent q ∈ FB

A. This is also equivalent to J being algebraically countablygenerated as a right module over A+ C 1B.

The following was not stated in [21].

Corollary 7.10. If A is an operator algebra, a closed subalgebra of a unitaloperator algebra B, then right and left FB-ideals in A are just the r-ideals and�-ideals in A of [13, 22], and B-HSA’s in A are just the HSA’s in A of thosereferences.

Proof. By Corollary 7.8 a right FB-ideal is the closure of an increasing unionof right ideals in A of the form zA for z ∈ FA. However this is the characterizationof r-ideals from [22]. Similarly for the left ideal case. A similar argument worksfor the HSA case using Corollary 7.14; alternatively, if D is a B-HSA then D hasa bai from FA. By Corollary 6.12, D has a cai. �

If A is a Banach algebra with unitization B it would be nice to say that theright FB-ideals in A are precisely the sets of form EA for a subset E ⊂ FB

A (orequivalently, E ⊂ rBA). One direction of this is obvious: just take E to be the baiin FB

A (resp. rBA). However the other direction is false in general Banach algebras,although it does hold in operator algebras [22] and commutative Banach algebras.(Another characterization of closed ideals with bai’s in commutative Banach alge-bras may be found in [60].)

That EA is a right FB-ideal in A if A is a commutative Banach algebra andE ⊂ FB

A , follows from Theorem 7.1 in [21] after noting that by Corollary 6.20 wemay replace A by A + C 1B. The key part of the proof of Theorem 7.1 in [21] isto show that for any finite subset G of E there exists an element zG ∈ FB

A ∩ EA

with GA = zGA. Indeed one can take zG to be the average of the elements in G.

Then the net (z1n

G ), indexed by the finite subsets G of E and n ∈ N, is easily seen

to be a left bai in EA from FBA . An application of this: for such subsets E of an

operator algebra or commutative Banach algebra A, the Banach algebra generatedby E has a bai in FB

A . This follows from the argument above since the zG above

are in the convex hull of E, hence the bai (z1n

G ) is in the Banach algebra generatedby E. In particular, if A is generated as a Banach algebra by rBA , then A has a bai,and this bai may be taken from rBA . (The present paragraph is a summary of theresults in [21, Section 7], and a generalization of these results to the case that A isnot approximately unital.)

Unless explicitly said to the contrary, all the remaining results in this sectionare again generalizations to general Banach algebras of results from [21]. Some


of these were proved earlier in the operator algebra case by the author and Read.Again for their proofs we can assume that A is unital, and appeal to the matchingresults in [21].

Lemma 7.11. Let A be a Banach algebra with unitization B. Let D be a closedsubalgebra of A. If D has a bai from FB

A, then for every compact subset K ⊂ D,there is an x ∈ FB

D such that K ⊂ xDx ⊂ xAx.

As in the proof sketched for Corollary 7.7 this leads to:

Theorem 7.12. Let A be a Banach algebra with unitization B, and let D bean inner ideal in A. Then D has a countable bai from FB

D (or equivalently, from

rBD) iff there exists an element z ∈ FBD with D = zAz. Thus D is of the form in

Lemma 7.5, and such D has a countable commuting bai from FBD, namely (z

1n ).

Any separable inner ideal in A with a bai from rBD is of this form.

From this most of the following generalization of the Aarne-Kadison theorem(see Theorem 2.1) is immediate. By a strictly real positive element in (v) below,we mean an element x ∈ A such that Re ϕ(x) > 0 for all states ϕ of A which donot vanish on A. In [22,25] we generalized some basic aspects of strictly positiveelements in C∗-algebras to operator algebras. The following is mostly in [21,25],and relies on ideas from [22].

Corollary 7.13 (Aarnes–Kadison type theorem). If A is a Banach algebrathen the following are equivalent:

(i) There exists an x ∈ rA with A = xAx.(ii) There exists an x ∈ rA with A = xA = Ax.(iii) There exists an x ∈ rA with s(x) a mixed identity for A∗∗.

If B is a unitization of A then items (i), (ii), or (iii) above hold with x ∈ rBA iff

(iv) A has a sequential bai from rBA.

The approximate identity in (iv) may be taken to be commuting, indeed it may be

taken to be (x1n ) for the last mentioned element x. If A is separable and has a bai

in rBA then A satisfies (iv) and hence all of the above. Moreover if A is an operatoralgebra then (i)–(iv) are each equivalent to:

(v) A has a strictly real positive element,

and any of these imply that the operator algebra A has a sequential real positive cai.

Again, r can be replaced by F throughout this result, or in any of the items (i)to (v).

The proof of Corollary 7.13 is mostly in [21,25], and relies partly on ideas from

[22]. In the operator algebra case, if (ii) holds with x ∈ FA then (( 12x)1n ) is a cai

for A in 12FA by [22, Section 3], and s(x) = 1A∗∗ . So x is a strictly real positive

element by [22, Lemma 2.10]. Conversely, if an operator algebra A has a strictlyreal positive element then it is explained in the long discussion before [25, Lemma3.2] how to adapt the proof of [22, Lemma 2.10] to show that (iv) holds, hence (ii),and hence A has a sequential real positive cai by e.g. [23, Corollary 3.5], or by ourearlier Corollary 7.10.

Corollary 7.14. The B-HSA’s in a Banach algebra A with unitization B areexactly the closures of increasing unions of HSA’s of the form zAz for z ∈ FB

A.

64 DAVID P. BLECHER

Acknowledgements. We thank the referee for his comments. Tragically, duringthe revision stage of this article we learned that our dear friend, brother, andcoauthor Charles Read had passed away while jogging in Canada, just over 48 hoursafter we visited at a week-long conference in Toronto. We are proud to know him,and are grateful that we had that last time on this earth to have happy fellowship,during which he seemed to be in good health and spirits (and also delivered a trulyamazing lecture). See you on the other side of the river, Charles!

References

[1] J. F. Aarnes and R. V. Kadison, Pure states and approximate identities, Proc. Amer. Math.Soc. 21 (1969), 749–752. MR0240633 (39 #1980)

[2] E. M. Alfsen, Compact convex sets and boundary integrals, Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57. MR0445271(56 #3615)

[3] P. Ara, F. Perera, and A. S. Toms, K-theory for operator algebras. Classification of C∗-algebras, Aspects of operator algebras and applications, Contemp. Math., vol. 534, Amer.Math. Soc., Providence, RI, 2011, pp. 1–71, DOI 10.1090/conm/534/10521. MR2767222

(2012g:46092)[4] W. B. Arveson, Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578–642.

MR0223899 (36 #6946)[5] W. B. Arveson, Subalgebras of C∗-algebras, Acta Math. 123 (1969), 141–224. MR0253059

(40 #6274)[6] L. Asimow and A. J. Ellis, Convexity theory and its applications in functional analysis,

London Mathematical Society Monographs, vol. 16, Academic Press, Inc. [Harcourt BraceJovanovich, Publishers], London-New York, 1980. MR623459 (82m:46009)

[7] J. Baker, A. T.-M. Lau, and J. Pym, Module homomorphisms and topological centres asso-ciated with weakly sequentially complete Banach algebras, J. Funct. Anal. 158 (1998), no. 1,186–208, DOI 10.1006/jfan.1998.3280. MR1641570 (99h:46081)

[8] C. J. K. Batty and D. W. Robinson, Positive one-parameter semigroups on ordered Banachspaces, Acta Appl. Math. 2 (1984), no. 3-4, 221–296, DOI 10.1007/BF02280855. MR753696(86b:47068)

[9] C. A. Bearden, D. P. Blecher, and S. Sharma, On positivity and roots in operator algebras,Integral Equations Operator Theory 79 (2014), no. 4, 555–566, DOI 10.1007/s00020-014-2136-y. MR3231244

[10] B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of C∗-algebras and von Neumann algebras; Operator Algebrasand Non-commutative Geometry, III. MR2188261 (2006k:46082)

[11] D. P. Blecher, Multipliers, C∗-modules, and algebraic structure in spaces of Hilbert spaceoperators, Operator algebras, quantization, and noncommutative geometry, Contemp. Math.,vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 85–128, DOI 10.1090/conm/365/06701.

MR2106818 (2005i:46066)[12] D. P. Blecher,Noncommutative peak interpolation revisited, Bull. Lond. Math. Soc. 45 (2013),

no. 5, 1100–1106, DOI 10.1112/blms/bdt040. MR3105002[13] D. P. Blecher, D. M. Hay, and M. Neal, Hereditary subalgebras of operator algebras, J. Op-

erator Theory 59 (2008), no. 2, 333–357. MR2411049 (2009m:46074)[14] D. P. Blecher and L. E. Labuschagne, Von Neumann algebraic Hp theory, Function spaces,

Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 89–114, DOI10.1090/conm/435/08369. MR2359421 (2008h:46027)

[15] D. P. Blecher and C. Le Merdy, Operator algebras and their modules—an operator spaceapproach, London Mathematical Society Monographs. New Series, vol. 30, The ClarendonPress, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR2111973(2006a:46070)

[16] D. P. Blecher and M. Neal, Metric characterizations of isometries and of unital operatorspaces and systems, Proc. Amer. Math. Soc. 139 (2011), no. 3, 985–998, DOI 10.1090/S0002-9939-2010-10670-6. MR2745650 (2012d:47212)


[17] D. P. Blecher and M. Neal, Open projections in operator algebras I: Comparison theory,Studia Math. 208 (2012), no. 2, 117–150, DOI 10.4064/sm208-2-2. MR2910983

[18] D. P. Blecher and M. Neal, Open projections in operator algebras II: Compact projections,Studia Math. 209 (2012), no. 3, 203–224, DOI 10.4064/sm209-3-1. MR2944468

[19] D. P. Blecher and M. Neal, Metric characterizations II, Illinois J. Math. 57 (2013), no. 1,25–41. MR3224559

[20] D. P. Blecher and M. Neal, Completely contractive projections on operator algebras, Preprint

2015.[21] D. P. Blecher and N. Ozawa, Real positivity and approximate identities in Banach algebras,

Pacific J. Math. 277 (2015), no. 1, 1–59, DOI 10.2140/pjm.2015.277.1. MR3393680[22] D. P. Blecher and C. J. Read, Operator algebras with contractive approximate identities,

J. Funct. Anal. 261 (2011), no. 1, 188–217, DOI 10.1016/j.jfa.2011.02.019. MR2785898(2012e:47226)

[23] D. P. Blecher and C. J. Read, Operator algebras with contractive approximate identities, II,J. Funct. Anal. 264 (2013), no. 4, 1049–1067, DOI 10.1016/j.jfa.2012.11.013. MR3004957

[24] D. P. Blecher and C. J. Read, Operator algebras with contractive approximate identities III,Preprint 2013 (ArXiv version 2 arXiv:1308.2723v2).

[25] D. P. Blecher and C. J. Read, Order theory and interpolation in operator algebras, StudiaMath. 225 (2014), no. 1, 61–95, DOI 10.4064/sm225-1-4. MR3299396

[26] D. P. Blecher and Z. Wang, Roots in operator algebras and Banach algebras, J. IntegralEquations Operator Theory 85 (2016), 63–90. DOI 10.1007/s00020-015-2272-z

[27] F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elementsof normed algebras, London Mathematical Society Lecture Note Series, vol. 2, CambridgeUniversity Press, London-New York, 1971. MR0288583 (44 #5779)

[28] L. G. Brown, P. Green, and M. A. Rieffel, Stable isomorphism and strong Morita equivalenceof C∗-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR0463928 (57 #3866)

[29] M. D. Choi and E. G. Effros, Injectivity and operator spaces, J. Functional Analysis 24 (1977),no. 2, 156–209. MR0430809 (55 #3814)

[30] C. K. Chui, P. W. Smith, R. R. Smith, and J. D. Ward, L-ideals and numerical rangepreservation, Illinois J. Math. 21 (1977), no. 2, 365–373. MR0430817 (55 #3822)

[31] M. Crouzeix, A functional calculus based on the numerical range: applications, Linear Mul-tilinear Algebra 56 (2008), no. 1-2, 81–103, DOI 10.1080/03081080701336610. MR2378304(2009g:47046)

[32] H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Mono-graphs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000.Oxford Science Publications. MR1816726 (2002e:46001)

[33] P. G. Dixon, Approximate identities in normed algebras. II, J. London Math. Soc. (2) 17(1978), no. 1, 141–151, DOI 10.1112/jlms/s2-17.1.141. MR485443 (80b:46055)

[34] R. S. Doran and J. Wichmann, Approximate identities and factorization in Banach modules,Lecture Notes in Mathematics, vol. 768, Springer-Verlag, Berlin-New York, 1979. MR555240(83e:46044)

[35] S. Drury, Principal powers of matrices with positive definite real part, Linear MultilinearAlgebra 63 (2015), no. 2, 296–301, DOI 10.1080/03081087.2013.865732. MR3273755

[36] E. G. Effros and E. Størmer, Positive projections and Jordan structure in operator algebras,Math. Scand. 45 (1979), no. 1, 127–138. MR567438 (82e:46076)

[37] J. Esterle, Injection de semi-groupes divisibles dans des algebres de convolution et construc-tion d’homomorphismes discontinus de C(K) (French), Proc. London Math. Soc. (3) 36(1978), no. 1, 59–85. MR0482218 (58 #2300)

[38] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Chapman &Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129, Chapman& Hall/CRC, Boca Raton, FL, 2003. MR1957004 (2004j:46030)

[39] Y. Friedman and B. Russo, Contractive projections on C0(K), Trans. Amer. Math. Soc. 273

(1982), no. 1, 57–73, DOI 10.2307/1999192. MR664029 (83i:46062)[40] Y. Friedman and B. Russo, Conditional expectation without order, Pacific J. Math. 115

(1984), no. 2, 351–360. MR765191 (86b:46116)[41] Y. Friedman and B. Russo, Solution of the contractive projection problem, J. Funct. Anal.

60 (1985), no. 1, 56–79, DOI 10.1016/0022-1236(85)90058-8. MR780104 (87a:46115)[42] T. W. Gamelin, Uniform Algebras, Second edition, Chelsea, New York, 1984.

66 DAVID P. BLECHER

[43] M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances andApplications, vol. 169, Birkhauser Verlag, Basel, 2006. MR2244037 (2007j:47030)

[44] P. Harmand, D. Werner, and W. Werner, M-ideals in Banach spaces and Banach alge-bras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR1238713(94k:46022)

[45] R. Harte and M. Mbekhta, On generalized inverses in C∗-algebras, Studia Math. 103 (1992),no. 1, 71–77. MR1184103 (93i:46097)

[46] R. Harte and M. Mbekhta, Generalized inverses in C∗-algebras. II, Studia Math. 106 (1993),no. 2, 129–138. MR1240309 (94k:46113)

[47] D. M. Hay, Closed projections and peak interpolation for operator algebras, Integral EquationsOperator Theory 57 (2007), no. 4, 491–512, DOI 10.1007/s00020-006-1471-z. MR2313282(2008f:46064)

[48] G. Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag,Berlin-New York, 1970. MR0438077 (55 #10996)

[49] R. V. Kadison, Order properties of bounded self-adjoint operators, Proc. Amer. Math. Soc.2 (1951), 505–510. MR0042064 (13,47c)

[50] R. V. Kadison, Isometries of operator algebras, Ann. Of Math. (2) 54 (1951), 325–338.MR0043392 (13,256a)

[51] R. V. Kadison, A representation theory for commutative topological algebra, Mem. Amer.Math. Soc., 1951 (1951), no. 7, 39. MR0044040 (13,360b)

[52] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator alge-bras, Ann. of Math. (2) 56 (1952), 494–503. MR0051442 (14,481c)

[53] R. V. Kadison, Non-commutative conditional expectations and their applications, Operatoralgebras, quantization, and noncommutative geometry, Contemp. Math., vol. 365, Amer.Math. Soc., Providence, RI, 2004, pp. 143–179, DOI 10.1090/conm/365/06703. MR2106820(2005i:46072)

[54] B. Fuglede and R. V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55(1952), 520–530. MR0052696 (14,660a)

[55] R. V. Kadison and G. K. Pedersen, Equivalence in operator algebras, Math. Scand. 27 (1970),205–222 (1971). MR0308803 (46 #7917)

[56] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Vol. 1,Graduate Studies in Mathematics, Amer. Math. Soc. Providence, RI, 1997.

[57] R. V. Kadison and I. M. Singer, Triangular operator algebras. Fundamentals and hyperre-ducible theory., Amer. J. Math. 82 (1960), 227–259. MR0121675 (22 #12409)

[58] J. L. Kelley and R. L. Vaught, The positive cone in Banach algebras, Trans. Amer. Math.Soc. 74 (1953), 44–55. MR0054175 (14,883e)

[59] A. T.-M. Lau and R. J. Loy, Contractive projections on Banach algebras, J. Funct. Anal.254 (2008), no. 10, 2513–2533, DOI 10.1016/j.jfa.2008.02.008. MR2406685 (2009h:47083)

[60] A. T.-M. Lau and A. Ulger, Characterization of closed ideals with bounded approximateidentities in commutative Banach algebras, complemented subspaces of the group von Neu-mann algebras and applications, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4151–4171, DOI10.1090/S0002-9947-2014-06336-8. MR3206455

[61] C.-K. Li, L. Rodman, and I. M. Spitkovsky, On numerical ranges and roots, J. Math.

Anal. Appl. 282 (2003), no. 1, 329–340, DOI 10.1016/S0022-247X(03)00158-6. MR2000347(2004g:47009)

[62] V. I. Macaev and Ju. A. Palant, On the powers of a bounded dissipative operator (Russian),

Ukrain. Mat. Z. 14 (1962), 329–337. MR0146664 (26 #4184)[63] B. Magajna, Weak∗ continuous states on Banach algebras, J. Math. Anal. Appl. 350 (2009),

no. 1, 252–255, DOI 10.1016/j.jmaa.2008.09.072. MR2476906 (2010b:46110)[64] R. T. Moore, Hermitian functionals on B-algebras and duality characterizations of C∗-

algebras, Trans. Amer. Math. Soc. 162 (1971), 253–265. MR0283572 (44 #803)[65] J. von Neumann, John von Neumann: selected letters, History of Mathematics, vol. 27, Amer-

ican Mathematical Society, Providence, RI; London Mathematical Society, London, 2005.With a foreword by P. Lax and an introduction by Marina von Neumann Whitman; Editedand with a preface and introductory comments by Miklos Redei. MR2210045 (2006m:01020)

[66] E. Ortega, M. Rørdam, and H. Thiel, The Cuntz semigroup and comparison of open pro-jections, J. Funct. Anal. 260 (2011), no. 12, 3474–3493, DOI 10.1016/j.jfa.2011.02.017.MR2781968 (2012d:46148)


[67] T. W. Palmer, Banach algebras and the general theory of ∗-algebras. Vol. I, Encyclopediaof Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994.Algebras and Banach algebras. MR1270014 (95c:46002)

[68] V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Ad-vanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR1976867(2004c:46118)

[69] G. K. Pedersen, Factorization in C∗-algebras, Exposition. Math. 16 (1998), no. 2, 145–156.

MR1630695 (99f:46086)[70] G. K. Pedersen, C∗-algebras and their automorphism groups, London Mathematical Society

Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR548006 (81e:46037)

[71] R. T. Prosser, On the ideal structure of operator algebras, Mem. Amer. Math. Soc. No. 45(1963), ii+28. MR0151863 (27 #1846)

[72] C. J. Read, On the quest for positivity in operator algebras, J. Math. Anal. Appl. 381 (2011),no. 1, 202–214, DOI 10.1016/j.jmaa.2011.02.022. MR2796203 (2012g:47217)

[73] A. G. Robertson and M. A. Youngson, Positive projections with contractive complements onJordan algebras, J. London Math. Soc. (2) 25 (1982), no. 2, 365–374, DOI 10.1112/jlms/s2-25.2.365. MR653394 (83e:46045)

[74] A. M. Sinclair, Bounded approximate identities, factorization, and a convolution algebra,J. Funct. Anal. 29 (1978), no. 3, 308–318, DOI 10.1016/0022-1236(78)90033-2. MR512247(80c:46054)

[75] A. M. Sinclair and A. W. Tullo, Noetherian Banach algebras are finite dimensional, Math.Ann. 211 (1974), 151–153. MR0355607 (50 #8081)

[76] J. G. Stampfli and J. P. Williams, Growth conditions and the numerical range in a Banachalgebra, Tohoku Math. J. (2) 20 (1968), 417–424. MR0243352 (39 #4674)

[77] E. Størmer, Positive projections with contractive complements on C∗-algebras, J. Lon-don Math. Soc. (2) 26 (1982), no. 1, 132–142, DOI 10.1112/jlms/s2-26.1.132. MR667251(83h:46074)

[78] B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kerchy, Harmonic analysis of operators on Hilbertspace, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR2760647

(2012b:47001)[79] K. Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

Reprint of the sixth (1980) edition. MR1336382 (96a:46001)[80] M. A. Youngson, Completely contractive projections on C∗-algebras, Quart. J. Math. Oxford

Ser. (2) 34 (1983), no. 136, 507–511, DOI 10.1093/qmath/34.4.507. MR723287 (85f:46112)

Department of Mathematics, University of Houston, Houston, Texas 77204-3008



Higher weak derivatives and reflexive algebras of operators

Erik Christensen

Dedicated to R. V. Kadison on the occasion of his ninetieth birthday.

Abstract. Let D be a self-adjoint operator on a Hilbert space H and x abounded operator on H. We say that x is n times weakly D−differentiable,if for any pair of vectors ξ, η from H the function 〈eitDxe−itDξ, η〉 is ntimes differentiable. We give several characterizations of n times weak dif-ferentiability, among which, one is original. These results are used to showthat for a von Neumann algebra M on H the algebra of n times weaklyD−differentiable operators in M has a natural representation as a reflexivesubalgebra of B(H ⊗ C(n+1)).

1. Introduction

Let D be a self-adjoint, usually unbounded, operator on a Hilbert space H andx a bounded operator on H, then Quantum Mechanics, [7] Operator Algebra [5] andNoncommutative Geometry [3] offer plenty of reasons why we should be interestedin operators that are formed as commutators [D, x] = Dx−xD. In noncommutativegeometry we want to find a set-up such that classical smooth structures may bedescribed in a language based on operators on a Hilbert space. A derivative isdescribed in terms of a commutator [D, x] and a higher derivative via an iteratedcommutator [D, [D, . . . , [D, x] . . . ]], so a basic question is to determine the set ofoperators for which such an iterated commutator makes sense. It is not clearwhen a commutator such as [D, x] is densely defined and bounded on its domainof definition, and for two bounded operators x, y such that [D, x] and [D, y] arebounded and densely defined the sum of the commutators and/or the commutator[D, xy] may not be densely defined, so the expression [D, x] does not define aderivation on a subalgebra of B(H) in a canonical way. In the article [2] we realizedthat the concept we named weak D−differentiability provides a set-up, which maybe used to decide for which bounded operators x the commutator [D, x] shouldbe defined. We say that a bounded operator x on H is weakly D−differentiableif for each pair of vectors ξ, η in H the function 〈eitDxe−itDξ, η〉 is differentiable.For a weakly D−differentiable operator x the commutator [D, x] is then definedand bounded on all of the domain of D, so it is possible to define a derivation δwfrom the algebra of weakly D−differentiable operators into B(H). We were laterinformed that the concept of weak D−differentiability, the algebra property of the

2010 Mathematics Subject Classification. Primary 46L55, 58B34; Secondary; 37A55, 47D06,81S05.


69

70 ERIK CHRISTENSEN

weakly differentiable operators and the derivation δw are well known by researchersin mathematical physics [1] and [4], so according to the notation of the book [1],see page 192, the mentioned algebra is C1(D,H). We will adopt this notationbut modify it such that it makes it possible to look at those elements of a C*-algebra A acting on H which are weakly D−differentiable. This subalgebra of Ais then denoted C1(A, D). First of all we will like to study the algebra of higherweak derivatives which with the notation of [1] is the algebra Cn(D,H) and thealgebra of n times weakly D−differentiable operators inside a C*-algebra A on H isCn(A, D) := Cn(D,H) ∩A. In section 4 we give several characterizations of thoseoperators that are n times weakly D−differentiable, and we would like to mentionhere, that a bounded operator x is n times weakly D−differentiable if and only iffor any k in {1, . . . , n} the k′th commutator [D, [D, . . . , [D, x] . . . ]] is defined andbounded on dom(Dk). This is known to many mathematicians, but we could notfind a reference where the details are easy to follow, so we have included a proofhere. This characterization of n times weak D−differentiability will be crucial forthe results of section 5 on reflexive algebras. We also give a characterization ofhigher weak differentiability based on an embedding of the higher commutators[D, [D, . . . [D, a] . . . ]] into a linear space consisting of infinite matrices of boundedoperators. This set-up is original, and we hope that it will turn out to be a usefulframe inside which some operator theoretical questions can be dealt with in away which avoids the tiresome considerations of the validity of products and sumsof operators. After the article [2] was accepted for publication and proof read,we realized, that the one parameter group of automorphisms of B(H) given byB(H) � x → eitDxe−itD ∈ B(H) is actually a so-called adjoint semigroup on adual Banach space. Adjoint semigroups were first studied in [8], and [6] contains asurvey of the general theory of adjoint semigroups. Our usage of the general theoryis limited, but several things could have been presented in an easier way in [2], ifwe had been able to make references to [6].

2. Weak and higher weak D−differentiability

In order to avoid confusion we will like to clear up a point which has not beenpresented in an optimal way in [2]. The Definition 1.1 of [2] defines a boundedoperator x to be weakly D−differentiable if there exists a bounded operator b onH such that for any pair of vectors ξ, η in H we have

limt→0

|〈(eitDxe−itD − x

t− b

)ξ, η〉 = 0.

This definition implies that for any ξ, η the function

t → 〈eitDxe−itDξ, η〉is differentiable at t = 0, and it is stated, but not explicitly proven that this latterproperty implies weak D−differentiability as defined via a weak derivative b. It isquite easy to see that the two sorts of weakD−differentiability are equivalent and allthe arguments are presented in [2], but the consequences are not made sufficientlyclear. The right formal definition of weak D−differentiability then becomes asfollows.

Definition 2.1. A bounded operator x on H is weakly D−differentiable if forany pair of vectors ξ, η in H the function t → 〈eitDxe−itDξ, η〉 is differentiable att = 0.

WEAK DERIVATIVES AND REFLEXIVITY 71

To see that our present definition of weak D−differentiability implies the ex-istence of a weak derivative, i.e. a bounded operator b such that Definition 1.1of [2] is satisfied, we refer the reader to the proof of (ii) ⇒ (iii) in Theorem 3.8of [2]. That step is the crucial part of the proof, and it is based on the uniformboundedness principle applied to all the operators

{eitDxe−itD − x

t: t �= 0}.

This set is bounded because any function such as t → 〈eitDxe−itDξ, η〉 is differen-tiable at t = 0, and hence the set of values

{〈eitDxe−itD − x

tξ, η〉 : t �= 0}

is bounded and the principle applies. The existence of b then follows from the restof Theorem 3.8 of [2]. We will quote that theorem below and define the higher weakderivatives, but first we will recall a couple of other forms of D−differentiability.

We say that a bounded operator x is uniformly D−differentiable if the func-tion t → eitDxe−itD is differentiable at t = 0, with respect to the norm topol-ogy on B(H). In analogy with the definition of weak D−differentiability we saythat x is strongly D−differentiable if for each vector ξ in H the function t →eitDxe−itDξ is differentiable at t = 0 with respect to the norm topology on H. Itfollows from [2] that weak and strongD−differentiability are equivalent but uniformD−differentiability is in general a stronger property.

The book [1] studies strongD−differentiability in its Chapter 5, and it mentionsthat this concept is equivalent to weak D−differentiability, which we prefer to workwith, because it seems to be closer to the classical concepts involving differentiablefunctions on R.Anyway we already have adopted the notation from [1], but modifiedit so that the C*-algebra A is part of the notation too, so we define:

Definition 2.2. Let A be a C*-algebra on a Hilbert space H and D a self-adjoint operator on H. Then the algebra of n times weakly D−differentiable oper-ators in A is denoted Cn(A, D).

The self-adjoint operator D defines a one parameter automorphism group αt

on B(H), which for a bounded operator x on H is defined by αt(x) := eitDxe−itD.For a weakly D−differentiable operator x in B(H) it then follows, that δw(x) is theweak operator derivative d

dtαt(x)|t=0, but there is also the possibility of having anorm derivative of αt(x) at 0, and in that case we let δu(x) denote that derivative.On the other hand, when speaking of higher derivatives, we quote from [2] thefollowing result, which tells that higher uniform derivatives are closely related toweak derivatives.

Theorem 2.3. Let x be a bounded operator on H and n ≥ 2. If x is n timesweakly D−differentiable then x is n− 1 times uniformly D−differentiable.

Proof. See Corollary 4.2 of [2]. �

We will quote Theorem 3.8 from [2] here, without description of all the languageused. Not all of the results below may be generalized to higher derivatives andfor those properties, which can be extended, we will give the necessary precisedefinitions, when needed.

72 ERIK CHRISTENSEN

Theorem 2.4. Let x be a bounded operator on H. The following properties areequivalent:

(i) x is strongly D−differentiable.(ii) x is weakly D−differentiable.(iii) x is D−Lipschitz continuous.(iv) The sesquilinear form S(i[D, x]) on the domain of D is bounded.(v) The infinite matrix m(i[D, x]) represents a bounded operator.(vi) The operator Dx− xD is defined and bounded on a core for D.(vii) The operator Dx−xD is bounded and its domain of definition is dom(D).

If x is weakly D−differentiable then

∀ξ, η ∈ H limt→0

〈(eitDxe−itD − x)ξ, η〉t

=〈δw(x)ξ, η〉

x domD ⊆ domD and δw(x)∣∣dom(D) =i(Dx− xD)

∀t ∈ R : ‖αt(x)− x‖ ≤‖δw(x)‖|t|.

The properties (iii) and (iv) from the theorem just above have no simple gener-alizations to higher derivatives and will not be discussed here at all. The remainingfive properties all suggest natural extensions to the setting of higher weak deriva-tives and higher commutators as well, and we will discuss this in the next section.

Before embarking into the study of higher weak derivatives we would like tomake the following observation explicit. The reason being, that although mostpeople know it, we do not have an exact reference at hand.

Lemma 2.5. If a bounded operator x on H is weakly D−differentiable then forany pair of vectors ξ, η in H the function 〈αt(x)ξ, η〉 is differentiable on R and

d

dt〈αt(x)ξ, η〉 = 〈αt(δw(x))ξ, η〉.

Proof. By definition the equality holds for t = 0, and arguments similar tothe ones given in the proof of Lemma 2.1 of [2] show that the identity may betranslated from t = 0 to any other real t. �

This lemma has an immediate consequence, which we formulate as a proposi-tion, since it is important, although its proof is trivial.

Proposition 2.6. A bounded operator x on H is n times weakly D−differen-tiable if and only if x is in dom(δnw) and if and only if for any pair ξ, η in H thefunction 〈αt(x)ξ, η〉 is in Cn(R).

If x is n times weakly differentiable then

dn

dtn〈αt(x)ξ, η〉 = 〈αt(δ

nw(x))ξ, η〉.

Proof. Follows from Lemma 2.5 by induction. �

We will end this section by introducing a norm on Cn(D,H).

Definition 2.7. For any x in Cn(D,H) the norm ‖x‖n is defined by

‖x‖n =

n∑j=0

1

j!‖δjw(x)‖.


This is not the same norm as the one defined in [1] Definition 5.1.1 at page195, but it is equivalent to that norm, and it follows from [2] Proposition 3.10 that(Cn(A, D), ‖.‖n

)is a Banach algebra.

3. Higher weak D−derivatives and iterated commutators

Having Proposition 2.6 one might think that our understanding of δw and itspowers is sufficiently well established for most purposes, but it is not. The problemis that we do not know how to relate higher weak derivatives to expressions involvingiterated commutators with iD. If x is in dom(δw) then it follows from Theorem 2.4that iDx− ixD is defined on all of dom(D) and δw(x) is the closure of iDx− ixD.If x is in dom(δ2w), then it is natural to look at the second iD commutator

iD(iDx− ixD)− (iDx− ixD)(iD),

but we know nothing about its domain of definition, possible boundedness andclosure. In this section we will show that the properties of the higher commutatorsare as nice as we can possibly hope for. We will show that for a bounded n timesweakly differentiable operator x, the n times iterated commutator between iD andx is defined on dom(Dn) and the closure of this operator equals δnw(x). We will basethe proof of this on the results of Theorem 2.4. In order to simplify the writingsbelow we define an operator d on the space of linear operators on H.

Definition 3.1.(i) A linear operator on H is a linear operator defined on a subspace of H

and with values in H. The space of all linear operators on H is denoted L.A product yz of operators in L is defined on those vectors in the domainof z which are mapped into the domain of y by z, and a sum is definedon the intersection of the domains of all the summands.

(ii) The operator d on L is defined for y in L by d(y) := (iD)y − y(iD).

We will start our investigation on higher commutators by making the followingobservation.

Lemma 3.2. Let x be a bounded operator in B(H) and n a natural number. Ifx is n times weakly differentiable then for any k in {1, . . . , n} :

δk−1w (x) : dom(D) → dom(D),

δkw(x)|dom(D) = i[D, δk−1w (x)] = d(δk−1

w (x)).

Proof. If x is n times weakly differentiable, then for any k in {1, . . . , n} wehave δk−1

w (x) is in dom(δw). Then Theorem 2.4 item (vii) presents the claimedproperties of δk−1

w (x). �

The statements in Lemma 3.2 show that δkw(x) is the closure of the commutator[iD, δk−1

w (x)], but if k > 1 then δk−1w (x) is defined as a closure of the commutator

[iD, δk−2w (x)], so we have no direct control over the operator [iD, δk−1

w (x)]. Thisis not sufficient for our purpose, so we want to look at the restriction of such acommutator to dom(Dk), and then show that on this domain the higher weak de-rivative may be computed without any closure operations, as a higher commutator,and that the closure of this algebraically defined commutator equals δkw(x).

74 ERIK CHRISTENSEN

Proposition 3.3. Let x be an n times weakly differentiable bounded operatoron H, then for k in {1, . . . , n}

(i) δk−1w (x)dom(D) ⊆ dom(D)

(ii) x dom(Dk) ⊆ dom(Dk),(iii) dom(dk(x)) = dom(Dk)(iv) δkw(x)|dom(Dk) = dk(x)(v) δkw(x) = closure(dk(x)).

Proof. The item (i) follows from Lemma 3.2. The following four items arerelated and we show them by induction on k. For k = 1 the results follow again fromitem (iv) of Theorem 2.4. Then suppose 1 < k ≤ n and that the statements are truefor natural numbers in the set {1, . . . , k − 1}. We start by proving (iii), so we willchoose a vector ξ in dom(Dk), then ξ is in dom(Dk−1) so dk−1(x)ξ = δk−1

w (x)ξ andby item (i) dk−1(x)ξ is in dom(D), and finally ξ is in dom(iDdk−1(x)). By assump-tions (iD)ξ is in dom(Dk−1) which equals dom(dk−1(x)) so ξ is in dom(dk−1(x)(iD))too, and dom(Dk) ⊆ dom(dk(x)). The opposite inclusion is trivially true since dk(x)is a sum of terms, where the last summand is (−i)kxDk.

With respect to item (iv), note that

Ddom(Dk) ⊆ dom(Dk−1) ⊆ dom(D),

so by the induction hypotheses the domain for dk−1(x)D equals dom(Dk) anddk−1(x)D = δk−1

w D∣∣dom(Dk). By (i) and the induction hypotheses Dδk−1

w (x) is

defined on dom(Dk) and equals Ddk−1(x) on that domain. Hence item (iv) follows.With respect to (v) we remark, that dom(Dk) is a core for D since it contains

the vectors in the core E , which was introduced in the proof of (v)⇒ (vi) in Theorem3.8 of [2]. Then δkw(x) is the closure of the commutator d(δk−1

w (x))|dom(Dk), butthe latter equals dk(x) so (v) follows.

To prove (ii) we remark, that from (i) and (iv) it follows that dk−1(x)dom(Dk) ⊆dom(D). On the other hand a closer examination of the expression dk−1(x)ξ for avector ξ in dom(Dk) shows that

(3.1) dk−1(x)ξ = (i)k−1k−1∑j=0

(k − 1

j

)(−1)jDk−1−jxDjξ

For j > 0 we have Djξ is in dom(Dk−j) and by assumption xDjξ is in dom(Dk−j)so Dk−1−jxDjξ is in dom(D). Then for j = 0 we find that Dk−1xξ may be writtenas a difference of two vectors in dom(D) and hence xξ is a vector in dom(Dk), anditem (ii) is proven. �

4. Equivalent Properties

In analogy with the results of Theorem 2.4 we want to show that higher orderweak differentiability may be characterized in several different ways. Some of theproperties we find are expressed in terms of infinite matrices of operators, so wewill include a short description of this set-up here.

In [2] we defined a sequence of pairwise orthogonal projections with sum I inB(H) by letting en denote the spectral projection for D corresponding to the inter-val ]n− 1, n]. Then we defined M to be all matrices (yrc) with r and c integers andyrc an operator in erB(H)ec. Any bounded operator x on H induces an element


m(x) in M which is defined as m(x)rc := erxec. The operator D has a representa-tion m(D) in M too, and it is defined as a diagonal matrix m(D)rc = 0, if r �= cand diagonal elements dr := m(D)rr := Der. Then for any element y = (yrc) inM, the commutator i[m(D), y] makes sense in M by

i[m(D), y]rc := i(dryrc − yrcdc),

and we may define a linear mapping dM : M → M by

∀y = (yrc) ∈ M : dM(y)rc := idryrc − iyrcdc.

By the computations above we get that the powers dnM are given as

(4.1) ∀n ∈ N ∀y = (yrc) ∈ M : dnM(y)rc = inn∑

k=0

(n

k

)(−1)n−kdkryrcd

n−kc .

We can now formulate our result on characterizations of higher weak differen-tiability.

Theorem 4.1. Let x be a bounded operator on H and n a natural number. Thefollowing properties are equivalent:

(i) x is in dom(δnw).(ii) x is n times weakly D−differentiable.(iii) x is n times strongly D−differentiable.(iv) ∀k ∈ {1, . . . , n}

x : dom(Dk) → dom(Dk)

dk(x) is defined and bounded on dom(Dk) with closure δkw(x).

(v) For k in {1, . . . , n} the infinite matrix dkM(m(x))) represents a boundedoperator.

(vi) There exists a core F for D such that for any k in {1, . . . , n} the operatordk(x) is defined and bounded on F .

Proof. We prove (i) ⇔ (ii), (ii) ⇔ (iii), (ii) ⇒ (iv) ⇒ (v) ⇒ (ii) and (ii) ⇔(vi).

(i) ⇔ (ii):Follows from Proposition 2.6.

(ii) ⇒ (iii):Follows by an induction based on the following induction step. Suppose 0 ≤ k < n,x is a bounded n times weakly differentiable operator, which is k times stronglydifferentiable, then δkw(x) is the k′th strong derivative by Theorem 2.4, and sincethis operator is weakly differentiable, the same theorem shows that δkw(x) is stronglydifferentiable with strong derivative δk+1

w (x).(iii) ⇒ (ii):

Follows from the Cauchy-Schwarz inequality.(ii) ⇒ (iv):

This follows from Proposition 3.3(iv) ⇒ (v):

Let 1 ≤ k ≤ n, then we are given that δkw(x) exists and is a bounded operator suchthat δkw(x)|dom(Dk) = dk(x). Let c be an integer then ecH ⊆ dom(Dk) so for any

76 ERIK CHRISTENSEN

integer r we get

erδkw(x)ec = erd

k(x)ec = ikk∑

j=0

(k

j

)(−1)k−jerD

jxDjec = dkM(m(x))rc,

hence dkM(x) is the matrix of a bounded operator and (v) follows.(v) ⇒ (ii):

Assume (v), i.e. that for any k in {1, . . . , n} there exists a bounded operator zkon H such that for any pair of integers r, c we have erzkec = dkM(x)rc. The casek = 1 is covered by Theorem 2.4. The proof may be found in [2], but we recall themain step, because we will use it repeatedly below. For any vector ξ from ecH weshowed that xξ is in dom(D). It then follows that for any integer r and a vector ξin ecH we have xξ is in dom(D) and

erz1ξ = i(drerxec − erxecdc)ξ = ier(Dx− xD)ξ.

and we concluded that x is weakly differentiable with δw(x) = z1. We may nowassume that 1 < k ≤ n and x is weakly differentiable of order k−1 with δjw(x) = zjfor 1 ≤ j ≤ k − 1. Then for ξ in ecH we get δk−1

w (x)ξ is in dom(D) so we have

erzkξ =i(drerdk−1M (x)ec − erd

k−1M (x)ecdc)ξ

=ier(Dδk−1w (x)− δk−1

w (x)D)ξ.

Hence δk−1w (x) is weakly differentiable and δkw(x) = zk, so x is n times weakly

differentiable and (ii) follows.(ii) ⇒ (vi):

For any n in N, the space dom(Dn) is a core for D, so (vi) follows from (iv), which,in turn, follows from (ii).

(vi) ⇒ (ii):Now suppose (vi) holds for a bounded operator x on H. Then for k in {1, . . . , n}there exist bounded operators yk = closure(dk(x)|F). Let us look at the case k = 1first. Then (iD)x − x(iD) is defined and bounded on the core F for D, so byTheorem 2.4 item (vi) x is in dom(δw) and y1 = δw(x). Let us now suppose that1 < k ≤ n and we know that yj = δjw(x), for 1 ≤ j ≤ k − 1, then for any ξ in F wecan find a sequence of vectors ξn in F such that ξn → ξ and Dξn → Dξ for n → ∞.Since dk(x) is bounded and defined on F we have

ykξ = limn→∞

dk(x)ξn = limn→∞

((iD)dk−1(x)ξn − dk−1(x)(iD)ξn

)= lim

n→∞

((iD)δk−1

w (x)ξn − δk−1w (x)(iD)ξn

).

Since the last part of these equations forms a convergent sequence we find thatlimn→∞

(iD)δk−1w (x)ξn exists and

limn→∞

(iD)δk−1w (x)ξn = ykξ + δk−1

w (x)(iD)ξ.

Hence δk−1w (x)ξ is in dom(D) and

ykξ = (iD)δk−1w (x)ξ − δk−1

w (x)(iD)ξ.

By Theorem 2.4 we get that δk−1w (x) is weakly differentiable and δkw(x) = yk, so x

is n times weakly differentiable, and the theorem follows. �


5. Reflexive representations of the algebras of higher weaklydifferentiable operators in a von Neumann algebra

In this section we will consider the case where we are dealing with a vonNeumann algebra M on H and study aspects of the algebras Cn(M, H) of ntimes higher weakly D−differentiable elements inside M, but unlike the case innoncommutative geometry we will not assume that Cn(M, D) is dense in M inany ordinary topology. The prototype of a von Neumann algebra, or rather thecommutative example which may give inspiration for general results on von Neu-mann algebras is the algebra of measurable essentially bounded functions on theunit circle, L∞(T, dθ), and in this setting Cn(M, D) is nothing but the n timesweakly D := −i d

dθ−differentiable functions, so we find here that for n ≥ 1 we haveCn(L∞(T, dθ), D) = Cn(C(T), D), and we may wonder if the von Neumann algebraproperty plays a role at all ? We have no answer, but this might be because ourunderstanding of the relations between noncommutative and commutative geome-try is still quite limited. Below we will describe the property called reflexivity of analgebra of bounded operators, but for the moment just say, that a von Neumannalgebra M is reflexive and that property is partly inherited by Cn(M, D), in thesense that this algebra has a representation as a reflexive algebra of bounded oper-ators on a Hilbert space. We will describe the reflexivity property in details below,but right now we will like to mention that reflexivity is a very strong property foran algebra of operators to have. This follows from von Neumann’s bicommutanttheorem which shows that if an algebra of bounded operators on H is self-adjointand reflexive then it is a von Neumann algebra. The algebras we will study are notself-adjoint, but sub-algebras of the upper triangular matrices in Mn+1(B(H)), sovon Neumann’s theorem does not apply directly in our situation.

We will remind you of the definition of reflexivity as it was defined by Halmosand described in the book [9].

Definition 5.1. Let H be a Hilbert space.

(i) Let S be a set of bounded operators on a Hilbert space H then Lat(S) isthe lattice of closed subspaces of H which are left invariant by each of theoperators in S.

(ii) Let G denote a collection of closed subspaces of H then Alg(G) is thealgebra of bounded operators on H which leave each of the subspaces inG invariant.

(iii) A subalgebra R of B(H) is said to be reflexive if R = Alg(Lat(R)).

One of the strong properties of a reflexive algebra of bounded operators ona Hilbert space K is that it is an ultraweakly closed subspace of B(K) and thenit becomes a dual space since all the ultraweakly continuous functionals on B(K)form the predual of B(K). Then the reflexive algebra has a predual which is a quo-tient of the predual of B(K). In the set-up for the classical commutative differentialgeometry such kinds of dualities are well known and widely used. The reflexivity isactually stronger than this duality property, but so far we have not been able to sin-gle out a property which solely depends on the reflexivity of a certain representationof the algebra Cn(M, D). We will now formulate the result:

Theorem 5.2. Let M be a von Neumann algebra on a Hilbert space H, Da self-adjoint operator on H and n a non-negative integer. There exists a unital

78 ERIK CHRISTENSEN

injective algebraic homomorphism Φn : Cn(M, D) → B(H ⊗ Cn+1) such that theimage Rn(M, D) := Φn(C

n(M, D)) is a reflexive algebra on H ⊗ Cn+1.For any x in Cn(M, D) : 1

n+1‖x‖n ≤ ‖Φn(x)‖ ≤ ‖x‖n.

Proof. For n = 0 we have C0(M, D) = M, and then C0(M, D) is a reflexivesubalgebra of B(H), so we define Φ0 := id

∣∣C0(M, D), and R0 := M. For n > 0 wewill construct a representation Φn of Cn(M, D) into the upper triangular matriceswith constant diagonals inside the (n+ 1)× (n+ 1) matrices over B(H) such thatfor an x in Cn(M, D) the representation is given by

Φn(x) :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

x δw(x)12δ

2w(x) . . . 1

n!δnw(x)

0 x δw(x) . . . 1(n−1)!δ

n−1w (x)

. . . . . . .

. . . . . . .0 . . . . δw(x)

12δ

2w(x)

0 . . . . x δw(x)0 . . . . 0 x

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠and the element in the j’th upper diagonal is 1

j!δjw(x).

We define Rn := Φn(Cn(M, D)). If D is bounded then δw(x) = [iD, x] and

it is well known that the mapping Φn is a homomorphism and Rn is an algebra.But now δw(x) is the closure of the commutator [iD, x] so elementary algebra doesnot apply right away. The short proof of the homomorphism property is then thatthe results of Theorem 4.1 show that the algebraic arguments are still valid whenrestricted to take place on the domain dom(Dn) only. We will like to show thiswith some more details because these arguments will be needed, when we want toshow the reflexivity of Rn. To set the stage we define Bn as the matrix in Mn+1(C)with ones in the first upper diagonal and zeros elsewhere:

Bn :=

⎛⎜⎜⎜⎜⎝0 1 0 . 00 0 1 . 0. . . . .0 . . 0 10 . . 0 0

⎞⎟⎟⎟⎟⎠ .

Then Bn is nilpotent and satisfies B(n+1)n = 0, which will be very useful in

the computations to come. First we can describe Φn(x) inside the tensor productB(H)⊗Mn+1(C) as

Φn(x) = x⊗ I +n∑

j=1

1

j!δjw(x)⊗Bj

n,

and we see from Theorem 4.1 that all the elements in the sum are defined aselementary operator theoretical products or sums of such products on the spacedom(Dn)⊗ C(n+1). We will then define Dn = dom(Dn)⊗ C(n+1), and the comingcomputations will all take place on this dense subspace of H⊗C(n+1). We will workwith matrices of unbounded operators and the first, denoted Sn is defined as

Sn := iD ⊗Bn, dom(Sn) = H ⊕ dom(D)⊕ · · · ⊕ dom(D).

In order to be able to talk on specific matrix elements we suppose that C(n+1)

is equipped with its canonical basis, and that the basis elements ej are numbered


by 0 ≤ j ≤ n. Then the matrix elements are indexed by {ij} with i, j ∈ {0, . . . , n}too.

For x in Cn(M, D) the Theorem 4.1 shows that for any j in {1, . . . , n} wehave xdom(Dj) ⊆dom(Dj) so for any set of natural numbers j1, . . . , jk with j1 +· · · + jk ≤ n, any set of operators x0, x1, . . . , xk in Cn(M, D) and any vector ξ indom(Dn), the vector ξ will be in the domain of definition for x0D

j1x1 . . . Djkxk.

We will lift this product to the matrices, and in order to do so we introduce thecanonical amplification ι(x) of B(H) into M(n+1)(B(H)) by ι(x) := x⊗ I. Then forx0, x1, . . . , xk in Cn(M, D) we can define a product of operators which always willbe defined on Dn by the following convention:

ι(x0)Sj1n ι(x1) . . . S

jkn ι(xk)

∣∣Dn

:=

{0∣∣Dn if j1 + · · ·+ jk > n((x0(iD)j1x1 . . . (iD)jkxk

)⊗B

(j1+···+jk)n

)∣∣Dn if j1 + · · ·+ jk ≤ n.

This means that Sn and ι(Cn(M, D)) generate an algebra with this specialproduct. The product is a bit more complicated than just the product of therestrictions to Dn of each of the factors. This is because the operator D doesnot map dom(Dn) into dom(Dn), so Sn does not map Dn into Dn but anyway allthe products mentioned make sense by first making the standard operator productand then restricting the outcome to Dn. We can then define Tn as the algebra ofmatrices defined on Dn with this product and generated by Sn and ι(Cn(M, D))The point of this is that we may now use standard algebra on this associative unitalalgebra and we define elements Tn and its inverse T−1

n by exponentiating Sn. Thenil-potency of Sn gives us the following formulas inside this algebra:

Tn := exp(Sn) = I ⊗ I +

n∑j=1

1

j!Sjn T−1

n := exp(−Sn) = I ⊗ I +

n∑j=1

1

j!(−Sn)

j .

In order to relate Sn and Tn to the unital algebra Rn we remind you thatin any unital associative algebra C with a nilpotent element s we may study thederivation ad(s) on C given by ad(s)(x) := [s, x] and we have that exp

(ad(s)

)(x) =

exp(s)x exp(−s). In our setting we then get that for any x in Cn(M, D) we have

ad(Sn)j(ι(x))

∣∣Dn =

{δjw(x)⊗Bj

n

∣∣Dn if j ≤ n

0∣∣Dn if j > n,

so the equalities above yield the following identities in the algebra Tn.Tnι(x)T

−1n = exp(Sn)ι(x) exp(−Sn)(5.1)

= exp(ad(Sn))(ι(x))

=(ι(x) +

n∑j=1

1

j!δjw(x)⊗Bj

n

)∣∣Dn

= Φn(x)∣∣Dn.

We can now find a family Ln of closed Rn invariant subspaces of H ⊗ C(n+1)

such that we will have Rn = Alg(Ln), and in this way the reflexivity of Rn willbe established. The proof is made by induction and for the case of n = 0 thefamily L0 is just the set Lat(M) of closed subspaces which are invariant under anyelement in M, and and it follows from von Neumann’s bicommutant theorem that

80 ERIK CHRISTENSEN

C0(M, D) = M = Alg(L0). We now assume that n ≥ 1 and we define a subset Ln

of Lat(Rn) which is so big that Alg(Ln) = Rn. Since we are forming an inductionargument it is convenient to think of the Hilbert spaces H ⊗ C(n+1), as a nestedfamily of closed subspaces of �2(N0, H) in the following way,

H0 := H ⊗ e0

Hn := H ⊗ e0 ⊕ · · · ⊕H ⊗ en,

and we will let K = �2(N0, H) and let En denote the orthogonal projection of Konto Hn. For each n we will also identify Hn with the abstract tensor productH ⊗C(n+1) in the way that an expression ξ ⊗ ej which appears in both spaces areidentified. In this way Rn may be identified with some upper triangular matriceswhose entries are 0 whenever any index is bigger than n, or described as a subspaceof the bounded operators on K which satisfies X = EnXEn. We then see that for0 ≤ j ≤ n the subspace Hj is invariant for Rn, and we also note that for any closedsubspace F in Lat(M) the subspace F ⊗ e0 is invariant for the algebra Rn too.We will point out 2 more, but closely related examples of closed subspaces of Hn

which are invariant for Rn, and we will denote these spaces Pn and Qn. First itis practical to redefine ι(x) to act on H ⊗ �2(N0) by ι(x) := x ⊗ I�2(N0

and alsoredefine Bn as the canonical image of Bn in B(K) under the embedding of Rn intoEnB(K)En. For a natural number n we define a subspace Pn of Hn by(5.2)

Pn := {Tn(ξ⊗en) : ξ ∈ dom(Dn)} = {ξ⊗en+n∑

j=1

1

j!(iD)jξ⊗en−j : ξ ∈ dom(Dn) },

so this space is just the graph of a certain operator Vn from dom(Dn)⊗ en to Hn.By the closedness of all the powers (iD)j we see that Vn is a closed operator, soPn is a closed subspace of Hn and by the relation (5.1) we get that for any x inCn(M, D) and any ξ in dom(Dn) we have

(5.3)xξ ∈ dom(Dn) since x ∈ Cn(M, D),

Φn(x)Tnξ ⊗ en = Tnι(x)ξ ⊗ en = Tn(xξ)⊗ en,

so Pn is an invariant subspace for Rn acting on Hn. If j < n then for any xin Cn(M, D) ⊆ Cj(M, D) we see by the construction of Φj(x) and Φn(x) thatΦn(x)

∣∣Hj = Φj(x) Hence for any j < n we also have that Pj is a closed invariantsubspace for Rn.

To construct the last invariant subspace we remind you that if we define D :=

D+ I then D is also a self-adjoint operator, and since exp(itD) = eit exp(itD) thecorresponding automorphism groups αt and αt are identical so for any n in N0 we

have Cn(M, D) = Cn(M, D) and δnw = δnw. In particular Rn = Rn so a closed

subspace of Hn, which is invariant for Rn is also invariant for Rn. We may thenrepeat the construction made for Pn but now based on D+I to obtain the invariant


subspace Qn which is obtained via the equations below:

Tn : = I ⊗ En +

n∑j=1

1

j!(i(D + I))j ⊗Bj

n

Qn : = {Tn(ξ ⊗ en) : ξ ∈ dom(Dn)}(5.4)

= {ξ ⊗ en +

n∑j=1

1

j!(i(D + I))jξ ⊗ en−j : ξ ∈ dom((D + I)n) }.

We need to remark that dom((D + I)n) equals dom(Dn), and that statement fol-lows from the binomial formula and the fact that for j ≤ n we have dom(Dn) ⊆dom(Dj).

We can then define the collection Ln of Rn invariant subspaces of Hn by

Ln := L0 ∪ {Hj : 0 ≤ j ≤ n } ∪ {Pj : 1 ≤ j ≤ n} ∪ {Qj : 1 ≤ j ≤ n}.It is clear that the algebra Alg(Ln) will contain the unit I of B(K), which can

never be an element Rn, whose matrices all have zero entries outside the upper(n+ 1)× (n+ 1) corner, but we will prove by induction that

Rn = {X ∈ Alg(Ln) : EnXEn = X.}The case n = 0 is already established, so let us assume that n > 0 and the statementis true for n−1, and let X be an operator in Alg(Ln) such that EnXEn = X. ThenH(n−1) is an invariant subspace for X so XE(n−1) = E(n−1)XE(n−1) and we findimmediately that XE(n−1) also leaves all the subspaces in L(n−1) invariant and the

induction hypothesis tells that there exists an operator x in C(n−1)(M, D) suchthat XE(n−1) = Φ(n−1)(x). Unfortunately we do not know that the operator x isin Cn(M, D) too, but we will show it now and then prove that X = Φn(x). Weknow that Pn and Qn are invariant subspaces for X and from the equations ( 5.2)and ( 5.4) we have descriptions of Pn and Qn which will become useful. Hence

let ξ be in dom(Dn), Tnξ ⊗ en and Tnξ ⊗ en be the corresponding vectors in Pn

and Qn respectively. The invariance of Pn under X has as its first consequencethat for the operator entry xnn of X we get xnnξ is in dom(Dn). If we look at the(n− 1)’st coordinate of the vector XTnξ⊗ en the invariance of Pn under X impliesthe equation

(5.5) x(iD)ξ + x(n−1)nξ = (iD)xnnξ.

By analogy we get a similar equation based on the invariance of Qn under X so weget

(5.6) x(i(D + I))ξ + x(n−1)nξ = (i(D + I))xnnξ.

By subtraction of those equations we get

(5.7) ∀ξ ∈ dom(Dn) : xξ = xnnξ,

so since both operators are bounded we have xnn = x. The equation (5.5) may thenbe applied to show that x(n−1)n = δw(x) and it is possible to continue along thisline to show that X = Φn(x), but we will instead address the first element, of thevector XTnξ⊗ en, since it seems to be easier to write down the details in this case.Let us return to the general setting we studied just in front of the equation (5.1)where we have an associative unital algebra B and an element s in B. We will thendefine operators L and R on B by left and right multiplications by s, so Lb := sb

82 ERIK CHRISTENSEN

and Rb := bs. then by the binomial formula we get, since L and R commute thatfor any b in B(5.8)n∑

j=0

1

(n− j)!

1

j!ad(s)j(b)s(n−j) =

n∑j=0

1

(n− j)!

1

j!(L−R)jR(n−j)b =

1

n!Lnb =

1

n!snb.

We will recall the commutator mapping d which we defined in Definition 3.1 asd(x) := [iD, x]. We know from above that x is in C(n−1)(M, D) so by Theorem 4.1for any j in {1, . . . , n− 1} we have xdom(Dj) ⊆ dom(Dj), from equation ( 5.7) wehave xdom(Dn) ⊆ dom(Dn), so all the expressions dj(x) are defined on dom(Dn).The algebraic identity (5.8) then applies and we get

(5.9)

n∑j=0

1

(n− j)!

1

j!dj(x)(iD)(n−j)

∣∣dom(Dn) =1

n!(iD)nx

∣∣dom(Dn).

Since x is in C(n−1)(M, D) we have

dj(x)∣∣dom(Dn) = δjw(x)

∣∣dom(Dn), for 0 ≤ j ≤ n− 1.

On the other hand the invariance of Pn shows that

(5.10)((

(n−1)∑j=0

1

(n− j)!

1

j!δjw(x)(iD)(n−j)) + x0n

)∣∣dom(Dn) =1

n!(iD)nx

∣∣dom(Dn).

By elementary algebra we then get that 1n!d

n(x)∣∣dom(Dn) = x0n

∣∣dom(Dn), so

by Theorem 4.1 we find that x is in Cn(M, D) and that 1n!δ

nw(x) = x0n, as expected.

Recall that by (5.3) Pn is invariant under the elements in Rn, and with this inmind we get that Pn must be invariant under Y := (X−Φn(x)), which is a columnmatrix such that yij = 0 whenever j �= n, and also satisfies ynn = 0, which is crucialfor the next argument. Given any vector ξ in dom(Dn) with corresponding vectorTn(ξ⊗en) in Pn we see that ynn = 0 implies that the n’th coordinate of Y Tn(ξ⊗en)is equal to 0, but then Y Tn(ξ⊗en) = 0, since the space Pn may be thought of as thegraph of an operator defined on the last coordinate, which here vanishes. On theother hand, for the given ξ in dom(Dn) we get 0 = Y Tn(ξ⊗ en) =

∑ni=0(yinξ)⊗ ei.

Hence for 0 ≤ i ≤ n we get yin = 0 and then X = Φn(x) for an x in Cn(M, D)and the reflexivity of Rn is proven. �

References

[1] W. O. Amrein, A. Boutet de Monvel, and V. Georgescu, C0-groups, commutator methods andspectral theory of N-body Hamiltonians, Progress in Mathematics, vol. 135, Birkhauser Verlag,Basel, 1996. MR1388037 (97h:47001)

[2] E. Christensen. On weakly D-differentiable operators. To appear in Expo. Math.,http://dx.doi.org/10.1016/j.exmath.2015.03.002

[3] A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.MR1303779 (95j:46063)

[4] V. Georgescu, C. Gerard, and J. S. Møller, Commutators, C0-semigroups and resolvent esti-mates, J. Funct. Anal. 216 (2004), no. 2, 303–361, DOI 10.1016/j.jfa.2004.03.004. MR2095686(2005h:47044)

[5] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I,Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich,Publishers], New York, 1983. Elementary theory. MR719020 (85j:46099)

[6] J. van Neerven, The adjoint of a semigroup of linear operators, Lecture Notes in Mathematics,vol. 1529, Springer-Verlag, Berlin, 1992. MR1222650 (94j:47059)


[7] J. von Neumann, Mathematische Grundlagen der Quantenmechanik (German), UnveranderterNachdruck der ersten Auflage von 1932. Die Grundlehren der mathematischen Wissenschaften,Band 38, Springer-Verlag, Berlin-New York, 1968. MR0223138 (36 #6187)

[8] R. S. Phillips, The adjoint semi-group, Pacific J. Math. 5 (1955), 269–283. MR0070976(17,64a)

[9] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York-Heidelberg,1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. MR0367682 (51 #3924)

Department of Mathematics, University of Copenhagen, Denmark



Parabolic induction, categories of representations andoperator spaces

Tyrone Crisp and Nigel Higson

To Richard Kadison with admiration on the occasion of his 90th birthday

Abstract. We study some aspects of the functor of parabolic induction within

the context of reduced group C∗-algebras and related operator algebras. Weexplain how Frobenius reciprocity fits naturally within the context of operatormodules, and examine the prospects for an operator algebraic formulation ofBernstein’s reciprocity theorem (his second adjoint theorem).

1. Introduction

Harish-Chandra famously decomposed the regular representation of a real re-ductive group G into an explicit integral of its isotypical parts. His program to doso had two parts:

(a) the classification of the so-called cuspidal representations of G and its Levisubgroups; and

(b) the construction, by the process of parabolic induction, of further representa-tions, sufficiently many in number to decompose the regular representation.

The cuspidal representations of a real reductive group G are the irreducible andunitary representations of G that are square-integrable, modulo center. Their clas-sification fits well with ideas from C∗-algebra K-theory and noncommutative geom-etry. Indeed the classification was an important source of inspiration for the formu-lation of the Baum-Connes conjecture. But the functor of parabolic induction hasreceived less attention from operator algebras and noncommutative geometry. Ourpurpose here is to continue the effort begun in [Cla13] and in [CCH16a,CCH16b]to address this imbalance, if only modestly.

A few years ago Pierre Clare explained in [Cla13] how parabolic induction fitsinto the theory of Hilbert C∗-modules and bimodules in a way that is very similarto Marc Rieffel’s well known treatment of ordinary induction [Rie74]. In jointwork with Clare [CCH16a,CCH16b] we studied parabolic induction as a functorbetween categories of Hilbert C∗-modules. Using a considerable amount of rep-resentation theory, due to Harish-Chandra, Langlands and others, we constructed

2010 Mathematics Subject Classification. Primary 22E45; Secondary 46L07, 46H15.The first author was partially supported by the Danish National Research Foundation through

the Centre for Symmetry and Deformation (DNRF92).The second author was partially supported by the US National Science Foundation through

the grant DMS-1101382.


85

86 TYRONE CRISP AND NIGEL HIGSON

an adjoint Hilbert C∗-bimodule, used it to define a functor of parabolic restrictionbetween categories of Hilbert C∗-modules, and proved that parabolic induction andrestriction are two-sided “local” adjoints of one another.

A drawback of our work was that the concept of local adjunction is significantlyweaker than the standard category-theoretic notion of adjunction. This shortcom-ing was unavoidable: there is no category-theoretic adjunction at the Hilbert C∗-module level. Moreover the natural candidates for the unit maps of the sought-foradjunctions are even not properly defined at the Hilbert C∗-module level.

The purpose of this article is to examine the extent to which the shortcomingsof the Hilbert C∗-module theory can be remedied by adjusting the context a lit-tle. To this end we shall study new categories consisting of group or C∗-algebrarepresentations on operator spaces. We shall prove that the new categories haveonly the familiar irreducible objects, so that they present a plausible context forrepresentation theory. Then we shall formulate and prove a simple theorem aboutadjoint pairs of functors between the new categories of operator space modules overC∗-algebras (as opposed to Hilbert C∗-modules). As we shall explain, this impliesin a very simple way (that does not require any sophisticated representation theory)a Frobenius reciprocity theorem for parabolic induction (the theorem is that thefunctor of parabolic induction has a left adjoint, which we shall describe explicitly,along with the adjunction isomorphism).

Secondly, we shall examine in detail the form of parabolic induction and re-striction at the level of Harish-Chandra’s Schwartz algebra in the particular casewhere G = SL(2,R). We shall summarize the tempered representation theory ofG in the form of a Morita equivalence between the Harish-Chandra algebra anda simpler and more accessible algebra. Our reason for doing this is to formulateand prove a “second adjoint theorem” for tempered representations in this case,along the lines of Bernstein’s fundamental second adjoint theorem (that parabolicinduction also has an explicit right adjoint) in the smooth representation theory ofreductive p-adic groups [Ber87,Ber92].

We shall say a good deal more elsewhere [CH] about our second adjoint theoremfor tempered representations. Our reason for introducing the result here is to use itas a test for measuring the potential usefulness of new operator-algebraic contextsfor representation theory. To this end, we shall conclude by using the explicitformulas obtained for the Harish-Chandra algebra to explore the prospects for anelaboration of the operator space module Frobenius reciprocity relation analyzed inSection 2 so as to include Bernstein’s second adjunction. We shall give one concretesuggestion about how this might be achieved in Section 4.3.

Note added by NH. It is an immense pleasure to contribute to this volumecelebrating Dick Kadison’s 90th birthday. The whole operator algebra communityowes Dick a great debt of gratitude for his decades-long leadership, but in mycase the debt is not only mathematical, but personal too. Dick gave me my firstjob and guided me through my first years as a mathematician. His support andencouragement helped me launch my career, and the advice he gave me then helpsme still. Thank you, Dick.

2. Categories of Operator Space Modules

We shall study operator space modules over C∗-algebras and, later on, overoperator algebras. For the most part we shall refer to the monograph [BLM04] for

PARABOLIC INDUCTION AND OPERATOR SPACES 87

background on operator spaces, but we shall repeat some of the basic definitionshere.

Before we start, let us explain our point of view. In the representation theoryof real reductive groups there is broad agreement about the concepts of irreduciblerepresentation that are appropriate for study, along with the associated conceptsof equivalence among irreducible representations. But representations that lie wellbeyond the irreducible representations are little-studied in representation theory.From the point of view of noncommutative geometry this is an awkward omission,since for example the K-theory studied in noncommutative geometry, and usedto formulate the Baum-Connes conjecture, involves representations that are farfrom irreducible. So it is of interest to explore some of the potentially convenientcategories of representations that operator algebra theory provides.

As we mentioned in the introduction, our immediate concern here is not K-theory but parabolic induction, together with adjunction theorems such as Frobe-nius reciprocity. But here, too, the choice of a category of representations matters.Our main observation is that operator spaces can offer a very convenient startingpoint from which to begin an examination of Frobenius reciprocity and relatedmatters, because the theorem assumes a particularly elementary form there.

To continue, recall that an operator space is a complex vector space X equippedwith a family of Banach space norms on the spaces Mn(X) of n× n matrices overX that satisfy the following two conditions:

(a) If x ∈ Mn(X) and a, b ∈ Mn(C), then

‖axb‖ ≤ ‖a‖‖x‖‖b‖.

(b) The norm of a block-diagonal matrix is the maximum of the norms of thediagonal blocks.

A linear map T : X → Y between operator spaces induces maps

Mn(T ) : Mn(X) −→ Mn(Y )

by applying T to each matrix entry, and we say that T is completely bounded (c.b.)if

supn

‖Mn(T )‖operator < ∞.

The supremum is the completely bounded norm. We shall also use the relatednotions of completely contractive and completely isometric map.

Example 2.1. Every Hilbert space H carries a number of operator space struc-tures. In this paper we shall consider only the column Hilbert space structure, inwhich H is identified with the concrete operator space B(C, H) of bounded op-erators from C to H. Every bounded operator between Hilbert spaces is com-pletely bounded as an operator between column Hilbert spaces, and the completelybounded norm is the operator norm. See [BLM04, 1.2.23].

Let X, Y and Z be operator spaces. A bilinear map Φ: X × Y → Z gives riseto bilinear maps

Mn(Φ): Mn(X)×Mn(Y ) −→ Mn(Z)

through the formula

Mn(Φ):([xij ], [yij ]

)�−→

[∑nk=1 Φ(xik, ykj)

].


The Haagerup tensor product X⊗hY is a Banach space completion of the algebraictensor product over C and an operator space, characterized by the property thatevery completely contractive Φ as above factors uniquely through a completelycontractive map X ⊗h Y → Z. See [BLM04, 1.5.4].

Definition 2.2. An operator algebra is an operator space A which is also aBanach algebra with a bounded approximate unit, such that the product in Ainduces a completely contractive map A⊗h A → A.

Definition 2.3. A (left) operator module over an operator algebra A is anoperator space X and a nondegenerate1 left A-module for which the module actionextends to a completely contractive map A ⊗h X → X. One similarly definesright operator modules, and operator bimodules. We shall denote by OpModA thecategory of left operator modules over A and completely bounded A-module maps.

2.1. Irreducible Operator Modules. If A is a C∗-algebra, then the cate-gory OpModA contains the category HilbA of nondegenerate Hilbert space repre-sentations of A as a full subcategory (each Hilbert space being given its columnoperator space structure). Our first observation is that if A is of type I, thenOpModA and HilbA have the same irreducible objects (that is, the same moduleshaving no nontrivial closed submodules):

Proposition 2.4. Let A be a type I C∗-algebra. Every irreducible operator A-module is completely isometrically isomorphic in OpModA to an irreducible Hilbertspace representation of A.

Proof. Let X be an irreducible operator A-module. By [BLM04, Theorem3.3.1] there is a nondegenerate representation of A on a Hilbert space H, a secondHilbert space K, and a completely isometric isomorphism from X to a closed A-submodule of the space B(K,H) of bounded operators from K to H. We shallrealize X as a subspace of B(K,H) in this way, and we may assume that X ·K isdense in H.

We are going to argue that the representation of A on H is a multiple of a singleirreducible representation of A. To begin, the representation of A on H extendsto the multiplier algebra M(A), and the restriction to the center Z(M(A)) is amultiple of a single irreducible representation of the center. For otherwise therewould exist elements z1, z2 ∈ Z(M(A)) with z1z2 = 0 yet

z1H �= 0 and z2H �= 0,

so that

z1X �= 0 and z2X �= 0.

The subspace

z1X ⊆ X

would then be a nontrivial submodule of the supposedly irreducible module X.Assume now that A is liminal (that is, A acts as compact operators in each

irreducible representation) and that furthermore the spectrum A is a Hausdorfftopological space. The Dauns-Hofmann theorem (see [DH68] or [Ped79, Section4.4]) identifies Z(M(A)) with the algebra of bounded, continuous, complex-valued

functions on A. The identification is as follows:

1Nondegenerate means that A ·X is dense in X.


(a) If I ⊆ Z(M(A)) is the maximal ideal corresponding to evaluation at [π] ∈ A,then IA is the kernel of π.

(b) In contrast, if I ⊆ Z(M(A)) is the maximal ideal corresponding to evaluationat a point at infinity, then IA = A.

In our present situation, we see that the action of Z(M(A)) on H must factor

through the quotient by a maximal ideal corresponding to a point of A, as in item(a), since the action of A on H is certainly nonzero. Therefore the action of A mustfactor through a quotient A/IA. Since the quotient is isomorphic to the compactoperators, the action of A is a multiple of a single irreducible representation, asrequired.

Next assume that A is a general liminal C∗-algebra. Let {Jα} be a compositionseries for A for which each quotient Jα+1/Jα has Hausdorff spectrum. Take the leastα for which JαX �= 0; by irreducibility we must then have JαX = X. Consider Xas an irreducible operator module over B = Jα/Jα−1, where as usual Jα−1 denotesthe closure of the union of all ideals in the composition series smaller than Jα. Theargument above shows that H is a multiple of a single irreducible representation ofB, and hence is a multiple of a single irreducible representation of A.

Finally, if A is a general type I C∗-algebra, we can apply the above argumentto a composition series for A with liminal quotients to show that H is a multipleof a single irreducible representation of A in this case too.

We have now shown in general that H is a multiple of a single irreduciblerepresentation of A, say

H ∼= M ⊗ L

where A acts trivially on M and irreducibly on L. We shall now show that there isa bounded operator S : K → M such that X consists precisely of all operators inB(K,M ⊗ L) of the form

(2.5) k �−→ S(k)⊗ �,

as � ranges over L. The map sending the operator (2.5) to ‖S‖ · � will then be acompletely isometric A-linear isomorphism from X to L.

For each k ∈ K and m ∈ M consider the completely bounded, A-linear mapfrom X to L defined by

(2.6) X � T �−→ m∗ · T · k ∈ L,

where m∗ : M ⊗ L → L is the operator m′ ⊗ � �→ 〈m,m′〉�. Fix k0 and m0 forwhich the operator (2.6) is nonzero. Then this operator is invertible: its kernelis a proper closed submodule of the irreducible module X, while its image is anonzero A-invariant subspace of L, which must equal L by Kadison’s transitivitytheorem [Kad57] (or by a direct argument in the present rather elementary type Isituation).

Applying Schur’s lemma to the irreducible representation L, we find that thereare scalars ck,m such that

m∗ · T · k = ck,m ·m∗0 · T · k0

for all k ∈ K and m ∈ M . Taking S ∈ B(K,M) to be the operator defined by〈m,S(k)〉 = ck,m, we have

T (k) = S(k)⊗ (m∗0 · T · k0)

for all T ∈ X and all k ∈ K. �


Remark 2.7. We do not know if the assumption in the previous propositionthat A be of type I is actually necessary.

2.2. Functors Between Operator Module Categories. If X is a rightoperator B-module, and if Y is a left operator B-module, then we can of courseform the algebraic tensor product of X and Y over B. The balanced Haageruptensor product X ⊗hB Y is a completion of the algebraic tensor product and anoperator space, characterized by the fact that any completely contractive bilinearmap

Φ: X × Y −→ Z

with Φ(xb, y) = Φ(x, by) extends to a completely contractive map from X ⊗hB Yto Z. The Haagerup tensor product is associative, and functorial with respect toc.b. bimodule maps. If X and Y carry left operator A-module and right operatorC-module structures, respectively, then X⊗hBY is an operator A-C-bimodule. See[BLM04, Section 3.4].

Now let A and B be operator algebras, and let E be an operator A-B-bimodule.We obtain a functor

OpModB −→ OpModA

from the Haagerup tensor product operation:

X �−→ E ⊗hB X.

When needed, we shall give the functor the same name—E—as the bimodule. Notethat composition of functors corresponds, up to natural isomorphism, to Haageruptensor product of bimodules.

If X is an operator A-module, then the module structure induces a completelyisometric isomorphism

(2.8) A⊗hA X∼=−→ X.

Similarly, we obtain a completely isometric isomorphism

(2.9) X ⊗hB B∼=−→ X

in the case of a right operator B-module structure. See [BLM04, Lemma 3.4.6].So tensoring with A or B, viewed as operator bimodules over themselves, gives theidentity functor up to natural isomorphism.

2.3. Adjunctions. Our aim is to study adjunction relations, in the usualsense of category theory, between the functors introduced above. So let A and Bbe operator algebras, and let E be an operator A-B-bimodule. In addition, let Fbe an operator B-A-bimodule. Following standard terminology, we say that F isleft adjoint to E, and that E is right adjoint to F , if there is a natural isomorphism

(2.10) CBB(F ⊗hA X,Y )∼=−→ CBA(X,E ⊗hB Y ),

as X ranges over all operator A-modules and Y ranges over all operator B-modules.The bijection is required to be simply a bijection of sets, but in fact it is au-

tomatically a uniformly (over X and Y ) completely bounded natural isomorphismof operator spaces, as the following simple lemmas make clear (the lemmas sim-ply place the unit/counit characterization of adjoint functors within the operatormodule context: compare [ML98, Chapter IV]).


Lemma 2.11. Associated to each natural isomorphism (2.10) there is a com-pletely bounded A-bimodule map

η : A −→ E ⊗hB F

(the unit of the adjunction) with the property that the composition

CBB(F ⊗hA X,Y )(2.10)−→ CBA(X,E ⊗hB Y )

∼=−→ CBA(A⊗hA X,E ⊗hB Y )

sends a morphism T : F ⊗hA X → Y to the composition

A⊗hA Xη⊗id−→ E ⊗hB F ⊗hA X

id⊗T−→ E ⊗hB Y.

Proof. The definition of η is very simple (and standard). Take X = A andY = F in the isomorphism (2.10) to obtain

CBB(F ⊗hA A,F )∼=−→ CBA(A,E ⊗hB F ).

Then define η to be the image on the right hand side of the canonical elementF ⊗hA A → F on the left given by the module action. The map η defined in thisway is a priori just a left A-module map, but the naturality of (2.10) implies it isa right A-module map too.

The proof that the isomorphism (2.10) is given by the formula in the lemma is astraightforward consequence of naturality of the isomorphism once again, togetherwith the following claim: the isomorphism

CBB(F ⊗hA X,F ⊗hA X)(2.10)−→ CBA(X,E ⊗hB F ⊗hA X)

∼=−→ CBA(A⊗hA X,E ⊗hB F ⊗hA X)

takes the identity operator on F ⊗hA X to η ⊗ idX . As for the claim, denote by

S : X −→ E ⊗hB F ⊗hA X

the image of the identity operator on F ⊗hA X under (2.10). From the commutingdiagram

CBB(F ⊗hA X,F ⊗hA X)

��

�� CBA(X,E ⊗hB F ⊗hA X)

��

CBB(F ⊗hA A,F ⊗hA X) �� CBA(A,E ⊗hB F ⊗hA X)

CBB(F ⊗hA A,F ⊗hA A)

��

�� CBA(A,E ⊗hB F ⊗hA A)

��

in which both squares are associated, by the naturality of (2.10), to a c.b. A-modulemap from A into X, we see that S is equal to η⊗ idX on the image of any A → X.But these images are dense in X, so the claim is proved. �

Similarly:

Lemma 2.12. Associated to each natural isomorphism (2.10) there is a com-pletely bounded B-bimodule map

ε : F ⊗hA E −→ B


(the counit of the adjunction) with the property that the inverse of the composition

CBA(X,E ⊗hB Y )(2.10)←− CBB(F ⊗hA X,Y )

∼=←− CBB(F ⊗hA X,B ⊗hB Y )

sends a morphism S : X → E ⊗hB Y to the composition

F ⊗hA Xid⊗S−→ F ⊗hA E ⊗hA Y

ε⊗id−→ B ⊗hB Y. �The unit and counit of an adjunction are linked by standard identities, and

conversely any appropriate pair of linked bimodule maps gives an adjunction:

Lemma 2.13. Given an adjunction (2.10) and maps

η : A −→ E ⊗hB F and ε : F ⊗hA E −→ B

as in the previous lemmas, the two compositions

A⊗hA Eη⊗id−−−→ E ⊗hB F ⊗hA E

id⊗ε−−−→ E ⊗hB B −→ E

and

F ⊗hA Aid⊗η−−−→ F ⊗hA E ⊗hB F

ε⊗id−−−→ B ⊗hB F −→ F

are the the canonical isomorphisms induced from the left and right A- and B-moduleactions on E and F , respectively. Conversely this data determines an adjunctionisomorphism. �

For the proof, compare for example [ML98, Chapter IV] once again.

Example 2.14. Let A be a closed subalgebra of B satisfying

AB = B = BA.

Let E = B, considered as an operator A-B-bimodule. The corresponding tensorproduct functor

OpModB −→ OpModAsimply associates to an operator B-module its restriction to an operator A-module.Then define F = B, considered as an operator B-A-bimodule. The associatedtensor product functor X �→ B ⊗A X is left adjoint to E. The maps

η : A −→ E ⊗hB F and ε : F ⊗hA E −→ B

given by the formulas η(a1a2) = a1 ⊗ a2 and ε(b1 ⊗ b2) = b1b2 are the unit andcounit of an adjunction.

2.4. An Adjunction Theorem from Hilbert C*-Modules. Hilbert C∗-modules provide a very simple set of instances of the ideas from the previous section.To see this, we need to first recall some elegant observations, due to Blecher [Ble97],that link operator spaces to Hilbert C∗-modules. See also [BLM04, Chapter 8], aswell as [Lan95] for an introduction to Hilbert C∗-modules.

Let E be a right Hilbert C∗-module over a C∗-algebra B. The matrix spaceMn(E) is naturally a Hilbert C∗-module over Mn(B), with inner product⟨

[eij ], [fij ]⟩=[∑

k〈eki, fkj〉],

and in this way we give E the structure of an operator space and a right operatorB-module.

A bounded, adjointable operator between Hilbert C∗-B-modules is automati-cally completely bounded with the same norm (in fact this is true for any boundedB-module map, whether or not it is adjointable).


We are especially interested in the situation where a Hilbert C∗-B-module E isequipped with a left action of a second C∗-algebra A by bounded and adjointableoperators. One sometimes calls E a C∗-correspondence from A to B, and everysuch correspondence is an operator A-B-bimodule.

Now if E is any operator space, then its adjoint E∗ is the complex conjugatevector space, equipped with the norms∥∥[eij ]∥∥Mn(E∗)

=∥∥[eji]∥∥Mn(E)

,

which endow E∗ with the structure of an operator space. See [BLM04, Section1.2.25]. If E is an operator A-B-bimodule, where A and B are C∗-algebras, thenE∗ is an operator B-A-bimodule via the formula

b · e∗ · a = (a∗ · e · b∗)∗.Let us apply this construction to the situation in which E is a Hilbert C∗-B-

module, as follows. Denote by KB(E) the C∗-algebra of B-compact operators onE, that is, the closed linear span of all bounded adjointable operators on E of theform

e1 ⊗ e∗2 : e �−→ e1〈e2, e〉.The tensor product notation is particularly apt in view of the following very elegantand useful calculation of Blecher.

Lemma 2.15. [BLM04, Corollary 8.2.15]. The above formula defines a com-pletely contractive map

κ : E ⊗hB E∗ −→ KB(E),

and this map is in fact a completely isometric isomorphism. �

The Haagerup tensor product also fits with Hilbert module theory in a secondway:2

Lemma 2.16. [CCH16a, Lemma 3.17]. If E is a C∗-A-B-correspondence,then the inner product induces a completely contractive map

E∗ ⊗hA E −→ B

of operator B-B-bimodules. �

The lemmas lead to the following simple, sufficient condition for a C∗-corre-spondence E to admit a left adjoint when viewed as a functor

E : OpModB −→ OpModA .

Theorem 2.17. Let A and B be C∗-algebras, and let E be a C∗-correspondencefrom A to B. If the action of A on E is through B-compact operators, then theoperator B-A-bimodule E∗ is left adjoint to E.

Proof. The action of A on E gives rise to a ∗-homomorphism

α : A −→ KB(E),

and hence, by Lemma 2.15, to a c.b. A-A-bimodule map

η : A −→ E ⊗hB E∗.

2A third very elegant connection, which like the first is due to Blecher, will be indicated inLemma 3.5.


On the other hand by Lemma 2.16 the inner product on E gives us a c.b. B-bimodulemap

ε : E∗ ⊗hA E −→ B.

We claim that these are the unit and counit, respectively of an adjunction. Ac-cording to Lemma 2.13 to prove this it suffices to show that the compositions

(2.18) A⊗hA Eη⊗id−−−→ E ⊗hB E∗ ⊗hA E

id⊗ε−−−→ E ⊗hB B −→ E

and

(2.19) E∗ ⊗hA Aid⊗η−−−→ E∗ ⊗hA E ⊗hB E∗ ε⊗id−−−→ B ⊗hB E∗ −→ E∗

are the the canonical isomorphisms induced from the left and right A- and B-moduleactions on E and E∗, respectively.

The composition

E ⊗hB E∗ ⊗hA Eid⊗ε−→ E ⊗hB B

∼=−→ E

is given on elementary tensors by the formula

e1 ⊗ e∗2 ⊗ e3 �→ e1〈e2, e3〉 = κ(e1 ⊗ e∗2)e3,

where κ is the completely isometric isomorphism of Lemma 2.15. On the otherhand, the map

A⊗hA Eη⊗id−→ E ⊗hB E∗ ⊗hA E

is given by the formula

a⊗ e ∈ E �−→ κ−1(α(a))⊗ e.

Combining these two computations, we find that the composition (2.18) is

A⊗hA E � a⊗ e �−→ κ−1(α(a))⊗ e �−→ α(a)e ∈ E,

as required. The second composition (2.19) is treated similarly. The composition

E∗ ⊗hA E ⊗hB E∗ ε⊗id−−−→ B ⊗hB E∗ −→ E∗


e∗1 ⊗ e2 ⊗ e∗3 �→ 〈e1, e2〉e∗3 = (e3〈e2, e1〉)∗ = (κ(e2 ⊗ e∗3)

∗e1)∗ ,

while the map

E∗ ⊗hA Aid⊗η−−−→ E∗ ⊗hA E ⊗hB E∗


e∗ ⊗ a �−→ e∗ ⊗ κ−1(α(a)).

So the composition (2.19) is

e∗ ⊗ a �−→ e∗ ⊗ κ−1(α(a)) �−→ (α(a)∗e)∗,

and the image is e∗α(a), as required. �


3. Operator Modules and Parabolic Induction

We turn now to representations of groups. Let G be a real reductive group.For definiteness, let us assume, more precisely, that G is the group of real pointsof a connected reductive group defined over R, as we did in [CCH16a], althoughwhat we have to say would certainly apply to a broader class of examples. On theother hand the special linear and general linear groups will suffice to illustrate theresults of this paper.

We shall be interested in (continuous) unitary representations of G, and usually,in particular, in representations that are weakly contained in the regular representa-tion, and so correspond to nondegenerate representations of the reduced C∗-algebraof G.

Let P be a parabolic subgroup of G, with Levi decomposition

P = LN.

For example if G is a general linear, or special linear, group, then up to conjugacyP is a subgroup of block-upper-triangular matrices, L is the subgroup of block-diagonal matrices, and N is the subgroup of block-upper-triangular matrices withidentity diagonal blocks. See [Kna96, Section VII.7] for the general definitions.

The functor of (normalized) parabolic induction,

IndGP : HilbC∗(L) −→ HilbC∗(G),

associates to a unitary representation π : L → U(H) (or equivalently, a nondegen-erate representation of the full group C∗-algebra) the Hilbert space completion ofthe space of smooth functions{

f : G → H : f(g�n) = π(�)−1δ(�)−12 f(g)

}in the inner product

〈f1, f2〉 =∫K

〈f1(k), f2(k)〉H dk,

where K is a maximal compact subgroup of G. Here δ : L → (0,∞) is the smoothhomomorphism defined by

δ(�) = det(Ad� : n → n

),

where n is the Lie algebra of N and Ad denotes the adjoint action.The presence of the normalizing factor δ−

12 ensures that the Hilbert space so

obtained is a unitary representation of G under the left translation action. If theoriginal representation is weakly contained in the regular representation, then sois the parabolically induced representation. For all this see for example [Kna86,Chapter VII].

3.1. Parabolic Induction and Hilbert C*-Modules. Pierre Clare beganthe study of parabolic induction from the point of view of modules and bimodulesover operator algebras in [Cla13].

Clare realized the functor of normalized parabolic induction as the tensor prod-uct with an explicit C∗-correspondence C∗

r (G/N), from C∗r (G) to C∗

r (L), which isobtained as a completion of the space of continuous, compactly supported functionson the homogeneous space G/N in a natural (normalized, using δ) inner productvalued in C∗

r (L). Thus he exhibited a natural isomorphism

IndGP H ∼= C∗r (G/N)⊗C∗

r (L) H


of functors from HilbC∗r (L) to HilbC∗

r (G). See [Cla13, Section 3] or [CCH16a,Section 4].

Remark 3.1. Actually Clare considered the full group C∗-algebra in [Cla13].Here we shall follow the approach in [CCH16a] and work with the reduced C∗-algebra, and the associated reduced version of Clare’s bimodule. The theorem thatwe shall present below holds in either context, but for later purposes it is moreappropriate for us to work with the reduced C∗-algebra.

The Hilbert module picture of parabolic induction as a tensor product allowsus to define parabolic induction of operator modules,

IndGP : OpModC∗r (L) −→ OpModC∗

r (G)

using the Haagerup tensor product:

IndGP X = C∗r (G/N)⊗hC∗

r (L) X.

Remark 3.2. By a famous theorem of Harish-Chandra [HC53, Theorem 6,p.230], every real reductive group is of type I; indeed it is liminal. So Proposi-tion 2.4, concerning irreducible objects in the categories OpModC∗(G) andOpModC∗

r (G) applies.

Within the context of operator modules it is natural and simple to consider, inaddition to C∗

r (G/N), the adjoint operator space C∗r (G/N)∗, which is an operator

C∗r (L)-C

∗r (G)-bimodule. We obtain from the tensor product formula

ResGP X = C∗r (G/N)∗ ⊗hC∗

r (G) X

a functorResGP : OpModC∗

r (G) −→ OpModC∗r (L),

that we shall call parabolic restriction.The considerations of Section 2.4 lead immediately and very simply to the

following Frobenius reciprocity theorem within the operator module context.

Theorem 3.3. Parabolic restriction is left-adjoint to parabolic induction, asfunctors on operator modules. Thus there is a natural isomorphism

CBC∗r (L)(Res

GP X,Y ) ∼= CBC∗

r (G)(X, IndGP Y )

for all operator C∗r (G)-modules X and all operator C∗

r (L)-modules Y .

Proof. In [CCH16a, Proposition 4.5] we showed that the action of C∗r (G) on

the C∗-correspondence C∗r (G/N) is by compact operators. The result is therefore

an immediate consequence of Theorem 2.17. �Remark 3.4. The same argument shows that C∗(G) acts by compact operators

on C∗(G/N), and so there is an analogue of Theorem 3.3 for operator modules overthe full group C∗-algebras.

3.2. Local Adjunction. Let us contrast the Frobenius reciprocity theoremproved in the previous section with the situation for categories of Hilbert C∗-modules.

In [CCH16a] we were able to show, using considerable input from representa-tion theory, that the operator bimodule C∗

r (G/N)∗ in fact carries the structure ofa C∗-correspondence. In other words its operator space structure is induced froma C∗

r (G)-valued inner product.


It needs to be stressed that this circumstance depends in a delicate way onissues in representation theory; in fact our explicit formula for the inner product isderived from Harish-Chandra’s Plancherel formula. There is for example no similarinner product within the context of full group C∗-algebras.

In any case, we can use Kasparov’s interior tensor product operation [Lan95,Chapter 4] to define parabolic induction and restriction functors

Ind: C*ModC∗r (L) −→ C*ModC∗

r (G)

and

Res : C*ModC∗r (G) −→ C*ModC∗

r (L)

between categories of (right) Hilbert C∗-modules and adjointable operators betweenHilbert C∗-modules.

Kasparov’s interior tensor product is related to the Haagerup tensor productin a very simple way:

Lemma 3.5. [Ble97, Theorem 4.3]. Let E be a C∗-A-B-correspondence andlet F be a C∗-B-C-correspondence. The natural completely bounded map

E ⊗hB F −→ E ⊗B F

from the Haagerup tensor product to the Kasparov tensor product is a completelyisometric isomorphism. �

See also [BLM04, Theorem 8.2.11]. But despite the lemma, and despite thetheorem proved in the previous section, it is not true that the two functors above areadjoint to one another. Instead, the best result available is that there are naturalisomorphisms

(3.6) KC∗r (L)(ResX,Y ) ∼= KC∗

r (G)(X, IndY )

between the spaces of compact adjointable operators. See [CCH16b, Theorem5.1].

In contrast to all this, our operator module result, Theorem 3.3, is stronger andrelies only on the fact that C∗

r (G) acts through compact operators on C∗r (G/N).

This is in turn an easy consequence of the geometry of G, involving no representa-tion theory; the essential point is that the homogeneous space G/P is compact.

3.3. SL(2,R). In order to explore the issues of the previous section a bitfurther, let us consider the special case of the group SL(2,R).

The general structure of the reduced C∗-algebra of a real reductive group issummarized in [CCH16a, Theorem 6.8]. We won’t repeat the general story here,but instead we shall focus on SL(2,R) alone. This example is also treated in[CCH16a, Example 6.10].3

Up to conjugacy there is a unique nontrivial parabolic subgroup inG=SL(2,R),namely the group P of upper triangular matrices, with Levi factor L the diagonalmatrices in SL(2,R). The (necessarily one-dimensional) irreducible unitary rep-resentations of L divide into two classes—the even representations where

[−1 00 −1

]3There is a long prior history of results on this topic (the reference [CCH16a] is certainly

not a primary source) and we won’t repeat that either, except to mention [BM76, Section 4],where the reader can find a prior set of full details for the SL(2,R) calculation.


acts as 1, and the odd representations where it acts as −1. There is accordingly adirect sum decomposition

C∗r (L)

∼= C∗r (L)even ⊕ C∗

r (L)odd

in which the even representations factor through the projection onto the even sum-mand, and the odd representations factor through the projection onto the oddsummand. Both summands are isomorphic to C0(R) as C

∗-algebras.Parabolically inducing the even and odd unitary representations of L, we obtain

the even and odd principal series representations of G.Apart from principal series, among the irreducible unitary representations of G

that are weakly contained in the regular representation there are also the cuspidalrepresentations. By definition, these are the irreducible unitary representations ofG that are weakly contained in the regular representation but do not embed in anyprincipal series representation; according to a basic calculation they are preciselythe irreducible square-integrable representations of G, also called the discrete seriesof G.

Associated to this division of the representations of C∗r (G) into three types

(cuspidal, plus even and odd principal series) there is a three-fold direct sum de-composition

C∗r (G) ∼= C∗

r (G)cuspidal ⊕ C∗r (G)even ⊕ C∗

r (G)odd.

Finally, there is a compatible direct sum decomposition

C∗r (G/N) ∼= C∗

r (G/N)even ⊕ C∗r (G/N)odd

under which the reduced C∗-algebras of both G and L act on the even and oddparts through the projections onto their respective even and odd summands.

In what follows we shall concentrate on the even summands. The odd sum-mands are similar, but a bit harder to describe in the case of C∗

r (G). However thesituation as regards adjunctions is actually simpler and less interesting for the oddsummands, and this is the reason that we shall concentrate on the even parts. Thecuspidal part of C∗

r (G) plays no role at all, since it acts trivially on C∗r (G/N).

There is a C∗-algebra isomorphism

C∗r (G)even ∼= C0

(R,K(H)

)Z/2Zwhere H is a separable infinite-dimensional Hilbert space, and the two-elementgroup Z/2Z acts on R by multiplication by −1, while it acts on K(H) trivially.

There is an isomorphism of Hilbert modules

C∗r (G/N)even ∼= C0(R, H)

under which

(a) The left action of C∗r (G)even becomes the obvious pointwise action under the

isomorphisms given above.(b) The right action of C∗

r (L)even is by pointwise multiplication under the identi-fication of C∗

r (L)even with C0(R), and the inner product is the pointwise innerproduct.

(c) The C∗r (G)-valued inner product on C∗

r (G/N)∗even takes values in the idealC∗

r (G)even, and is given by

〈f1, f2〉C∗r (G) =

12f1 ⊗ f∗

2 + 12w(f1)⊗ w(f2)

∗,


where f1, f2 ∈ C0(R, H), where

w(f)(x) = f(−x),

and where the tensors on the right hand side are to be viewed as rank oneadjointable operators on C0(R, H).

From all of this, and keeping in mind the obvious Morita equivalence

C0

(R,K(H)

)Z/2Z ∼Morita

C0(R)Z/2Z,

we find that the problem of formulating an adjunction theorem for the C∗-corres-pondence C∗

r (G/N) comes down to the same for the data

A = C0(R)Z/2Z, B = C0(R), E = C0(R),

with E being regarded as a C∗-A-B-correspondence in the obvious way.Frobenius reciprocity in the operator-module setting (Theorem 3.3) reduces

here to the simple case considered in Example 2.14: the unit

η : A −→ E ⊗hB E∗

is the inclusion (the tensor product is canonically isomorphic to B via the product),while the counit

ε : E∗ ⊗hA E −→ B

is the product.In contrast, the local adjunction isomorphism (3.6) in the Hilbert C∗-module

setting is equivalent to the assertion that the conjugate operator space structureon E∗ coincides with one induced by an A-valued inner product, namely the innerproduct

〈f1, f2〉A = 12f

∗1 f2 +

12w(f

∗1 f2).

The failure of the local adjunction isomorphism to extend to an isomorphism on alladjointable operators is a consequence of the fact that the counit ε defined above isa completely bounded map of B-bimodules, but not an adjointable map of Hilbertmodules when the Haagerup tensor product is identified with Kasparov’s internaltensor product using Lemma 3.5.

4. The Second Adjoint Theorem

For smooth representations of reductive p-adic groups, Bernstein made the re-markable discovery that parabolic induction has not only a left adjoint, but alsoa right adjoint. The right adjoint, like the left adjoint, is given by parabolic re-striction, but with respect to the opposite parabolic subgroup (the transpose). See[Ber87], or, for an exposition, [Ren10, Chapter VI].

Bernstein’s second adjoint theorem plays an important foundational role in therepresentation theory of p-adic groups, leading to a direct product decompositionof the category of smooth representations into component categories. See for ex-ample [Ren10, Chapter VI] again. Similar structure can be seen in the temperedrepresentation theory of both real and p-adic reductive groups, and one of the mainmotivations for the work presented in [CCH16b,CCH16a] was to obtain some-thing similar to Bernstein’s theorem in categories of representations related to thereduced group C∗-algebra.

The local adjunction isomorphism of [CCH16b] that we described in Sec-tion 3.2 is a partial solution. But it is not altogether satisfactory, since in the


p-adic context Bernstein’s theorem is a geometric foundation from which repre-sentation theory may be built up,4 whereas our local adjunction theorem requiredan extensive acquaintance with tempered representation theory to formulate andprove.

So the question remains whether or not a suitable counterpart of Bernstein’ssecond adjoint theorem can be developed in an operator-algebraic context. Weshall investigate this issue in detail elsewhere; our purpose here is to present twocomputations in the simple case of the group SL(2,R) that together indicate apossibly interesting role here for operator algebras and operator modules.

4.1. Harish-Chandra’s Schwartz space. If G is a real reductive group, asbefore, then its Harish-Chandra algebra is a Frechet convolution algebra HC(G) ofsmooth, complex valued functions on G that is perhaps easiest to present here as adistinguished subalgebra of C∗

r (G) that is closed under the holomorphic functionalcalculus.

The definition of HC(G) is a bit involved. Moreover it is not by any meansobvious, even after one has mastered the definitions, that HC(G) is closed underconvolution multiplication (see for example [Wal88, Section 7.1] for the details).We shall avoid these difficulties here by using a Fourier-dual description of HC(G)that will suffice for our present limited purposes; see the next section.

In any case, we shall study the following module category. In the context ofFrechet spaces, in this section and the next, the symbol ⊗ will denote the completedprojective tensor product of Frechet spaces.

Definition 4.1. Let A be a Frechet algebra (that is, a Frechet space equippedwith a (jointly) continuous and associative multiplication operation). A smoothFrechet module over A is a Frechet space V which is equipped with a continuousA-module structure, such that the evaluation map

A⊗A V → V a⊗ v �→ av

is an isomorphism.

Remark 4.2. The tensor product A ⊗A V used in the above definition is thequotient of the completed projective tensor product A⊗ V by the closed subspacegenerated by the balancing relators

a1a2 ⊗ v − a1 ⊗ a2v

with a1, a2 ∈ A and v ∈ V .

Definition 4.3. We denote by SFModA the category of smooth Frechet mod-ules over A, with continuous A-linear maps as morphisms.

If E is a smooth A-B-Frechet bimodule, then the tensor product constructionin Remark 4.2 gives us a tensor product functor

E : SFModB −→ SFModA .

We shall study parabolic induction from the perspective of such functors in the nextsection. We should remark that if A is a Frechet algebra, then it is not necessarilytrue that the multiplication map

A⊗A A −→ A4In this context see the recent article [BK13] for a striking geometric conceptualization of

Bernstein’s original theorem.


is an isomorphism, but this is true for the Harish-Chandra algebras that we shallbe studying.

4.2. The Harish-Chandra algebra of SL(2,R). We shall now specializeto G = SL(2,R) and its parabolic subgroup P = LN of upper triangular matrices.

The Harish-Chandra algebra for L admits a decomposition

HC(L) = HC(L)even ⊕HC(L)odd,that is compatible with the decomposition of the reduced group C∗-algebra. Boththe even and odd summands are isomorphic as Frechet algebras to the space S(R)of Schwartz functions on the line, with pointwise multiplication.

Similarly there is a decomposition

HC(G) = HC(G)cuspidal ⊕HC(G)even ⊕HC(G)odd

that is compatible with the decomposition of the reduced C∗-algebra in Section 3.3.Once again we shall concentrate on the even parts. There is an isomorphism

HC(G)even ∼= S (R,K(H))Z/2Z

in which the algebra appearing on the right is as follows.

(a) The Hilbert space H has a preferred orthonormal basis indexed by even integers(the Hilbert space carries an SO(2) representation, and the basis vectors areweight vectors).

(b) The right-hand algebra consists of continuous functions f from R into thecompact operators on H, invariant under the same Z/2Z action as before.

(c) If p is any continuous seminorm on the space of Schwartz functions on the line,and if fij denotes the ij-matrix entry of f with respect to the given orthonormalbasis of H, then p(fij) is of rapid decay in i and j.

Compare [Art75] and [Var89, Chapter 8].Finally there is the bimodule HC(G/N), which consists of suitable rapid decay

functions on G/N , as in [Wal92, Section 15.3]. There is a decomposition

HC(G/N) = HC(G/N)even ⊕HC(G/N)odd

as before, and there is an isomorphism

HC(G/N)even ∼= S(R, H),

where on the right hand side are the functions f : R → H whose component func-tions fj are of rapid decay with respect to any Schwartz space seminorm, as in (c)above.

Since there is again a Morita equivalence

S (R,K(H))Z/2Z ∼

MoritaS(R)Z/2Z

(that is, an equivalence of SFMod categories) we are finally reduced to studyingadjunction theorems in the following Frechet context:

A = S(R)Z/2Z, B = S(R), E = S(R),with E being assigned the structure of a smooth A-B-bimodule in the obvious way.

So far this is of course an uninteresting reworking of the computations thatwe made in Section 3.3. And the situation with regard to Frobenius reciprocity issimilarly predictable: if we define

F = S(R),


with its obvious B-A-bimodule structure, then, exactly as before:

Theorem 4.4. The bimodule maps

η : A −→ E ⊗B F and ε : F ⊗A E −→ Bdefined by

η(a1a2) = a1 ⊗ a2 and ε(f ⊗ e) = fe

are the unit and counit of an adjunction. �

But the situation with regard to “Bernstein reciprocity,” or the assertion thatE also has a right adjoint, is much more interesting. Surprisingly, in view of thefact that in most respects the Frechet algebra HC(G) behaves much like C∗

r (G),there is a striking difference between the two regarding the second adjoint theorem,which in fact does hold in the Harish-Chandra context.

We wish to define a candidate unit map

B −→ F ⊗A Eas follows:

(4.5) b1b2 �→ b1x⊗ b2 + b1 ⊗ xb2

for b1, b2 ∈ B (we are writing x for the function x �→ x). It is not immediatelyobvious that the formula is well-defined. But the following calculation shows thatthis is so:

Lemma 4.6. The quantity in F ⊗A E described in (4.5) depends only on theproduct b1b2 ∈ B, and the formula defines a continuous B-bimodule homomorphism.

Proof. Let us first show that if b ∈ B, thenb1bx⊗ b2 + b1b⊗ xb2 = b1x⊗ bb2 + b1 ⊗ xbb2.

We can write

b = a1 + a2x,

where a1, a2 ∈ A, and it suffices to consider separately the cases where a1 = 0 anda2 = 0. The latter is easy, since the tensor products are over A. As for the former,we calculate that

b1(a2x)x⊗ b2 + b1(a2x)⊗ xb2 = b1 ⊗ a2x2b2 + b1x⊗ a2xb2

= b1x⊗ a2xb2 + b1 ⊗ xa2xb2,

as required (we used the fact that x2a2 ∈ A). So the formula defines a continuousmap

B ⊗B B −→ F ⊗A E ,and the lemma follows from the easily verified fact that the multiplication map

B ⊗B B −→ Bis an isomorphism. �

Remark 4.7. Bernstein constructed the unit map for his second adjunctionusing the geometry of the homogeneous space G/N , and in particular the fact thatif P = LN is the opposite parabolic subgroup, then the product map

N × L×N −→ G


embeds the left hand side as an open subset of G. See for example [Ber92, Section3.1]. It is not immediately apparent, but the unit described here is essentially thesame, and differs only in that we have used the function x �→ x in place of (thereciprocal of) Harish-Chandra’s c-function from the theory of spherical functions.The c-function arises when one calculates Bernstein’s unit map from the spectral,or Fourier dual, perspective.

We can now prove a counterpart of Bernstein’s second adjoint theorem:

Theorem 4.8. The bimodule map given by the formula (4.5) is the unit mapfor an adjunction

HomB(Y,F ⊗B X) ∼= HomA(E ⊗B Y,X).

Proof. In order to prove the theorem we need to find a suitable counit map

E ⊗B F → A.

We shall use the formula

(4.9) e⊗ f �→ 1

x(ef)−

in which the superscript “−” on the right means that we take the odd part of thefunction ef ∈ B (a superscript “+” will likewise denote the even part of a function).

The composition

E ⊗B B −→ E ⊗B F ⊗A E −→ A⊗A E −→ Eis given by the formula

e⊗ b1b2 �→ e⊗ b1x⊗ b2 + e⊗ b1 ⊗ xb2

�→ 1

x(eb1x)

− ⊗ b2 +1

x(eb1)

− ⊗ xb2

�→ 1

x(eb1x)

−b2 +1

x(eb1)

−xb2

= (eb1)+b2 + (eb1)

−b2

= eb1b2,

and this is the standard multiplication map, as required. In addition the composi-tion

B ⊗B F −→ F ⊗A E ⊗B F −→ F ⊗A A −→ Fis given by the formula

b1b2 ⊗ f �→ b1x⊗ b2 ⊗ f + b1 ⊗ xb2 ⊗ f

�→ b1x⊗ 1

x(b2f)

− + b1 ⊗1

x(fb2x)

−

�→ b1(b2f)− + b1

1

x(fb2x)

−

= b1(b2f)− + b1(b2f)

+,

which gives us the standard module multiplication map once again, as required. �Remark 4.10. In the present context of Harish-Chandra spaces, the bimodules

HC(G/N) and HC(G/N) are in fact isomorphic to one another, so it is not possibleto detect the use of the opposite parabolic subgroup, except indirectly throughthe geometric role it plays in giving the formula for the unit map, as indicated inRemark 4.7.


4.3. Bernstein’s Theorem and Operator Spaces. In this final section weshall adapt the Schwartz algebra computations of the previous section to the contextof operator algebras.

The formula (4.9) for the Bernstein counit does not make sense for arbitrarycontinuous functions, and so does not make sense at the level of (reduced) groupC∗-algebras. We will show however that the Bernstein reciprocity theorem of theprevious section can be recovered after replacing C∗-algebras with non-self-adjointoperator algebras.

Given f ∈ C0(R), we shall continue to use the notation

f = f+ + f−

for the decomposition of f into its even and odd parts. We shall also denote by

w : C0(R) −→ C0(R)

the involution given by the formula

w(f)(x) = f(−x).

Let us now fix a smooth function c on the line (with a singularity at 0 ∈ R) withthe following properties:

(a) c is odd,(b) c(x) = 1/x for x near 0 ∈ R, and(c) c(x) = 1 for large positive x.

The notation is supposed to call to mind Harish-Chandra’s c-function, which is theultimate source of the function c(x) = 1/x that appears in the previous section;see Remark 4.7. We are simplifying matters somewhat here by insisting that 1/cis a smooth function, bounded at infinity (in the natural construction of the unitmap, involving the actual c-function from representation theory, the boundednesscondition does not hold). But this is a relatively minor issue; see Remark 4.17below.

Definition 4.11. We shall denote by B ⊆ C0(R) the space of those functionsf ∈ C0(R) for which the product c · f− extends to a continuous (and necessarilyeven) function on R. Equivalently B consists of those functions in C0(R) whoseodd part is differentiable at 0 ∈ R.

Lemma 4.12. The formula

δ(b) = c · b−

defines a w-twisted derivation from B into C0(R), so that

δ(b1b2) = δ(b1)b2 + w(b1)δ(b2).

for all b1, b2 ∈ B. As a result the formula

(4.13) b �−→[

b 0δ(b) w(b)

]defines an algebra embedding of B into the algebra 2× 2 matrices over C0(R). �

Lemma 4.14. The image of B in M2(C0(R)) under the embedding (4.13) is anorm-closed subalgebra. �


We shall equip the algebra B with the operator algebra structure it receivesfrom the embedding (4.13). In addition, let A be the C∗-algebra of even functionsin C0(R). It embeds completely isometrically into B, and

AB = B = BA

(indeed the closures are superfluous).Let E = B, considered as an operator A-B-bimodule, and let F = B considered

as an operatorB-A-bimodule. Frobenius reciprocity, or the assertion that the tensorproduct functor

F : OpModA −→ OpModB

is left-adjoint to the tensor product functor

E : OpModB −→ OpModA

holds as in Example 2.14. But in addition these modules satisfy the followingversion of Bernstein reciprocity:

Theorem 4.15. The tensor product functor E is left-adjoint to the tensor prod-uct functor F : there is a natural isomorphism

CBB(Y, F ⊗hA X) ∼= CBA(E ⊗hB Y,X).

Proof. We want to define a unit map

η : B −→ F ⊗hA E

by the formula

(4.16) b1b2 �−→ b1c⊗ b2 + b1 ⊗

b2c.

The formula gives a well-defined map by the argument of Lemma 4.6, which applieshere because every element of B is of the form a1 + a2/c. The map is completelybounded because the function 1

c is a bounded multiplier of C0(R). Clearly η is aB-bimodule map.

In addition, define a counit map

ε : E ⊗hB F −→ A

by the formula

e⊗ f �−→ c(ef)−.

This is certainly an A-bimodule map. It can be viewed as the composition

B ⊗hB B �� Bδ �� A

in which the first map is just the multiplication map on B, which is completelybounded. As for δ, it is the restriction to B of the completely bounded map

T �−→[0 1

]T

[10

]from M2(C0(R)) to C0(R), and so it too is completely bounded.

The verification, now, that η and ε are the unit and counit of an adjunction isexactly as in the proof of Theorem 4.8. �


Remark 4.17. If the function 1/c was unbounded (as it would be if we wereto use the natural, representation-theoretic c-function), then our formula (4.16) forthe unit map η would give an unbounded, densely-defined B-bimodule map. Itsdomain, an ideal in B, would be an operator algebra in its own right, and we couldrepeat the above argument with this algebra in place of the original B.

References

[Art75] J. Arthur, A theorem on the Schwartz space of a reductive Lie group, Proc. Nat. Acad.Sci. U.S.A. 72 (1975), no. 12, 4718–4719. MR0460539 (57 #532)

[Ber87] J.N. Bernstein, Second adjointness for representations of reductive p-adic groups,Preprint, 1987.

[Ber92] J.N. Bernstein. Representations of p-adic groups. Notes by K.E. Rumelhart, 1992.[BK14] J. Bernstein and B. Krotz, Smooth Frechet globalizations of Harish-Chandra modules,

Israel J. Math. 199 (2014), no. 1, 45–111, DOI 10.1007/s11856-013-0056-1. MR3219530[BK13] R. Bezrukavnikov and D. Kazhdan. Geometry of second adjointness for p-adic groups.

Represent. Theory 19 (2015), 299–332. MR3430373[Ble97] D. P. Blecher, A new approach to Hilbert C∗-modules, Math. Ann. 307 (1997), no. 2,

253–290, DOI 10.1007/s002080050033. MR1428873 (98d:46063)

[BLM04] D. P. Blecher and C. Le Merdy, Operator algebras and their modules—an operatorspace approach, London Mathematical Society Monographs. New Series, vol. 30, TheClarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications.MR2111973 (2006a:46070)

[BM76] R. Boyer and R. Martin, The regular group C∗-algebra for real-rank one groups, Proc.Amer. Math. Soc. 59 (1976), no. 2, 371–376. MR0476913 (57 #16464)

[CCH16a] P. Clare, T. Crisp, and N. Higson. Parabolic induction and restriction viaC*-algebras and Hilbert C∗-modules. Composito Math., FirstView (2016), DOI10.1112/S0010437X15007824

[CCH16b] P. Clare, T. Crisp, and N. Higson. Adjoint functors between categories of Hilbert C∗-modules. To appear in J. Inst. Math. Jussieu (2016).

[Cla13] P. Clare, Hilbert modules associated to parabolically induced representations, J. Oper-ator Theory 69 (2013), no. 2, 483–509, DOI 10.7900/jot.2011feb07.1906. MR3053351

[CH] T. Crisp and N. Higson, A second adjoint theorem for SL(2, R), arXiv:1603.08797,2016.

[DH68] J. Dauns and K. H. Hofmann, Representation of rings by sections, Memoirs of theAmerican Mathematical Society, No. 83, American Mathematical Society, Providence,R.I., 1968. MR0247487 (40 #752)

[HC53] Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I,Trans. Amer. Math. Soc. 75 (1953), 185–243. MR0056610 (15,100f)

[Kad57] R. V. Kadison, Irreducible operator algebras, Proc. Nat. Acad. Sci. U.S.A. 43 (1957),273–276. MR0085484 (19,47e)

[Kna86] A. W. Knapp, Representation theory of semisimple groups, Princeton Mathematical

Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based onexamples. MR855239 (87j:22022)

[Kna96] A. W. Knapp, Lie groups beyond an introduction, Progress in Mathematics, vol. 140,Birkhauser Boston, Inc., Boston, MA, 1996. MR1399083 (98b:22002)

[Lan95] E. C. Lance, Hilbert C∗-modules, London Mathematical Society Lecture Note Series,vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator alge-braists. MR1325694 (96k:46100)

[ML98] S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts inMathematics, vol. 5, Springer-Verlag, New York, 1998. MR1712872 (2001j:18001)

[Ped79] G. K. Pedersen, C∗-algebras and their automorphism groups, London MathematicalSociety Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Pub-lishers], London-New York, 1979. MR548006 (81e:46037)

[Ren10] D. Renard, Representations des groupes reductifs p-adiques (French), Cours Specialises[Specialized Courses], vol. 17, Societe Mathematique de France, Paris, 2010.MR2567785 (2011d:22019)


[Rie74] M. A. Rieffel, Induced representations of C∗-algebras, Advances in Math. 13 (1974),176–257. MR0353003 (50 #5489)

[Var89] V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups,Cambridge Studies in Advanced Mathematics, vol. 16, Cambridge University Press,Cambridge, 1989. MR1071183 (91m:22018)

[Wal88] N. R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132,Academic Press, Inc., Boston, MA, 1988. MR929683 (89i:22029)

[Wal92] N. R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132,Academic Press, Inc., Boston, MA, 1992. MR1170566 (93m:22018)

Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany


Department of Mathematics, Penn State University, University Park, Pennsylvania

16802



Spectral multiplicity and odd K-theory-II

Ronald G. Douglas and Jerome Kaminker

Dedicated to Dick Kadison, colleague and mentor on his 90th birthday

Abstract. Let {Dx} be a family of unbounded self-adjoint Fredholm opera-tors representing an element of K1(M). Consider the first two components ofthe Chern character. It is known that these correspond to the spectral flow of

the family and the index gerbe. In this paper we consider descriptions of theseclasses, both of which are in the spirit of holonomy. These are then studiedfor families parametrized by a closed 3-manifold. A connection between themultiplicity of the spectrum (and how it varies) and these classes is developed.

1. Introduction

In a previous paper, [7], we studied the set of unbounded, self-adjoint, Fred-holm operators with compact resolvent, filtered by the maximum dimension ofeigenspaces. One goal was to relate the vanishing of the components of the Cherncharacter of their K-theory class to bounds on the multiplicity of the spectrum. Inthe present note we will give a simple uniform description of the first two terms inthe odd Chern character and study them in the case of certain families over a closed3-manifold. This short paper is preliminary to carrying out a similar analysis forthe component of the Chern character in degree 5.

The authors would like to thank Alan Carey for valuable discussions and hos-pitality while visiting Australian National University.

2. Invariants of families of operators

We will consider families of operators {Dx} where x ∈ M , M a closed smoothmanifold. The operators will be unbounded, self-adjoint, Fredholm operators withcompact resolvent. We follow the notation and ideas of the paper, [7], and referthere for details. The (spectral) graph of the family is the closed subset Γ({Dx}) ={(x, λ)|λ ∈ sp(Dx)} ⊆ M × R. If the graph is not connected, then the familydetermines the trivial element in K1(M). In [7], the approach taken was to adapttopological obstruction theory to determine when one can deform the family toone with a disconnected graph. The obstructions met along the way were relatedto the components of the Chern character of the K-theory class defined by thefamily. Under the assumption that the multiplicity of the spectrum of the familywas bounded by 2, it was determined that the first obstruction corresponded to

2010 Mathematics Subject Classification. Primary 19K56, 58J30, 46L87.Key words and phrases. K-theory, unbounded selfadjoint Fredholms.


109

110 RONALD G. DOUGLAS AND JEROME KAMINKER

spectral flow and the second to the index gerbe. In the next two subsections wewill give elementary definitions of these classes. To this end, we note that there isa finite cover of the parameter space M , {Ui}, i = 1, . . . , N , and real numbers λi,such that λi is not in the spectrum of Dx, for x ∈ Ui. Moreover, we can and willassume that the sets in the cover and their finite intersections are contractible. Wewill build the invariants relative to this data.

2.1. Spectral flow. We will define a Cech cocycle in Z1({Ui},Z), where Zdenotes the presheaf of Z valued functions on M . Fix an open set Ui in the coverand, for x ∈ Ui, let PUi

(x) be the orthogonal projection onto the subspace of Hspanned by the eigenspaces of Dx for eigenvalues greater than λi. Note that PUi

(x)varies continuously in x. Suppose that Uj is another element of the cover andthat x ∈ Ui ∩ Uj . If λi > λj , then PUi

(x) − PUj(x) is a finite rank projection.

Thus one may associate to the (ordered) pair of projections their codimension,dim(PUi

(x), PUj(x)) ∈ Z. However, for simplicity in stating and proving the next

proposition, we will use the notion of essential codimension, ec(PUi(x), PUj

(x)),which agrees with the usual notion of codimension in the case above. We referto [2] for the definitions and properties. To this end, we define c(Ui, Uj)(x) =ec(PUi

(x), PUj(x)). Since Ui ∩Uj is connected, the value is independent of x. This

leads to the following result.

Proposition 2.1. The function c is a 1-dimensional cocycle in Z1({Ul},Z)whose image, sf({Dx}) = [c] ∈ H1(M,Z) ⊆ H1(M,Q) is equal to the spectral flowclass of the family {Dx}, which is a rational multiple of the first component of theChern character of the K-theory class represented by the family.

Proof. Using the definition of spectral flow described in [7], the result followsby a direct modification. �

2.2. Index gerbe. We will construct a 3-dimensional integral cohomologyclass on M and show that it agrees with the Dixmier-Douady invariant of the indexgerbe. This is based on the work of Carey, Mickelsson and others, [4, 5, 9, 11].We again start with the family of projections defined over the sets in the opencover, {Ui}. Fix a point x ∈ Ui. The projection PUi

(x) determines a quasi-freerepresentation of the CAR algebra, αUi

(x) : CAR(H) → B(FPUi(x)(H)), where

FPUi(x)(H) = F(PUi

(x)H)⊗ F((I − PUi(x))H) is the Fermionic Fock space. As a

reference to that theory we refer to [1]. If x ∈ Ui ∩ Uj , then we have two differentrepresentations, αUi

(x) and αUj(x), which are equivalent. This follows since the

projections yielding them differ by a finite rank projection and one can constructa canonical intertwining unitary operator, (see Appendix),

SPUj(x),PUi

(x) : FPUj(x)(H) → FPUi

(x)(H),

satisfying

(1) SPUj(x),PUi

(x)αUi(x)S∗

PUj(x),PUi

(x) = αUj(x).

Note that SPUj(x),PUi

(x) is defined only up to a choice of a scalar z ∈ S1. This

ambiguity disappears in the following,

(2) SPUj(x),PUi

(x) = AdSPUj(x),PUi

(x): B(FPUj

(x)(H)) → B(FPUi(x)(H)).

SPECTRAL MULTIPLICITY AND ODD K-THEORY-II 111

Next, suppose that x ∈ Ui ∩ Uj ∩ Uk, for sets Ui, Uj , Uk in the cover. Thenthere exists a function g(Ui, Uj , Uk) : Ui ∩ Uj ∩ Uk → S1 such that one has

(3) SPUj(x),PUi

(x) ◦ SPUi(x),PUk

(x) ◦ SPUk(x),PUj

(x) = g(Ui, Uj , Uk)(x) ∈ S1,

since the composition on the left intertwines an irreducible representation of CAR(H),and hence is a scalar multiple of the identity. We will show that g(Ui, Uj , Uk)(x)can be defined so as to be a continuous function of x. To see this, note first that theprojections onto the finite dimensional differences, Hi,j(x), are norm continuous.Fix a point x0 ∈ Ui ∩ Uj ∩ Uk. We claim there is a continuous family of unitaries,RUi,Uj

(x) with the property that RUi,Uj(x)(Hi,j(x)) = Hi,j(x0). Choosing an or-

thonormal basis for Hi,j(x0), a volume element can be determined in a continuousmanner for each Hi,j(x). This is what is needed to define the intertwining unitariesin a continuous way. See the appendix for more on this matter. We will be usingsheaf cohomology with respect to the presheaf of S1-valued functions on M , [3].

Proposition 2.2. The family, {g(Ui, Uj , Uk)} defines a cocycle in C2({Ui}, S1)

and, hence, a cohomology class [g] ∈ H2({Ui}, S1).

Proof. This is a direct computation. �There is a natural map H2({Ui}, S1) → H3(M,Z), which is an isomorphism if

the cover is well chosen. We will denote the image of [g] by G({Dx}) ∈ H3(M,Z).One can check that it depends on the family {Dx} only up to homotopy. Everyelement in this group is the Dixmier-Douady element of an appropriate equivalenceclass of gerbes. We next identify this class.

Theorem 2.3. The class G({Dx}) is equal to the Dixmier-Douady invariantof the index gerbe of the family as defined by Lott, [9].

Proof. Following Lott, we use the definition of a gerbe, as described byHitchin, [10]. Thus, there are line bundles LUi,Uj

over Ui ∩ Uj possessing thenecessary properties. It is easily seen that tensoring the bundle of Fock spacesover Ui ∩ Uj with LUi,Uj

is the same as applying the family of unitary operatorsSPUj

(x),PUi(x), (see Appendix). The result follows from this observation. �

Remark 2.3.1. As noted in [9] this class maps to the component in degree 3of the Chern character of the family.

Remark 2.3.2. A triple (ωi, θij , gijk), where ωi ∈ Ω2(Ui), θij ∈ Ω1(Ui ∩ Uj),and gijk : Ui ∩ Uj ∩ Uk → S1, satisfying certain conditions, determines a Delignecohomology class inH3

D(X,Z(3)), as in [3] p. 216. It is shown there that equivalenceclasses of gerbes with curving and connection are in 1-1 correspondence with thegroup H3

D(X,Z(3)). Thus, from the exact sequence,

0 −−−−−−→ Ω2(X)/Ω2Z(X)

i−−−−−→ H3D(X,Z(3))

π−−−−−−→ H3(X,Z) −−−−−−→ 0

one sees that there are many Deligne classes with the same Dixmier-Douady invari-ant.

In the present context there is a natural choice of connection obtained by pullingback the Chern connection from the Hilbert space bundle over the Grassmannian.If our family is obtained geometrically as a family of Dirac operators on odd-dimensional closed manifolds, then it follows from [9] that there is a natural choicefor curving yielding a specific Deligne class.


3. A low dimensional example

In this section we will analyze certain families over a closed 3-manifold. Firstrecall that when the spectral flow of the family is zero one is able to decompose thegraph into a union of levels indexed by the integers. Among the families that havethis property we single out those satisfying:

Condition (A): Level G0 can have a non-empty intersection with only G1 orG−1, and G0 ∩G−1 ∩G1 = ∅.

For such families we will provide a complete classification using the methodsof [7]. It is possible, under more general circumstances, to reduce a family to onesatisfying Condition (A), but it is more complicated and we leave that to a laterpaper.

We make some preliminary observations. Recall the addition operation forfamilies. If {Dx} and {D′

x} are families on the Hilbert spacesH andH′ respectively,then the sum of the families is the direct sum on the Hilbert space H ⊕ H′. Thegraph of the sum is the union of the graphs of each summand, and the multiplicityof eigenvalues is the sum of the multiplicities on the intersection of the graphsand the multiplicity for the individual families on the symmetric difference of thegraphs. The inverse of a family is obtained by reversing the signs of the eigenvalues,or by replacing {Dx} by {−Dx}.

Let {Dx} be a family parametrized by a smooth, closed, connected mani-fold, M . Consider sf({Dx}) ∈ H1(M,Z) = [M,S1]. Let α : M → S1 repre-sent this class. Let {D′

z} be a family on S1 with spectral flow equal to 1. Then{Dx} − α∗({D′

z}) has spectral flow zero. Note that, in general, one can’t do ananalogous construction to eliminate the index gerbe class in H3(M,Z) because ofthe possibility of torsion. However, if M is a closed 3-manifold then this reductioncan be done and we will make use of this in what follows.

Now, assume we are given a family, {Dx}, parametrized by a smooth closed3-dimensional manifold M . Let Γ = Γ({Dx}) ⊆ M ×R with projection π : Γ → Mbe the graph. We will assume that sf({Dx}) = 0, so that we have a spectraldecomposition,

(4) Γ =⋃k∈Z

Gk.

Condition (A) implies that the family has multiplicity ≤ 2 on G0.

Remark 3.0.3. It is not apparent that one can establish (A) for arbitraryfamilies using the “moves” developed in [7]. Although we are able to apply themto a family with multiplicity less than or equal to 3 to obtain one satisfying (A),the general case will require more elaborate techniques which will be addressedlater. Thus, we will assume now that (A) holds and describe next how we can usethe methods of [7] to ensure that the set π(G0 ∩G1), is contained in a ball whosecomplement contains π(G0 ∩G−1).

Proposition 3.1. Given a family satisfying (A), one can apply the procedureof [7] to obtain a family satisfying the additional property that there is a closed ball,B ⊆ M , satisfying π(G0 ∩G1) ⊆ B and π(G0 ∩G−1) ⊆ M \B.


Proof. We proceed as in [7, p. 325]. Since the multiplicity is at most 2 onG0, we can find open sets with disjoint closures, V and W contained in M , withπ(G0 ∩G1) ⊆ W and π(G0 ∩G−1) ⊆ V . Triangulate M finely enough so that anysimplex which meets π(G0∩G1) is contained in W and any which meets (G0∩G−1)is contained in V . Now we proceed, inductively over skeleta, to deform the familyso that G0 is separated from G1 and G−1, over the 0, 1 and 2 skeletons. At thisstage, G0∩G1 and G0∩G−1 for the deformed families are contained in the interiorsof 3-simplices, and hence in sets homeomorphic to 3-balls. Now, if we remove fromM the 3-balls containing G0∩G−1, the result is path connected. Thus we may findpaths between the 3-balls containing G0 ∩ G1 and avoiding the 3-balls containingG0 ∩ G−1, and then thicken them to be tubes. This can be done in such a waythat the result is a single closed 3-ball containing G0 ∩ G1 with G0 ∩ G−1 in itsexterior. �

We continue our construction. Let gi : M → Gi be the cross section map whichsends x to pr2(({x}×R)∩Gi), i.e. it sends x to the eigenvalue of Dx on Gi. Thereis a 2-plane bundle, E, over the 2-sphere boundary of B, whose fiber at a point xis the space of eigenvectors of g0(x) and g1(x) in H. This bundle extends over Band, hence, is trivial. There is a splitting of E|∂B as the direct sum of two linebundles, L0 and L1. The splitting is determined by the orthogonal eigenspaces forg0(x) and g1(x), for x in some ε neighborhood, W , of B, with W ⊆ M \G0 ∩G−1.Note that on W we have g0(x) < g1(x). Identifying ∂B with S2 the line bundle L0

is determined by the homotopy class of its clutching map κ : S1 → U1, and henceby an integer k. Moreover, the integer associated to L1 is −k. Unless k = 0, thesplitting will not extend over the ball, B. We fix this integer k associated to thefamily{Dx} and the ball B.

There is a tautologous family of operators over S3, [7,8,11]. It is obtained byidentifying x ∈ S3 with an element x ∈ SU(2). One then associates to x the Diracoperator on S1 twisted by the 2-plane bundle on S1 determined by x. This familyrepresents the element 1 ∈ K1(S3) ∼= Z. For an integer n, the family determined

by using xn will be denoted by {∂(n)x }. It is a family corresponding to the integer

n ∈ K1(S3).Next, observe that there is a degree one map c : M → S3 which sends the

interior of the ball B, int(B), to S3 \ {(0, 0,−1)} and M \ int(B) to (0, 0,−1).

Consider the family ˆ{Dx} = {Dx} + c∗({∂(−k)x }) over M where k is the integer

obtained above. We will show that this family is equivalent to a trivial family and,

hence, [{Dx}] = c∗([{∂(k)x }]) ∈ K1(M). This will provide canonical representatives

for classes in K1(M).

Proposition 3.2. The family ˆ{Dx} can be deformed to a family ˜{Dx} which

satisfies G0 ∩ G1 = ∅. Moreover, ˜{Dx} is equal to {Dx} + c∗({∂(−k)x }) outside a

neighborhood U ⊇ B, with U disjoint from G0 ∩G−1.

Proof. To verify this claim we will use the methods of [7]. Shrink B toB′ ⊆ B. We may arrange, by an initial homotopy of our families, that g0(x) =0 for x ∈ B′ for {Dx}, and g′0(x) = 0 for x ∈ c(B′) for c∗({∂−k

z }). Now we

apply the “flattening” operation, [7, p. 322], to both {Dx} and c∗({∂(−k)z }) on

B′ relative to B. This yields two new families, homotopic to the original ones,but with the eigenvalues g0(x), g1(x), g

′0(x), and g′1(x), constantly 0 on B′. The


span of their eigenspaces defines a 4-dimensional vector bundle E over B′. Morespecifically, the graph of the sum of the flattened families has the following structureon π−1(B′) ∩ G0. The multiplicity of the 0 eigenvalue is 4, so there is a (trivial)4-plane bundle, E, over B′ with a natural splitting on the boundary into a directsum of four line bundles, L0, L1, L

′0, and L′

1, where L′0, and L′

1 are associated with

c∗({∂(−k)z }) and L0, L1 with {Dx}. Now, L0 and L′

0 are determined by integerswhich are negatives of each other. Thus, the sum L0 ⊕ L′

0 is trivial, as is L1 ⊕ L′1.

This implies that the splitting E ∼= (L0 ⊕ L′0)⊕ (L1 ⊕ L′

1) extends across π−1(B′).

We may then use the “scaling operation” from [7, p. 324] to deform the family byincreasing the eigenvalue on the subspaces spanned by (L1 ⊕ L′

1) to a small value,ε > 0 on B′, decaying to 0 on B \B′, and leaving the family unchanged outside B

. One then obtains that G0 ∩ G1 = ∅ as required. �

Note that a family ˜{Dx} for which G0 ∩ G1 = ∅ satisfies that g0(x) < g1(x) forall x ∈ M , and hence the family is trivial.

We thus obtain,

Theorem 3.3. For a family satisfying condition (A), one has

(5) {Dx} " c∗({∂kz }),

where k is the degree of the bundle as above.

This yields the following theorem.

Theorem 3.4. Any family over a closed connected 3-manifold which satisfiescondition (A) is equivalent to one of the form α∗(sf) + c∗({∂k

z }), where sf is thefamily over the circle with spectral flow 1, and α : M → S1 is the map representingsf({Dx}).

Remark 3.4.1. This result can be deduced strictly using algebraic topology.However, we have shown something stronger. While the “moves” we used to passfrom the given family to the one in standard form imply homotopy of the families,the converse may not hold. Thus, if we define K1(M) to be the group generatedby families of operators modulo flattening and scaling, there is a surjection

(6) K1(M) → K1(M) → 0.

It would be interesting to know if it is not always injective.

Remark 3.4.2. The method of studying K1(M) we use, actually starts witha subset of M × R with an eigenspace associated to each point–that is, one hasan enhanced graph. Over sets of constant multiplicity one has a vector bundle.This could be viewed as a generalized local coefficient system on M and one couldthen try to define twisted cohomology groups. A theory along these lines wasdeveloped in [8]. It would also be useful to consider a notion of concordance,apriori weaker than homotopy, for enhanced graphs and obtain a result statingthat the concordance is trivial if a family of generalized characteristic classes agree.

Remark 3.4.3. Suppose we have a family, {Dx}, over a closed, oriented 3-manifold which satisfies condition (A). We can deform the family as in Proposition3.1 and proceed further so that G0 ∩G1 consists of a single point, {x}. Then k{x}defines a 0-dimensional homology class which is Poincare dual to the 3-dimensionalclass G({Dx}). This is analogous to a result of Cibotaru, [6].


4. Appendix

For the sake of exposition we will review the construction, c.f.[12], of the in-tertwining operators in the case at hand– that is, we assume we are given twoprojections P,Q with P − Q of dimension N. Given a Hilbert space H, we willdenote by H the conjugate Hilbert space. Decompose H using P and Q,

(7) H = PH⊕ (1− P )H = QH⊕ (1−Q)H.

Then we have

FQ(H) = F(QH)⊗F((P −Q)H)⊗F((I − P )H)

FP (H) = F(QH)⊗F((P −Q)H)⊗F((I − P )H).(8)

One defines an isomorphism SQ,P : FP (H) → FQ(H) via

(9) SQ,P = IF(QH) ⊗ SQ,P ⊗ IF((I−P )H),

where SP,Q : F((P −Q)H) → F((P −Q)H) is defined as follows. Choose a volumeelement ω ∈ F((P − Q)H), that is, a non-zero element of the top exterior power,

ΛN ((P −Q)H). Then SP,Q, on Λk((P −Q)H), is the composition,

Λk((P −Q)H)Θ−→ ΛN−k((P −Q)H)∗ →

→ ΛN−k(((P −Q)H)∗) → ΛN−k((P −Q)H),(10)

where the first map is Θ(x)(y) =< x ∧ y, ω > and the latter two are canonicalisomorphisms. Thus, the composite isomorphism depends only on the choice of ω.

One may view (10) as defining an isomorphism

(11)

N⊕k=0

(Λk((P −Q)H)

)⊗ ΛN ((P −Q)H) →

N⊕k=0

ΛN−k((P −Q)H).

If the projections, P and Q, depend continuously on x ∈ M , then (11) extendsto an isomorphism of bundles,

(12) F((P −Q)H)⊗ LP,Q → F((P −Q)H).

The map SQ,P is obtained by fixing a non-zero cross-section of LP,Q, and itdepends on that choice. In the cases considered in the paper one has that LP,Q istrivial.

References

[1] H. Araki, Bogoliubov automorphisms and Fock representations of canonical anticommu-tation relations, Operator algebras and mathematical physics (Iowa City, Iowa, 1985),Contemp. Math., vol. 62, Amer. Math. Soc., Providence, RI, 1987, pp. 23–141, DOI10.1090/conm/062/878376. MR878376 (88g:81043)

[2] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of C∗-algebras and K-homology,Ann. of Math. (2) 105 (1977), no. 2, 265–324. MR0458196 (56 #16399)

[3] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, ModernBirkhauser Classics, Birkhauser Boston, Inc., Boston, MA, 2008. Reprint of the 1993 edi-tion. MR2362847 (2008h:53155)

[4] A. Carey, Private communication.[5] A. Carey and J. Mickelsson, A gerbe obstruction to quantization of fermions on odd-

dimensional manifolds with boundary, Lett. Math. Phys. 51 (2000), no. 2, 145–160, DOI10.1023/A:1007676919822. MR1774643 (2001h:58029)

[6] D. Cibotaru, The odd Chern character and index localization formulae, Comm. Anal. Geom.19 (2011), no. 2, 209–276, DOI 10.4310/CAG.2011.v19.n2.a1. MR2835880


[7] R. G. Douglas and J. Kaminker, Spectral multiplicity and odd K-theory, Pure Appl. Math.Q. 6 (2010), no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer, 307–329,DOI 10.4310/PAMQ.2010.v6.n2.a2. MR2761849

[8] P. Kronheimer and T. Mrowka, Monopoles and three-manifolds, New Mathematical Mono-graphs, vol. 10, Cambridge University Press, Cambridge, 2007. MR2388043 (2009f:57049)

[9] J. Lott, Higher-degree analogs of the determinant line bundle, Comm. Math. Phys. 230(2002), no. 1, 41–69, DOI 10.1007/s00220-002-0686-3. MR1930571 (2003j:58052)

[10] N. Hitchin, Lectures on special Lagrangian submanifolds, Winter School on Mirror Symme-try, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud.Adv. Math., vol. 23, Amer. Math. Soc., Providence, RI, 2001, pp. 151–182. MR1876068(2003f:53086)

[11] J. Mickelsson, Current algebras and groups, Plenum Monographs in Nonlinear Physics,Plenum Press, New York, 1989. MR1032521 (90m:22044)

[12] A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, The ClarendonPress, Oxford University Press, New York, 1986. Oxford Science Publications. MR900587(88i:22049)

Department of Mathematics, Texas A&M University, College Station, Texas 77843-

3368


Department of Mathematics, IUPUI, Indianapolis, Indiana

Department of Mathematics, University of California Davis, Davis, California

95616



On the classification of simple amenable C*-algebras withfinite decomposition rank

George A. Elliott and Zhuang Niu

Dedicated to Richard V. Kadison on the occasion of his ninetieth birthday

Abstract. Let A be a unital simple separable C*-algebra satisfying the UCT.Assume that dr(A) < +∞, A is Jiang-Su stable, and K0(A) ⊗ Q ∼= Q. ThenA is an ASH algebra (indeed, A is a rationally AH algebra).

1. Introduction

Let A be a simple separable nuclear unital C*-algebra. In [20], Matui andSato showed that A ⊗ UHF can be tracially approximated by finite dimensionalC*-algebras (i.e., is TAF) if A is quasidiagonal with unique trace.

In this note, this result is enlarged upon as follows: the condition on the tracesimplex is removed, at the cost of assuming the UCT, (still) finite nuclear dimension,and (still) that all traces are quasidiagonal—e.g., by assuming finite decompositionrank—see [2]—and (so far) of restricting the K0-group to have torsion-free rankequal to one.

Theorem 1.1. Let A be a simple unital separable C*-algebra satisfying theUCT. If A⊗Q has finite decomposition rank and K0(A)⊗Q ∼= Q, then A⊗Q ∈ TAI(see Definition 2.11). In particular, A⊗Z is classifiable, where Z is the Jiang-Sualgebra ([12]).

This theorem can also be regarded as an abstract version (still in a specialcase) of the classification result of [10] and [7], where any simple unital locallyapproximately subhomogeneous C*-algebra is shown to be rationally tracially ap-proximated by Elliott-Thomsen algebras (1-dimensional noncommutative CW com-plexes) ([7]) and hence to be classifiable ([10]).

2. The main result and the proof

In this note let us use Q to denote the UHF algebra with K0(Q) ∼= Q, and letus use tr to denote the canonical tracial state of Q.

2010 Mathematics Subject Classification. Primary 46L35, 46L05.Key words and phrases. Classification of C*-algebras, C*-algebras with finite decomposition

rank, ASH algebras.The research of the first author was supported by a Natural Sciences and Engineering Re-

search Council of Canada (NSERC) Discovery Grant, and the research of the second author wassupported by a Simons Foundation Collaboration Grant.

©2016 American Mathematical Society

117

118 GEORGE A. ELLIOTT AND ZHUANG NIU

Definition 2.1 (N. Brown, [3]). Let A be a unital C*-algebra, and denote byTqd(A) the tracial states with the following property: For any (F , ε), there is aunital completely positive map φ : A → Q such that

(1) |τ (a)− tr(φ(a))| < ε, a ∈ F , and(2) ‖φ(ab)− φ(a)φ(b)‖ < ε, a, b ∈ F .

Remark 2.2. In the original definition of a quasidiagonal trace (Definition3.3.1 of [3]), the UHF algebra Q was replaced by a matrix algebra. It is easy to seethat these two approaches are equivalent.

Recall the tracial approximate uniqueness result of [6] and [16].

Theorem 2.3 (Theorem 4.15 of [6]; Theorem 5.3 of [16]). Let A be a simple,unital, exact, separable C*-algebra satisfying the UCT. For any finite subset F ⊆ Aand any ε > 0, there exist n ∈ N and a K-triple (P,G, δ) with the following property:For any admissible codomain B, and any three completely positive contractionsφ, ψ, ξ : A → B which are δ-multiplicative on G, with ξ unital, φ and ψ nuclear,and φ#(p) = ψ#(p) in K(B) for all p ∈ P, and such that φ(1) and ψ(1) are unitarilyequivalent projections, there exists a unitary u ∈ Un+1(B) such that∥∥∥∥u∗

(φ(a)

n · ξ(a)

)u−

(ψ(a)

n · ξ(a)

)∥∥∥∥ < ε, a ∈ F .

One may arrange that u∗(φ(1)⊕ n · 1)u = ψ(1)⊕ n · 1.

Remark 2.4. In the theorem above, n · ξ(a) (or n · 1) denotes the direct sumof n copies of ξ(a) (or 1). This notation is also used in the proof of Corollary 2.6below.

Remark 2.5. In the theorem above (and also Corollary 2.6 below), one assumesby convention that the finite subset G is sufficiently large and δ is sufficiently smallthat [φ(p)] is well defined for any p ∈ P if a map φ is δ-multiplicative on G.

When B = Q, in fact one does not have to consider all the K-theory withcoefficients. More precisely, one has

Corollary 2.6. Let A be a simple, unital, exact, separable C*-algebra sat-isfying the UCT. For any finite subset F ⊆ A and any ε > 0, there exist n ∈ Nand a K-triple (P,G, δ), with P ⊆ Proj∞(A), with the following property: For anythree completely positive contractions φ, ψ, ξ : A → Q which are δ-multiplicative onG, with φ(1) = ψ(1) = 1Q − ξ(1) a projection, [φ(p)]0 = [ψ(p)]0 in K0(Q) for allp ∈ P, and tr(φ(1)) = tr(ψ(1)) < 1/n, where tr is the canonical tracial state of Q,there exists a unitary u ∈ Q such that

‖u∗(φ(a)⊕ ξ(a))u− ψ(a)⊕ ξ(a)‖ < ε, a ∈ F .

Proof. Applying Theorem 2.3 to F and ε > 0, one obtains n0 ∈ N and a

K-triple (P,G, δ) with the property of Theorem 2.3. Set

n0 + 1 = n and P ∩ Proj∞(A) = P.

Let us show that n and (P,G, δ) have the desired property.Let φ, ψ, ξ : A → Q be completely positive contractions which are δ-multiplica-

tive on G, with φ(1) = ψ(1) = 1Q − ξ(1) a projection, [φ(p)]0 = [ψ(p)]0 in K0(Q)for all p ∈ P, and tr(φ(1)) = tr(ψ(1)) < 1/n.

CLASSIFICATION OF AMENABLE C*-ALGEBRAS 119

Decompose ξ approximately on F ⊆ A as a repeated direct sum

ξ′ ⊕ · · · ⊕ ξ′︸︷︷︸n0

,

where ξ′ : A → Q is again a completely positive contraction which is (necessarily, ifthe approximation is sufficiently good) δ-multiplicative on G, and (ξ′⊕· · ·⊕ξ′)(1A) =ξ(1A). Since tr(φ(1)) = tr(ψ(1)) < 1/n, one has that

tr(ξ(1)) > (n− 1)/n = n0/n,

and so

φ(1) = ψ(1) $ e,

where e = ξ′(1). Then the maps φ⊕ ξ and ψ ⊕ ξ have the forms

φ⊕ (n · ξ′), ψ ⊕ (n · ξ′) : A → Mn0+1(eQe),

respectively. Note that eQe is stably isomorphic to Q, and therefore

K0(eQe,Z/kZ) = {0}, k ∈ N \ {0}, and K1(eQe,Z/kZ) = {0}, k ∈ N ∪ {0}.Together with the assumption [φ(p)]0 = [ψ(p)]0 in K0(Q) for all p ∈ P, this implies

φ#(p) = ψ#(p) ∈ K(eQe), p ∈ P.

Thus, it follows from Theorem 2.3 that there is a unitary w ∈ Mn0+1(eQe) suchthat ∥∥∥∥w∗

(φ(a)

n0 · ξ′(a)

)w −

(ψ(a)

n0 · ξ′(a)

)∥∥∥∥ < ε, a ∈ F ,

and

(2.1) w∗(φ(1)⊕ n0 · e)w = ψ(1)⊕ n0 · e.Note that

φ(1)⊕ (n0 · e) = ψ(1)⊕ (n0 · e) = 1Q.

By (2.1), a straightforward calculation shows that

u := 1Qw1Q

is a unitary of Q. Clearly, if the approximation of ξ by ξ′⊕· · ·⊕ξ′ on F is sufficientlygood, then, in Q,

‖u∗(φ(a)⊕ ξ(a))u− ψ(a)⊕ ξ(a)‖ < ε, a ∈ F ,

as desired. �

Definition 2.7. Recall that an abelian group G is said to be of (torsion free)rank one if G⊗Q ∼= Q.

Lemma 2.8. Let Δ be a compact metrizable Choquet simplex. Then, for anyfinite subset F ⊆ Aff(Δ) and any ε > 0, there exist m ∈ N and unital (pointwise)positive linear maps � and θ,

Aff(Δ)�

�� Rm θ �� Aff(Δ),

where the unit of Rm is (1, . . . , 1), such that

‖θ(�(f))− f‖∞ < ε, f ∈ F .


Proof. By Theorem 5.2 of [14] and its corollary, there is an increasing se-quence of finite-dimensional subspaces of Aff(Δ) with dense union, containing thecanonical order unit 1 ∈ Aff(Δ), and such that each map Rmk → Rmk+1 andRmk ↪→ Aff(Δ) is positive, with respect to the canonical (pointwise) order rela-tions:

Rm1 ��

�� Rm2 ��

�� · · · ��

�� Aff(Δ).

(The authors are indebted to David Handelman for reminding us of [14].)Without loss of generality, one may assume that F ⊆ Rm1 , and hence one only

has to extend the identity map of Rm1 to a positive unital map � : Aff(Δ) → Rm1 .Write Rm1 = Re1⊕Re2⊕· · ·⊕Rem1

, and consider the unital positive functionals

ρi : Rm1 � (x1, x2, . . . , xm1

) �→ xi ∈ R, i = 1, . . . ,m1.

By the Riesz Extension Theorem ([21]), each ρi can be extended to a unital positivelinear functional ρi : Aff(Δ) → R. Then the map

� : Aff(Δ) � f �→ (ρ1(f), ρ2(f), . . . , ρm1(f)) ∈ Rm1

has the desired property. �Lemma 2.9. Let C = lim−→(Cn, ιn) be a unital inductive system of C*-algebras

such that C is simple. Let (Rm, ‖·‖∞ , u) be a finite-dimensional ordered Banachspace with order unit u, and let γ : Rm → Aff(T(C)) be a unital positive linearmap. Then, for any finite set F ⊆ Rm and any ε > 0, there are n and a unitalpositive linear map γn : Rm → Aff(T(Cn)) such that

‖γ(a)− ιn,∞ ◦ γn(a)‖ < ε, a ∈ F .

Proof. Denote by ei, i = 1, . . . ,m, the standard basis of Rm, and write

u = c1e1 + · · ·+ cmem,

where c1, . . . , cm > 0. Since C is simple, each affine function γ(ei) is strictly positiveon T(C). Since T(C) is compact, there is δi such that

(2.2) γ(ei)(τ ) > δi, τ ∈ T(C), 1 ≤ i ≤ m.

Without loss of generality, one may assume that F = {e1, e2, . . . , em}.Pick Cn and e′1, e

′2, . . . , e

′m−1 ∈ Aff(T(Cn)) such that

‖ιn,∞(e′i)− γ(ei)‖∞

< min{ε, δi2,

cmδm2(c1 + · · ·+ cm−1)

,cmε

2(c1 + · · ·+ cm−1)}, 1 ≤ i ≤ m− 1.

In particular, by (2.2),

ιn,∞(e′i)(τ ) ≥ δi/2, τ ∈ T(C), 1 ≤ i ≤ m− 1.

Setting

e′m :=1

cm(1− c1e

′1 − · · · − cm−1e

′m−1) ∈ Aff(T(Cn)),

one has

‖ιn,∞(e′m)− γ(em)‖∞ = ‖ 1

cm(1− c1ιn,∞(e′1)− · · · − cm−1ιn,∞(e′m−1))

− 1

cm(1− c1γ(e1)− · · · − cm−1γ(em−1))‖∞

≤ min{δm/2, ε}.


In particular, by (2.2),

ιn,∞(e′m)(τ ) ≥ δm/2, τ ∈ T(C).

Then, considering instead the images of e′1, e′2, . . . , e

′m in a building block further

out (replacing n by the later index), one may assume that

e′i(τ ) > δi/4, τ ∈ T(Cn), 1 ≤ i ≤ m.

In particular, all the affine functions e′i ∈ Aff(T(Cn)) are positive. Define γn :Rm → Aff(T(Cn0

)) byγn(ei) = e′i, 1 ≤ i ≤ m.

It is clear that γn satisfies the condition of the lemma. �

Theorem 2.10. Let A be a separable simple unital exact C*-algebra satisfyingthe UCT. Assume that T(A) = Tqd(A) and that K0(A) is of rank one. Then, forany finite set F ⊆ A⊗Q and any ε > 0, there are unital completely positive linearmaps φ : A⊗Q → I and ψ : I → A⊗Q, where I is an interval algebra, such that

(1) φ is F-δ-multiplicative, ψ is an embedding, and(2) |τ (ψ ◦ φ(a)− a)| < ε, a ∈ F , τ ∈ T(A⊗Q).

Proof. If T(A) = ∅, then the conclusion holds trivially (with I = {0}).Otherwise, assuming, as we may, that A ∼= A ⊗Q, we have K0(A) ∼= Q (as order-unit groups).

Apply Corollary 2.6 to A with respect to (F ·F , ε/4) to obtain n and (P,G, δ).Since K0(A) = Q (unique unital identification), we may suppose that P = {1A}.

By Theorem 3.9 of [23], there is a simple unital inductive limit C = lim−→(Ci, ιi)

such that K0(C) = Q (unital identification), Ci = Mki(C([0, 1])), the maps ιi are

injective, and there is an isomorphism

Ξ : Aff(T(A)) ∼= Aff(T(C)).

By Lemma 2.8, there is an approximate factorization, by means of unital posi-tive maps,

Aff(T(A))�

�� Rm θ �� Aff(T(A)),

such that‖θ(�(f))− f‖∞ < ε/16, f ∈ F .

Therefore, by Lemma 2.9, after discarding finitely many terms of the sequence(Ci, ιi), there is a unital positive linear map

γ : Aff(T(A))�

�� Rm �� Aff(T(C1))

such that

(2.3) ‖(ι1,∞)∗(γ(f))− Ξ(f)‖∞ < ε/8, f ∈ F .

Denote by γ∗ : T(C1) → T(A) the affine map induced by γ on tracial simplices.Since γ factors through Rm (so that γ∗ factors through a finite dimensional simplex),there are τ1, . . . , τm ∈ T(A) and continuous functions c1, c2, . . . , cm : [0, 1] → [0, 1]such that

(2.4) γ∗(τt) = c1(t)τ1 + c2(t)τ2 + · · ·+ cm(t)τm, t ∈ [0, 1],

and

c1(t) + c2(t) + · · ·+ cm(t) = 1, t ∈ [0, 1],


where τt ∈ T(C1) is determined by the Dirac measure concentrated at t ∈ [0, 1].Since τ1, τ2, . . . , τm ∈ Tqd(A), there are unital completely positive linear maps

φk : A → Q, k = 1, 2, . . . ,m, such that each φk is G-δ-multiplicative, and

(2.5) |tr(φk(f))− τk(f)| < ε/16m, f ∈ F .

For each t ∈ [0, 1], there is a open neighbourhood U such that for any s ∈ U ,one has

|ck(s)− ck(t)| < 1/4mn.

(Recall that n is the constant from Corollary 2.6, as in the second paragraph of theproof.) Since [0, 1] is compact, there is a partition 0 = t0 < t1 < · · · < tl−1 < tl = 1such that

(2.6) |ck(s)− ck(tj)| < 1/4mn, s ∈ [tj−1, tj ].

Moreover, we may assume that this partition is fine enough that

(2.7) |γ(f)(τt)− γ(f)(τtj )| < ε/8, f ∈ F , t ∈ [tj−1, tj ].

For each j = 0, 1, . . . , l, pick rational numbers rj,1, rj,2, . . . , rj,m ∈ [0, 1] suchthat

rj,1 + · · ·+ rj,m = 1

and

(2.8) |rj,k − ck(tj)| < min{ε/16m, 1/4mn}, k = 1, . . . ,m.

Write rj,k = qj,k/p where qj,k, p ∈ N, and then define

ϕj := (φ1 ⊕ · · · ⊕ φ1︸︷︷︸qj,1

)⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm︸︷︷︸qj,m

) : A → Q.

Note that it follows from (2.4), (2.5), and (2.8) that

(2.9)∣∣tr(ϕj(f))− γ∗(τtj )(f)

∣∣ < ε/4, f ∈ F .

By (2.8), (2.6), one has that

(2.10)|qj,k − qj+1,k|

p<

1

mn, k = 1, . . . ,m, j = 0, . . . , l − 1.

For each j = 0, . . . , l − 1, compare the direct sum maps

ϕj = (φ1 ⊕ · · · ⊕ φ1︸︷︷︸qj,1

)⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm︸︷︷︸qj,m

)

and

ϕj+1 = (φ1 ⊕ · · · ⊕ φ1︸︷︷︸qj+1,1

)⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm︸︷︷︸qj+1,m

),

and consider the common direct summand of these two maps,

ψj := (φ1 ⊕ · · · ⊕ φ1︸︷︷︸min{qj,1,qj+1,1}

)⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm︸︷︷︸min{qj,m,qj+1,m}

).

By (2.10), one has

|tr(1− ψj(1))| =1

p

m∑k=1

|qj,k − qj+1,k| <1

n.


On the other hand, since ϕj and ϕj+1 are unital, one has

[(ϕj % ψj)(1A)]0 = 1− tr(ψj(1A)) = [(ϕj+1 % ψj)(1A)]0.

Recall that P = {1A}. By the conclusion of Corollary 2.6 there is a unitary uj+1

such that ∥∥ϕj(f)− u∗j+1ϕj+1(f)uj+1

∥∥ < ε/4, f ∈ F · F , 0 ≤ j ≤ l − 1.

Define v0 = 1, and set

ujuj−1 · · ·u1 = vj , j = 1, . . . , l.

Then, for any 0 ≤ j ≤ l − 1 and any f ∈ F · F , one has

‖Ad(vj) ◦ ϕi(f)−Ad(vj+1) ◦ ϕj+1(f)‖= ‖(uj · · ·u1)

∗ϕi(f)(uj · · ·u1)− (uj+1 · · ·u1)∗ϕj+1(f)(uj+1 · · ·u1)‖

=∥∥ϕj(f)− u∗

j+1ϕj+1(f)uj+1

∥∥ < ε/4.

Replacing each homomorphism ϕj by Ad(vj)◦ϕj for j = 1, . . . , l, and still denotingit by ϕj , one has

(2.11) ‖ϕj(f)− ϕj+1(f)‖ < ε/4, f ∈ F · F , 0 ≤ j ≤ l − 1.

Define a unital completely positive linear map φ : A → C1 by

φ(f)(t) :=tj+1 − t

tj+1 − tjϕj(f) +

t− tjtj+1 − tj

ϕj+1(f), if t ∈ [tj , tj+1].

Then, by (2.11), the map φ is F-ε-multiplicative. By (2.9) and (2.7), one has

(2.12) ‖φ∗(f)− γ(f)‖∞ < ε/2, f ∈ F .

Note that A and C have cancellation for projections, and also K+0 (A)=K+

0 (C)=Q+ (unital identification) and Aff(T(A)) ∼= Aff(T(C)). By Theorem 4.4 and Corol-lary 6.8 of [9] (see also Theorem 2.6 of [5] and Theorem 5.5 of [4], expressedin terms of W instead of Cu), it follows that the Cuntz semigroup of A andthe Cuntz semigroup of C are isomorphic. Applied to the canonical unital mapCu(C1) → Cu(C) ∼= Cu(A), Theorem 1 of [22] implies that there is a unital homo-morphism ψ : C1 → A giving rise to this map, and in particular such that

(2.13) ψ∗ = Ξ−1 ◦ (ι1,∞)∗ on Aff(T(C1)).

Since the ideal of Cu(C1) killed by the map Cu(C1) → Cu(C) ∼= Cu(A) is zero,as the map C1 → C is an embedding, it follows that the map C1 → A is also anembedding. By (2.12), (2.13), and (2.3), one then has

‖φ∗ ◦ ψ∗(f)− f‖∞ < ε, f ∈ F ,

as desired. �

Recall that

Definition 2.11 ([15], [8]). Let S be a class of unital C*-algebras. A C*-algebra A is said to be tracially approximated by the C*-algebras in S, and onewrites A ∈ TAS, if the following condition holds: For any finite set F ⊆ A, anyε > 0, and any non-zero a ∈ A+, there is a non-zero sub-C*-algebra S ⊆ A suchthat S ∈ S, and if p = 1S , then

(1) ‖pf − fp‖ < ε, f ∈ F ,(2) pfp ∈ε S, f ∈ F , and


(3) 1− p is Murray-von Neumann equivalent to a subprojection of aAa.

Denote by I the class of interval algebras, i.e.,

I = {C([0, 1])⊗ F : F is a finite dimensional C*-algebra}.TAI, then, is the class of C*-algebras which can be tracially approximated byinterval algebras.

For TAI algebras, based on Winter’s deformation technique ([25] and [18])and on [19], one has the following classification theorem.

Theorem 2.12 (Corollary 11.9 of [17]). Let A,B be unital separable amenablesimple C*-algebras satisfying the UCT. Assume that A, B are Jiang-Su stable,and assume that A ⊗ Q ∈ TAI and B ⊗ Q ∈ TAI. Then A ∼= B if and only ifEll(A) ∼= Ell(B).

The following is the main result of this note, which asserts that certain abstractC*-algebras are covered by the classification theorem above.

Theorem 2.13. Let A be a separable simple unital C*-algebra satisfying theUCT. Assume that A ⊗ Q has finite nuclear dimension, T(A) = Tqd(A), andK0(A)⊗Q = Q (identification of order-unit groups). Then A⊗Q ∈ TAI.

Proof. This follows from Theorem 2.10 above and Theorem 2.2 of [24] di-rectly. �

Proof of Theorem 1.1. By Proposition 8.5 of [2], as A⊗Q has finite decom-position rank, T(A⊗Q) = Tqd(A⊗Q). Furthermore, by [13], A⊗Q is stably finiteand nuclear and so by [1] and [11], T(A) �= ∅. Then K0(A⊗Q) = Q (as order-unitgroups), and the statement follows from Theorem 2.13. (The classifiability of A⊗Zholds by Theorem 2.12.) �

References

[1] B. Blackadar and M. Rørdam, Extending states on preordered semigroups and the existenceof quasitraces on C∗-algebras, J. Algebra 152 (1992), no. 1, 240–247, DOI 10.1016/0021-8693(92)90098-7. MR1190414 (93k:46049)

[2] J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White, and W. Winter. Covering dimension ofC*-algebras and 2-coloured classification. 06 2015. URL: http://arxiv.org/abs/1506.03974.

[3] N. P. Brown, Invariant means and finite representation theory of C∗-algebras, Mem.Amer. Math. Soc. 184 (2006), no. 865, viii+105, DOI 10.1090/memo/0865. MR2263412(2008d:46070)

[4] N. P. Brown, F. Perera, and A. S. Toms, The Cuntz semigroup, the Elliott conjecture, anddimension functions on C∗-algebras, J. Reine Angew. Math. 621 (2008), 191–211, DOI10.1515/CRELLE.2008.062. MR2431254 (2010a:46125)

[5] N. P. Brown and A. S. Toms, Three applications of the Cuntz semigroup, Int. Math. Res. Not.IMRN 19 (2007), Art. ID rnm068, 14, DOI 10.1093/imrn/rnm068. MR2359541 (2009a:46104)

[6] M. Dadarlat and S. Eilers, On the classification of nuclear C∗-algebras, Proc. LondonMath. Soc. (3) 85 (2002), no. 1, 168–210, DOI 10.1112/S0024611502013679. MR1901373(2003d:19006)

[7] G. A. Elliott, G. Gong, H. Lin, and Z. Niu. The classification of simple separable unital locallyASH-algebras. 06 2015. URL: http://arxiv.org/abs/1506.02308.

[8] G. A. Elliott and Z. Niu, On tracial approximation, J. Funct. Anal. 254 (2008), no. 2, 396–440, DOI 10.1016/j.jfa.2007.08.005. MR2376576 (2009i:46116)

[9] G. A. Elliott, L. Robert, and L. Santiago, The cone of lower semicontinuous traces ona C∗-algebra, Amer. J. Math. 133 (2011), no. 4, 969–1005, DOI 10.1353/ajm.2011.0027.MR2823868 (2012f:46120)


[10] G. Gong, H. Lin, and Z. Niu. Classification of finite simple amenable Z-stable C*-algebras.01 2015. URL: http://arxiv.org/abs/1501.00135.

[11] U. Haagerup, Quasitraces on exact C∗-algebras are traces (English, with English and Frenchsummaries), C. R. Math. Acad. Sci. Soc. R. Can. 36 (2014), no. 2-3, 67–92. MR3241179

[12] X. Jiang and H. Su, On a simple unital projectionless C∗-algebra, Amer. J. Math. 121 (1999),no. 2, 359–413. MR1680321 (2000a:46104)

[13] E. Kirchberg and W. Winter, Covering dimension and quasidiagonality, Internat. J. Math.

15 (2004), no. 1, 63–85, DOI 10.1142/S0129167X04002119. MR2039212 (2005a:46148)[14] A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are L1 spaces and their repre-

senting matrices, Acta Math. 126 (1971), 165–193. MR0291771 (45 #862)[15] H. Lin, Tracially AF C∗-algebras, Trans. Amer. Math. Soc. 353 (2001), no. 2, 693–722 (elec-

tronic), DOI 10.1090/S0002-9947-00-02680-5. MR1804513 (2001j:46089)[16] H. Lin, Stable approximate unitary equivalence of homomorphisms, J. Operator Theory 47

(2002), no. 2, 343–378. MR1911851 (2003c:46082)[17] H. Lin, Asymptotic unitary equivalence and classification of simple amenable C∗-algebras,

Invent. Math. 183 (2011), no. 2, 385–450, DOI 10.1007/s00222-010-0280-9. MR2772085(2012c:46157)

[18] H. Lin, Localizing the Elliott conjecture at strongly self-absorbing C∗-algebras, II, J. ReineAngew. Math. 692 (2014), 233–243. MR3274553

[19] H. Lin and Z. Niu, Lifting KK-elements, asymptotic unitary equivalence and clas-sification of simple C∗-algebras, Adv. Math. 219 (2008), no. 5, 1729–1769, DOI10.1016/j.aim.2008.07.011. MR2458153 (2009g:46118)

[20] H. Matui and Y. Sato, Decomposition rank of UHF-absorbing C∗-algebras, Duke Math. J.163 (2014), no. 14, 2687–2708, DOI 10.1215/00127094-2826908. MR3273581

[21] M. Riesz. Sur le probleme des moments. iii. Ark. F. Mat. Astr. O. Fys, 17(16):1–52, 1923.[22] L. Robert, Classification of inductive limits of 1-dimensional NCCW complexes, Adv. Math.

231 (2012), no. 5, 2802–2836, DOI 10.1016/j.aim.2012.07.010. MR2970466[23] K. Thomsen, Inductive limits of interval algebras: the tracial state space, Amer. J. Math.

116 (1994), no. 3, 605–620, DOI 10.2307/2374993. MR1277448 (95f:46099)[24] W. Winter. Classifying crossed product C*-algebras. 08 2013. URL: http://arxiv.org/abs/

1308.5084.[25] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C∗-algebras, J. Reine

Angew. Math. 692 (2014), 193–231. MR3274552

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

M5S 2E4


Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071



Topology of natural numbers and entropy of arithmeticfunctions

Liming Ge

Dedicated to Professor Richard V. Kadison on the occasion of his ninetieth birthday.

Abstract. Compactification of natural numbers is studied in association withtheir arithmetics. Stone-Gelfand-Naimark’s theory on function algebras andabelian C*-algebras is applied to characterize the associated compact Haus-dorff spaces. Entropy for arithmetic functions is introduced. It is shown thatzero entropy functions form a C*-algebra. Sarnak’s Mobius Disjointness Con-jecture is studied in association with this algebra and its maximal ideal space.Connections between geometry and number theory are further discussed.

1. Introduction and definitions

Operator Algebras, a little younger than Dick Kadison and being nurturedby von Neumann, Kadison and several others, has matured and become an im-portant branch of Mathematics. S. S. Chern once commented that Number The-ory is an applied science—any good mathematics should have deep connectionsand applications in it. Although Operator Algebras has its roots in QuantumPhysics, its connections with Number Theory are evident from its basic examples.Most of the constructions of operator algebras come from groups, group algebrasand group actions on manifolds or measure spaces. The set of natural numbersN = {0, 1, 2, . . .} and its compactifications provide a vast class of spaces both topo-logically and measure-theoretically. Then operations on natural numbers give riseto actions of the semigroup, either additive or multiplicative, of natural numbers onthe spaces. Thus noncommutative tools can easily be applied to study such actions.It is our attempt in this article to apply operator algebra tools to the study of N-dynamics given by compact Hausdorff spaces obtained through compactificationsof N in association with its arithmetic structures.

Suppose X is a compact Hausdorff space and T a map from N to X witha dense range. Denote by l∞(N) the algebra of all uniform bounded, complex-valued functions on N. It is an abelian C*-algebra (as well as a maximal abelianvon Neumann algebra) acting on l2(N), the Hilbert space of all square summablefunctions on N. Then T induces an injective homomorphism, denoted by T again,

Key words and phrases. Compactification, arithmetic functions, C*-algebras, Mobiusfunction.

Supported in part by NNSF (No: 11321101) of China, Morningside Center of Mathematicsand Mathematical Sciences Center (Yau Center of Mathematics) at Tsinghua University .


127

128 LIMING GE

from C(X) (all bounded continuous functions on X) into l∞(N), i.e., for f ∈ C(X)and n ∈ N, (Tf)(n) = f(T (n)). In this case, C(X) can be viewed as a unital C*-subalgebra of l∞(N). Conversely, for any unital C*-subalgebra A of l∞(N), thereis a compact Hausdorff space X such that A ∼= C(X) by Stone-Gelfand-Naimarktheory. The space X is also known as the maximal ideal space of A, or, equivalentlythe pure (or, multiplicative) state space of A. For l∞(N), we shall use βN to denoteits maximal ideal space, also known as Stone-Cech compactification of N. It is easyto see that any multiplicative state (i.e., a homomorphism into C given by the pointevaluation at an element in βN) on l∞(N) is also a multiplicative state on A whichcorresponds to an element in X. This correspondence gives rise to a canonical mapfrom βN to X. Since βN is the maximal compactification of N, the above map fromβN to X is always continuous. From the density of N in βN and that of T (N) inX, one easily checks that this map is also surjective. Thus any compactification Xof N (given by a map T : N → X with a dense range) is given uniquely by a unitalC*-subalgebra of l∞(N)—here we do not require that the map T be injective (or,one-to-one).

To understand certain arithmetics of N through its topologies, one has to takeinto consideration some additional structures of N. There are two elementary alge-braic operations on N: addition and multiplication. Naturally N is equipped witha discrete topology. Throughout the paper, we shall use A to denote the map onN given by n → n+1, for all n ∈ N. Then Am : n → n+m is well defined for eachm ∈ N. Similarly, for each m ∈ N, we define Mm : n → mn, for all n ∈ N. Withthe discrete topology, any map T from N to any Hausdorff space Y is continuousincluding those to N itself. Moreover if the space Y is second countable, such a mapmay have a dense range. The maps A and Mm on N may or may not induce mapson T (N) (in Y ). When they do, then A : Tn → T (n+1) andMm : Tn → T (nm) arewell-defined on T (N). In general, these maps may not be extendable to continuousmaps on Y . We shall be interested in the case when such a map or maps can beextended continuously onto Y . From discussions in the previous paragraph, insteadof Y , it is natural to consider the algebra C(Y ) of all bounded continuous functionson it. The density of T (N) in Y induces an embedding of C(Y ) into l∞(N).

Now, suppose A is a C*-subalgebra of l∞(N) and A ∼= C(X), where X is acompact Hausdorff space. In this paper, we shall assume that all C*-algebras areunital. We again use T to denote the map from N to X, i.e., T (n) : f → f(n) isthe multiplicative state of point evaluation at n. In general, T (n) → T (n+ 1) maynot be well-defined. If T (n) → T (n + 1) is indeed a well-defined map (on T (N),denoted by A, again) and it can be extended to a continuous map from X intoitself, then we call the C*-algebra A an anqie1 of N. Without ambiguity, we mayalso call the N-dynamical system (X,A) an anqie of N (with a specified transitivepoint T (0) in X). One can easily see that a C*-subalgebra A of l∞(N) is an anqieif (and only if), for all f ∈ A, Af ∈ A, where (Af)(n) = f(n + 1), for all n ∈ N.Thus, an anqie A is a C*-subalgebra of l∞(N) that is closed under the action A(corresponding to the map n → n+1 in N). In addition, if each multiplication mapMm by m ∈ N \ {0}(= N∗) also extends to a continuous map on X, or equivalently,A is also closed under actions induced by all maps Mm (m ∈ N∗), then we call A(or X, as an (N,N∗)-dynamical system) a hyper-anqie of N.

1The word “anqie”, pronounced like “an-chy”, is the Chinese translation of the word “angel.”

NATURAL NUMBERS AND ENTROPY OF ARITHMETIC FUNCTIONS 129

Topologically, suppose X is a compact Hausdorff space and A a continuousmap on X. If A has a transitive point x0 in X (i.e., {Anx0 : n ∈ N} is dense inX), then the map T : n → Anx0 gives rise to a map from N to X with a denserange. The map A on X coincides with the map n → n + 1 on N and the densityof T (N) in X induces an embedding of C(X) into l∞(N). Thus C(X) (or (X,A))is an anqie of N. Thus an N-dynamics (X,A) is an anqie if it is (point) transitive.The anqie structure depends on the choice of a transitive point.

In topological dynamics, there are several notions of transitivity. The widelyused one is defined in terms of open sets: (X,A) is transitive if, for any two opensubsets U, V in X, there is an n such that An(U) ∩ An(V ) �= ∅. In most cases,this definition is equivalent to point transitivity we use in this paper: (X,A) iscalled transitive if there is an x0 ∈ X such that {Anx0 : n ∈ N} is dense in X.Transitive dynamics have been well studied by many mathematicians. Our focusis not the transitivity of an N-dynamics, but the arithmetics associated with it.More specifically, we are interested in number theoretic implications or propertiesof anqies or N-dynamics generated by specific arithmetic functions.

Arithmetic functions we encounter in this paper are well known ones in numbertheory or other areas of mathematics. The anqies they generate often give rise tonice geometrical objects with smooth structures. A major tool we shall use in thispaper to study anqies is the notion of “entropy”. Roughly speaking, the entropywe shall associate with an anqie A is the topological entropy of the additive mapA. Surprisingly, this dynamical entropy resembles more of Shannon’s entropy forrandom variables. This might hint a connection between the two philosophicallydifferent notions of entropies and reveal certain underlying complexity in a systemwith or without an action. Our results, even though primitive and preliminary, willshow certain connections between geometry and number theory. The natural bridgebetween the two is analysis and algebras. We hope to bring more geometrical andanalytical tools into the study of number theory, and strongly believe that geometryand number theory are the two faces of one angel (see also [3]).

Our paper is organized as follows. Section 2 contains two basic constructionsof anqies. Many of our examples are well known in classical dynamics and operatoralgebras. Topological characterizations are established in terms of the generatingarithmetic functions of anqies. An additive (or arithmetic) entropy, which we call“anqie entropy”, of arithmetic functions is introduced in Section 3. Examples, com-putations and basic properties of the entropy are also discussed. Further propertiesof anqie entropy are explored in Section 4. Among them, we show that “orthog-onality” (or certain independence) of arithmetic functions implies the additivityof anqie entropy. Also our entropy has a lower semi-continuity property with re-spect to uniform convergence of functions. These properties resemble those in bothShannon’s and Kolmogorov’s entropies. As a consequence, we shall see that allzero entropy functions form a C*-algebra. In Section 5, GNS constructions areperformed with respect to invariant mean states on anqies. With respect to vectornorms given by the states, different notions of periodicity are introduced. Asymp-totically periodic functions can be weakly approximated by zero entropy functions.Zero entropy functions can be further weakly approximated by zero entropy func-tions with finite ranges. Some number theoretically connections with anqies andtopological spaces are discussed in Section 6. Arithmetic compactification of N isstudied. As an application, we show that the Mobius function μ is disjoint from

130 LIMING GE

the C*-algebra generated by “essentially periodic” functions and also with finiteproducts of translations of μ2. Detailed proofs of most of our results and somefurther studies will appear in [4] and [5].

Most of the content in this article resulted from a course taught at TsinghuaUniversity in Spring 2014 and seminar talks at the Chinese Academy of Sciences,Morningside Center of Mathematics and UNH. The author wishes to thank YunpingJiang, Jianya Liu, Zeev Rudnick, Dong Wang, Fei Wei, Boqing Xue, Xiangdong Ye,Wei Yuan, Xin Zhang, Yitang Zhang and many others for collaboration and fruitfuldiscussions. Special thanks go to Professors Yuan Wang, Le Yang and S. T. Yaufor their constant support and encouragement.

2. Constructions and examples of anqies

For the basics and preliminary results on C*-algebras, topological dynamics,number theory and related topics, we refer to [7], [6], [11] and [1]. As usual,N = {0, 1, 2, . . .} and N∗ = {1, 2, . . .}. From the definition of anqies in Section 1, wesee that there are two standard methods to obtain anqies (or hyper-anqies) of N: oneis to construct A-invariant (or, A- and Mm-invariant, respectively) C*-subalgebrasof l∞(N); the other is to construct (point) transitive N-dynamical systems (X,A)(and, for hyper-anqies, Mm’s need be extendable to continuous maps as well). Inthis paper, we shall concentrate mostly on anqies, and very little on hyper-anqies.Through simple examples, we shall see the relations between the two constructions.The following example has been studied extensively by many in several areas ofmathematics. It is a motivating example for us as well.

Example 2.1. Let θ be an irrational number with 0 < θ < 1. Define T :n → e2πinθ, a map from N into the unit circle S1 = {z ∈ C : |z| = 1} in thecomplex plane C. It is easy to see that T has a dense range in S1 and A : e2πinθ →e2πi(n+1)θ = e2πiθe2πinθ, known as an irrational rotation map, induces a continuousmap z → e2πiθz on S1. Thus (S1, A) (or, C(S1) ⊂ l∞(N) determined by T ) is ananqie of N. Also it is not hard to show that multiplication maps Mm (m ∈ N∗)induce continuous maps z → zm on S1. So the irrational rotation action by θ onS1 gives rise to a hyper-anqie of N.

The above map A is metric preserving when S1 is equipped with its usualmetric. The following example (suggested by Xin Zhang) is also well known.

Example 2.2. Let A be the tent map on [0, 1] defined by A(x) = 2x when0 ≤ x ≤ 1

2 and 2(1 − x) when 12 < x ≤ 1. It is well known that ([0, 1], A)

is transitive (and also topologically mixing, see, e.g., [1] for more details). Thechoices of different transitive points give rise to many embeddings of C[0, 1] intol∞(N) and thus many anqies. By using dyadic rational approximation, Dr. Zhangshowed us many transitive points given by sums like

∑n

rn2n , where rn are rationals

in (0, 1).

Topologically, the tent map has winding number equal to zero on S1 (when 0and 1 are identified in [0, 1]). Measure theoretically, it is unitarily equivalent tothe map A : z → z2 on S1 which has a winding number equal to 2. For any θirrational, the embedding T : n → e2πi2

nθ of N into S1 gives rise to an anqie C(S1)with the additive map A also given by z → z2 (extending T (n) → T (n+1)). As anarithmetic function, T (n) = e2πi2

nθ and similar exponentials have been extensively


studied in additive number theory. Obviously, T (n) is closely related to the tentmap in dynamics.

At topological level, there is no simple way to recognize hyper-anqies from anN-dynamics. Geometrically (see [3]), the unit circle S1 and the unit disk D in Care naturally associated with N (and Z, the ring of integers). The unit interval [0, 1]might be a little far from N in association with its arithmetics. It seems hard toobtain a hyper-anqie from C[0, 1]. The above tent map and more general piece-wisemonotone (or linear) maps of [0, 1] (into itself) have been studied by many people,especially in C*-dynamics. They seem to provide many examples of anqies for usas well.

Now, we return to the C*-algebra construction of anqies. Our starting point isa C*-subalgebra of l∞(N).

First of all, we know that l∞(N) is not a separable C*-algebra. Thus it isnot countably generated as a C*-algebra (it is indeed singly generated as a vonNeumann algebra). Correspondingly, the maximal ideal space βN of l∞(N) is notmetrizable. C*-subalgebras of l∞(N) we will be interested in are often countablygenerated. When a C*-subalgebra A of l∞(N) is countably generated, the maximalideal space X of A is metrizable. Moreover if A is singly generated by a boundedarithmetic function f , its maximal ideal space X is homeomorphic to the closureof f(N) in C. Now, we assume that A is an anqie of N. Then f ∈ A impliesthat Af ∈ A (again (Af)(n) = f(n + 1), for n ∈ N). One can easily check thatthe smallest anqie containing a bounded arithmetic function f is the unital C*-subalgebra of l∞(N) generated by f,Af,A2f, . . .. We shall call this C*-algebra theanqie of N generated by f , and denote it by Af . We shall use Xf to denote the

maximal ideal space of Af . Let f(N) denote the closure of f(N) in C. Since Af

contains f , there is always a continuous map from Xf onto f(N). But these twospaces may not be the same.

The following theorem gives a description of Xf in term of f(N) and a repre-sentation of A (corresponding to the Bernoulli shift on a product space).

Theorem 2.3. Suppose that f is a function in l∞(N) and D = f(N). Denoteby DN the Cartesian product of D indexed by N. Assume that B is the Bernoullishift on DN defined by B : (a0, a1, a2, . . .) → (a1, a2, a3, . . .). Note that DN is acompact Hausdorff space endowed with the product topology. Let Df be the closureof {(f(n), f(n+1), . . .) : n ∈ N} in DN, and Xf the maximal ideal space of the anqieAf generated by f . Then Df

∼= Xf and the restriction of B on Df is identifiedwith A on Xf .

The proof of the theorem is straight forward. A generalization of the abovetheorem and its detailed proof are given in [5]. Since D is a metric space (inducedby the usual metric on C), the product space DN can be equipped with a metricequivalent to the product topology. The following metric on DN is one of such

and will be often used: d({aj}, {bj}) =∑∞

j=0|aj−bj |

2j . Then Df (⊆ DN) becomes a

compact metric space and we also know that C(Df ) ∼= Af . For different f ’s, weget many interesting examples of anqies (Xf , A). Sometimes, Xf ’s might be thesame as topological spaces, but have very different anqie structures. For example,let f1(n) = e2πinθ and f2(n) = e2πi2

nθ with θ irrational (see Example 1 and theparagraph after Example 2). Then Xf1

∼= Xf2∼= S1. But the map A is metric

preserving on Xf1 but not on Xf2 .

132 LIMING GE

Now we introduce some basic definitions related to anqies (similar definitionscan also be introduced for hyper-anqies). Let A ⊆ l∞(N) be an anqie of N. AnA-invariant C*-subalgebra B of A is called a subanqie. Two anqies A1,A2 ⊆ l∞(N)are called isomorphic if there is an A-preserving (i.e., dynamics preserving) *-isomorphism between A1 and A2. When two anqies are given by two compactHausdorff spaces X,Y with respective continuous maps A1, A2 and their respectivetransitive points x0 ∈ X and y0 ∈ Y , we can define anqie homomorphisms andisomorphisms between (X,A1) and (Y,A2) accordingly. From the transitivity ofA2, we know that all anqie homomorphisms are surjective continuous maps.

Given any S ⊆ l∞(N), if A is the smallest A-invariant unital C*-subalgebra ofl∞(N) containing S, then we say anqie A is generated by S. In fact, in this case,A is generated by {Anf : n ∈ N, f ∈ S} as a unital C*-algebra. Similarly, A is ahyper-anqie generated by S if A is the smallest unital C*-algebra containing Anfand Mmf for all m ∈ N∗, n ∈ N and f ∈ S.

Since all anqies are A-invariant unital C*-subalgebras of l∞(N), such inclusionsinduce continuous onto maps from βN to the maximal ideal spaces of the anqies.Thus all anqies as transitive topological N-dynamics (X,A) can be viewed as con-tinuous, dynamical preserving images of (βN, A). From C*-algebraic point of view,the additive map A is the same for all anqies. This restriction prevents us fromintroducing exterior operations between anqies in any natural way, but meantimealso gives us advantages to study certain relationships among them. The followingproposition helps us understand the connection between anqie inclusions and theircorresponding topological dynamics.

Proposition 2.1. Suppose A is a subanqie of B, or equivalently there is aninjective, dynamics preserving *-homomorphism from A into B. Let X and Y be themaximal ideal spaces of A and B, respectively. Then there is an induced continuoussurjective map from Y onto X (which is also dynamics preserving).

The proof of the above proposition follows from the fact that each algebraichomomorphism from B to C is again an algebraic homomorphism from A to C andthat every maximal ideal in A extends to a maximal ideal (may not be unique) inB.

To summarize, all anqies are A-invariant C*-subalgebras of l∞(N), or equiv-alently, they are transitive N-dynamics that are surjective dynamical images of(βN, A) (as an N-dynamics). As we know, there is a vast literature in transitiveN-dynamics that can be used freely to study anqies. Our hope is to build a bridgebetween number theory and geometry through arithmetic functions so that analyt-ical, geometrical and other tools can be used to study problems in number theory.Or, number theoretical results can help in understanding analysis, geometry anddynamics since many questions in these areas are related to arithmetics.

3. Anqie entropy of arithmetic functions

From previous sections, we have seen that, for a given compact Hausdorff spacesuch as S1, there might be different anqie structures given by different transitive N-actions. Even with the same action, different transitive points may also determinedifferent anqies. To understand the dynamics on a given space, it is natural tostudy the complexity of the system. A good invariant to describe such complexityis the notion of (topological) entropy. In this section, we shall introduce an entropyfor anqies.


The notion of “entropy” was first introduced and studied by physicists in the19th century. It measures some uncertainty or disorderness of a thermodynami-cal system. A mathematical notion of entropy was first introduced by Shannon.After Shannon’s introduction of a measurement of certain information containedin random variables (or signals), von Neumann made the following comment: Youshould call it entropy, for two reasons. In the first place, your uncertainty func-tion has been used in statistical mechanics under that name, so it already has aname. In the second place, and more important, no one knows what entropy reallyis, so in a debate you will always have the advantage. Von Neumann’s commentmay be viewed as an indication that Shannon’s entropy may not have much todo with the one in physics. Shannon’s entropy agrees with Fisher’s informationmeasure, can be described by certain simple axioms and is the foundation for in-formation theory. Not long after Shannon’s entropy, another mathematical notionof entropy was introduced by Kolmogorov for measure preserving transformations(or automorphisms) on probability spaces. This entropy measures the complexityof the dynamics determined by the transformations. In comparison, Kolmogorov’sentropy captures the essence of the same notion in physics in many ways and is anexcellent invariant for measure-preserving ergodic Z-actions (or automorphisms) onprobability spaces.

There are many generalizations of these two seemingly distinct notions of en-tropy. One direction is often spatial and the other dynamical, e.g., topologicalentropy is one of the generalizations on the dynamical side. Is there a deep con-nection between the two?

The “entropy” for anqies defined in this section is based formally on the topolog-ical entropy of a dynamical system associated with an anqie (X,A), i.e., h(A)—thetopological entropy of A on X. From its definition, one obtains many properties ofa dynamical entropy. When we use anqie generators, i.e., certain arithmetic func-tions, to express the entropy of the anqies in a form similar to Shannon’s entropy,many properties of Shannon’s entropy for random variables are also preserved.

Here, let us first recall the definition of topological entropy for an N-dynamics(X,T ), where X is a compact Hausdorff space and T a continuous map on it.

Definition 3.1. Let X be a compact Hausdorff space and T a continuous mapon X. Suppose U and V are two open covers for X. Denote by U ∨ V the opencover containing all intersections of elements from U and V (i.e., U ∨ V = {A ∩B :A ∈ U , B ∈ V}), and by min(W) the minimal number of open sets in W that willcover X. Define

h(T,U) = limn

1

n{log

(min(U ∨ T−1(U) ∨ · · · ∨ T−n+1(U))

)},

h(T ) = supU

{h(T,U) : U is an open cover of X} .

Then h(T ) is called the (topological) entropy of T .

When X is a compact metric space with metric d, then h(T ) can be definedequivalently by the following:

Proposition 3.1. Suppose (X, d) is a compact metric space and T a con-tinuous map on X. For each n ∈ N, define a metric dn on X by dn(x, y) =max

0≤j≤n−1d(T jx, T jy). For each δ > 0, let Ωn(δ) be the δ-covering number of X with

134 LIMING GE

respect to metric dn, i.e., the minimal number of (open) δ-balls needed to cover X.

Then h(T ) = supδ

lim supn

log(Ωn(δ))

n.

If a finite cover U of X and T generate the topology on X, then h(T ) = h(T,U).We refer to [11] for basics on topological dynamics and entropy.

Now we are ready to introduce the notion of entropy for anqies.

Definition 3.2. Suppose A ⊆ l∞(N) is an anqie. Then we define the anqieentropy of A to be the topological entropy h(A) of the additive map A extendingthe map n → n + 1 on N to the maximal ideal space of A. We use Æ(A) todenote it, i.e., Æ(A) = h(A). If A is generated by arithmetic functions {fi}i (as ananqie), we often use Æ({fi}i) to denote Æ(A), which is called the anqie entropyof arithmetic functions {fi}i, or “the entropy of {fi}i” when no confusion arises.

We list some simple but very useful facts of Æ in the following lemma. Theirproofs follow easily from similar properties in topological entropy.

Lemma 3.3. Suppose A,B ⊆ l∞(N) are anqies. Then we have the following:1. Æ(A) ≥ 0;2. If A is a subanqie of B, then Æ(A) ≤ Æ(B);3. If f1, f2, . . . and g1, g2, . . . generate the same anqie, then Æ(f1, f2, . . .)= Æ(g1, g2, . . .);4. For any f1, . . . , fn ∈ l∞(N) and any polynomials φj ∈ C[x1, . . . , xn] with 1 ≤j ≤ m, we have

Æ(φ1(f1, . . . , fn), . . . , φm(f1, . . . , fn)) ≤ Æ(f1, . . . , fn).

We may replace the above polynomials φj ’s by any continuous functions definedon the maximal ideal space of the anqie generated by f1, . . . , fn. For example, iff ≥ 0, then we can choose φ(x) =

√x and then we have Æ(

√f) ≤ Æ(f).

For anqies generated by a single arithmetic function, Theorem 2.1 provides uswith a method to compute anqie entropy.

Example 3.4. Let f(n) = e2πinθ, for n ∈ N, where θ is an irrational number.We have seen (Section 2, Example 1) that Xf = S1 and A : z → e2πiθz, z ∈ S1 ⊂ C.Then A is metric preserving. Thus h(A) = 0 and so Æ(f) = 0.

One-dimensional transitive dynamics has been extensively studied by manypeople [1]. We know that there are transitive N-dynamics on [0, 1] or S1 with anypositive entropy including infinity. Although, for a given transitive N-dynamics(X,A), as an anqie C(X) and the embedding C(X) ⊆ l∞(N) depend on the choiceof a transitive point, anqie entropy Æ(C(X)) does not. For example, the tentmap on [0, 1] has topological entropy log 2 (see Section 2, Example 2). We knowthat C([0, 1]), as an anqie whose dynamics is given by the tent map and with anytransitive point, has anqie entropy log 2. Suppose f ∈ C([0, 1]) that generatesC([0, 1]) as a unital C*-algebra. Then, for any given transitive point of the tentmap, the restriction of f on its dense orbit gives rise to a bounded arithmeticfunction. By our definition, we have Æ(f) = log 2, here f may represent differentarithmetic functions.

From Property (2) in the above lemma, we easily conclude that Æ(l∞(N)) = ∞.The following result is useful to compute anqie entropy when a generating

function has a finite range. Here we follow the notation introduced in Theorem 2.1.


Theorem 3.5. Suppose f ∈ l∞(N) has a finite range, i.e., f(N) is a finiteset. Let Df denote the image of Xf in f(N)N, and f(N)n the Cartesian productof f(N) with itself n times. Define Φn : f(N)N → f(N)n to be the projection mapfrom f(N)N onto its first n coordinates. Then Φn(Df ) is a finite set. Let |Φn(Df )|be the cardinality of Φn(Df ). Then we have Æ(f) = limn

log |Φn(Df )|n . Moreover,

Æ(f) = 0 if and only if the Hausdorff dimension of Df is zero.

The proof of this theorem follows from a direct computation (see [5] for details).From the theorem, we know that Æ(f) ≤ log |f(N)| when f(N) is a finite set. Theequality holds when Φn(Df ) = f(N)n. In general, it is very hard to computethe exact value of Æ(f) even when f takes finitely many values. From [10], weknow that both Æ(μ) and Æ(μ2) are positive, where μ is the Mobius function (wemay define μ(0) = 0 when needed). Arithmetic functions with finite ranges arevery important in number theory. We will show in [5] that Æ(χS) = 0 for manycharacteristic functions defined on subsets S of N, e.g., when S is the set of primesor the set of all prime powers.

The result in the following example follows easily from a result in topologicaldynamics: If T : X → X is a continuous map with X compact and X1 ⊂ X is aT -invariant closed subset so that X \X1 is countable, then h(T ) = hX1

(T ).

Example 3.6. Let f(n) = e2πi√n, n ∈ N. It is known that f(N) is dense in

S1. Let Af be the anqie generated by f and Xf the maximal ideal space of Af . Infact, it is not hard to show that Xf is homeomorphic to the following subset of C:

X = {e− 1n f(n) : n ∈ N} ∪ S1,

and the induced action of the additive map A on X (still denote by A) is given

by A : e−1n f(n) → e−

1n+1 f(n + 1) and A is the identity map on S1. Since S1 is

A-invariant and X \ S1 is countable, it follows that Æ(f) = 0.

When we examine the map M2 : n → 2n and its extension to X in the above

example, we see formally that M2 : e−1n e2πi

√n → e−

12n e2πi

√2n, which is not, or

cannot be extended to, a continuous map on X (or S1 as a subset of X). Thus thisanqie is not itself a hyper-anqie. Similar results hold when

√n is replaced by nr

for 0 < r < 1.In the following section, anqie entropy is compared with Shannon’s and Kol-

mogorov’s entropies. More properties of anqie entropy will be obtained and moreexamples are given.

4. Anqie independence and semi-continuity of anqie entropy

From the definition of anqie entropy we see that it is given by the topologicalentropy of an N-dynamics with a transitive map corresponding to the addition inN. Naturally it possesses all properties of a dynamical entropy. Notation-wise, ananqie entropy is also defined based on anqie generators which are functions definedon N. This notation is similar to Shannon’s entropy for random variables. Indeed,we shall see that anqie entropy does share a lot of nice properties with Shannon’sentropy. For example Æ(f1, . . . , fn) = Æ(f1) + · · · + Æ(fn) when f1, . . . , fn are“independent”, and in general, we have that Æ(f1, . . . , fn) ≤ Æ(f1) + · · ·+Æ(fn).Most interestingly, anqie entropy behaves well with respect to limits of functions.This is similar to a property in dynamics when the limit is taken with respect tomaps (on a given space).

136 LIMING GE

Independence has been generalized in several different ways, even in a noncom-mutative setting (see, e.g., [2]). Topologically, a natural way to describe “indepen-dence” is through tensor products of function spaces, or C*-algebras in general.

Definition 4.1. Suppose A and B are two anqies (i.e., A-invariant C*-sub-algebras of l∞(N)). We call them anqie independent (or simply “independent”) ifthe C*-algebra generated by A and B (in l∞(N)) is canonically isomorphic to A⊗Bas a C*-algebra tensor product. Two families of arithmetic functions are calledanqie independent if the anqies they generate, respectively, are independent.

C*-algebra tensor products are an important and subtle topic. As far as we areconcerned here, most C*-algebras involved are abelian ones (and thus amenable).Based on this, we see that all anqies are always mutually commuting. Their rela-tions are completely determined by their inclusions in l∞(N). Now we state one ofthe main results in this section.

Theorem 4.2. For any arithmetic functions f1, . . . , fn, g1, . . . , gm ∈ l∞(N), wealways have

Æ(f1, . . . , fn, g1, . . . , gm) ≤ Æ(f1, . . . , fn) +Æ(g1, . . . , gm).

The above equality holds if f1, . . . , fn and g1, . . . , gm are two anqie independentfamilies.

The proof of the above theorem follows easily from Proposition 2.1 and somebasic facts in dynamics.

The following simple example shows some aspect of our independence and howit is related to certain arithmetic structures related to natural numbers. For sim-plicity of notation, we sometimes denote e2πix by e(x).

Example 4.3. For any k ∈ N∗, define fk(j) = e( jk ), for j ∈ N. Then fk isa periodic function of period k. It is not hard to show that fn and fm are anqieindependent if and only if (n,m) = 1.

Before we continue with more examples, let us recall a result of Weyl’s which isnot only very useful in our computations when exponential functions are involved,but also provides another angle to see “independence”.

Proposition 4.1. (Weyl’s Criterion) Suppose αn = (xn1, . . . , xnk), n ≥ 1, isa sequence of real vectors in Rk. Then the following are equivalent:(a) {αn} is uniformly distributed modulo 1 in the following sense: Let α′

n =({xn1}, . . . , {xnk}), where {x} denotes the fractional part of a real number x. Forany [aj , bj ] ⊆ [0, 1], j = 1, . . . , k, then we have

limn

1

n

∣∣∣∣∣∣{m : α′m ∈

k∏j

[aj , bj ], 1 ≤ m ≤ n}

∣∣∣∣∣∣ =k∏j

(bj − aj).

(b) For any continuous function f : Rk/Zk → R,

limn

1

n

n∑m=1

f(α′m) =

∫Rk/Zk

f(x1, . . . , xk)dx1 · · · dxk.


(c) For any (l1, . . . , lk) ∈ Zk \ {0},

limn

1

n

n∑m=1

e2πi(l1xm1+···+lkxmk) = 0.

Example 4.4. Let f(n) = e(n2θ), for n ≥ 0 and θ irrational. Denote by Af

the anqie generated by f . It is well known that f(N) = S1 and Af(N) = S1. Now

A2f(n) = e((n+ 2)2θ) = e(((n+ 1)2 + 2(n+ 1) + 1)θ) = e(3θ)Af(n)e(2nθ).

Since z → z is a continuous function on S1 and the C*-algebra generated byf contains all continuous functions on S1, we have that g(n) = e(2nθ) ∈ Af .Inductively one can show that Af is generated by f and g, and thus also by fand Af as a C*-algebra. A simple application of Weyl’s criterion shows that Af

∼=C(S1×S1). We can also write g(n) = e(((n+1)2−n2−1)θ) = (Af)(n)f(n)e(−θ).

Thus A2f(n) = e(2θ)(Af(n))2f(n). If we use (z1, z2) to denote the coordinatesgiven by f and Af in S1 × S1, then A : (z1, z2) → (z2, e(2θ)z1z

22). This A is

the composition of a toral automorphism with a matrix representation

(0 1−1 2

)followed by an irrational rotation (by 2θ) on the second coordinate. Then we knowthat Æ(f) = h(A) = 0. Moreover, one can show that Af is a hyper-anqie.

Note that, for f(n) = e(2nθ), the situation is much easier than the aboveexample. We can easily show that Æ(f) = log 2 in this case.

In the rest of this section, we discuss the continuity of anqie entropy. For anyf ∈ l∞(N) and λ �= 0, we know that Æ(λf) = Æ(f). Letting λ tend to 0, wehave that λf tends to 0 in l∞(N). Thus we cannot expect a general continuityproperty for Æ with respect to uniform convergence of arithmetic functions inl∞(N). For topological entropy on a given compact (metric) space, we do havea lower semi-continuity property for the entropy, where the limit is taken over afamily of continuous maps. Here we shall discuss the limit of anqie entropy over afamily of arithmetic functions.

Given fk in l∞(N), assume that fk converges uniformly to f . Maximal idealspaces of the anqies generated by fk’s may be quite different from one another.When fk’s differ by small perturbations (e.g., we may perturb each fk by an arbi-trarily “small” function gk with ‖gk‖l∞ ≤ 1

1+k ), then the new sequence of functionswill have the same limit as the original. But their resulting anqie entropies may bevery different from the original ones. Surprisingly, we still have the following result.

Theorem 4.5. Anqie entropy Æ is lower semi-continuous on l∞(N), i.e., forany f ∈ l∞(N) with Æ(f) < ∞ and ε > 0, there is a δ > 0, such that, wheneverg ∈ l∞(N) with ‖f − g‖l∞ < δ, we have Æ(g) > Æ(f)− ε.

The proof of the theorem relies on the characterizations of topological dynamics(Xf , A) and (Xg, A) associated with f and g, respectively (see Theorem 2.1). Wemay choose a closed disk D in C containing both f(N) and g(N). Then Xf and Xg

correspond to closed subsets of DN. Then the proof of our theorem is similar tothat for the lower semi-continuity for topological entropy. The following corollaryis equivalent to the theorem.

Corollary 4.1. If fk ∈ l∞(N) converges uniformly to f , then lim infk Æ(fk) ≥Æ(f).

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From Lemma 3.1, we see that zero entropy functions are closed under algebraicoperations. Now they are also closed under uniform limit. Thus we have thefollowing:

Theorem 4.6. Let E0(N) = {f ∈ l∞(N) : Æ(f) = 0} be all bounded arithmeticfunctions with vanishing anqie entropy. Then E0(N) is a unital C*-subalgebra ofl∞(N).

We shall use E0(N) to denote the maximal ideal space of the C*-algebra E0(N)and will call it the E0-compactification (or, arithmetic compact-ification) of N. ThenE0(N) = C(E0(N)). A similar construction can be carried out for Z and we obtaina compactification of Z. For simplicity, we sometimes use E0 to denote E0(N) inthis paper. The following theorem is also useful.

Theorem 4.7. For any g ∈ E0 and f ∈ l∞(N), we always have Æ(f + g) =Æ(f).

The proof of the theorem follows easily from Lemma 3.1: Æ(f + g)+Æ(−g) ≥Æ(f) and the fact that Æ(f + g) ≤ Æ(f) +Æ(g).

Suppose f and g are two anqie independent, real-valued arithmetic functions.Then we know that Æ(f, g) = Æ(f) + Æ(g). Since f + ig generates a C*-algebracontaining both f and g, we have Æ(f + ig) = Æ(f)+Æ(g). We do not expect theconverse to be true due to the above theorem. But it is interesting to ask: to whatextent, will the additivity of anqie entropy determine the relations between two (ortwo families of) arithmetic functions?

5. Invariant means on N and GNS constructions

Before we go into applications of anqies in number theory, we need some prepa-rations. In number theory, we are often concerned with certain estimates or limitsof expressions of the form 1

x

∑n≤x f(n). For this purpose, we shall consider states

on l∞(N) given by certain limits along “ultrafilters” from a sum like 1N

∑N−1n=0 f(n).

Then the inner product of two functions f and g given by the states involves exactly

(certain limits of) sums like 1N

∑N−1n=0 f(n)g(n).

Recall that elements in βN \ N are called free ultrafilters. Free ultrafilters areclosely related to invariant means on locally compact groups introduced by vonNeumann. In a similar way, we can define invariant mean states on l∞(N).

Definition 5.1. Let A again be the homomorphism on l∞(N) induced byA : n → n + 1 on N, i.e., Af(n) = f(n + 1). A state ϕ on l∞(N) is called A-invariant, or “invariant” for short, if ϕ(Af) = ϕ(f), for all f ∈ l∞(N).

Invariant states may or may not be related to “average values” of functions.Here we give an example to illustrate certain phenomena.

Example 5.2. Let S = ∪∞n=1{2n − n, 2n − n + 1, . . . , 2n − 1} be a subset of

N. Suppose Sn = {j ∈ S : 0 ≤ j ≤ n}. Define Fn(f) = 1|Sn|

∑j∈Sn

f(j), for

f ∈ l∞(N). Choose ω ∈ βN \ N and define Fω(f) = limn→ω Fn(f). Then Fω isan A-invariant state on l∞(N). If χS is the characteristic function supported on S,then Fω(χS) = 1. But the relative density of S in N is zero. Thus Fω(f) may not

depend on the average 1n

∑n−1j=0 f(j).

On the other hand, there are invariant states depending on average values offunctions.


Definition 5.3. Suppose ω ∈ βN \N is a given free ultrafilter. For any n ∈ N

and any f in l∞(N), we define En(f) =1n

∑n−1j=0 f(j). Then, for each f given, the

function n → En(f) gives rise to another function in l∞(N). The limit of En(f)at ω is denoted by Eω(f). Then Eω is a state defined on l∞(N) and is called anA-invariant mean state, or “a mean state” (or, “a mean” for short).

In the rest of the section, we shall use E to denote a given mean state on l∞(N)(depending on a free ultrafilter). When restricted to characteristic functions onsubsets of N, E gives rise to a finitely additive, A-invariant “probability measure”on N, and, for a real-valued uniform bounded function f on N, we always have:

lim infn→∞

1

n

n−1∑j=0

f(j) ≤ E(f) ≤ lim supn→∞

1

n

n−1∑j=0

f(j).

When En(f) has a limit, then E(f) = limn→∞ En(f). For example, when a subsetS of N has density zero in N, E(χS) = 0.

Now we perform the GNS construction on l∞(N) with respect to E: Let 〈f, g〉 =E(gf) be a semi-inner product defined on l∞(N). Denote by K the subalgebra ofl∞(N) containing all f so that E(|f |2) = 〈f, f〉 = 0. Then K is a closed (two-sided)ideal in l∞(N), A = l∞(N)/K is a unital C*-algebra, and 〈 , 〉 induces an inner

product on A. When f ∈ l∞(N), we may use f (or simply f if there is no ambiguity)

to denote the coset f +K in A. For f , g ∈ A, we still use 〈f , g〉 = E(gf) to denote

the inner product on A and ‖f‖E = 〈f , f〉 12 for the (Hilbert space) vector norm on

A. The completion of A under this norm is denoted by HE . The action of A onHE is given by left multiplication (with elements in A) on A, as a subspace of HE .

With this so called GNS construction, many notions in operator algebras canbe borrowed freely to our setting. Two anqies A,B are called orthogonal (or per-pendicular) (with respect to E) if E(fg) = E(f)E(g), for all f ∈ A, g ∈ B. Twoarithmetic functions f and g are orthogonal if E(fg) = 0 (most often, one of E(f)and E(g) is already equal to 0).

Remark 5.4. Many notions in this section are defined in terms of E (dependingon the choice of a free ultrafilter), e.g., orthogonality. But when we refer to thenotions, we rarely keep track on which mean state E we refer to. Therefore, whenwe refer to a notion in future defined by a mean state E, we usually mean that theproperties used to define the notion are independent of the choices of E (or unlessit is clearly stated otherwise). For example, if f and g are orthogonal, E(fg) = 0holds for all mean states E.

It is not hard to see that there are functions in K that can take all possible(nonzero) anqie entropy values. Elements in K can be viewed as “tiny” arithmeticfunctions. We have seen that anqie entropy is stable under perturbations by zeroentropy functions. It is important to understand similar perturbations by other,especially “tiny” elements.

Definition 5.5. An arithmetic function f ∈ l∞(N) is said to have minimalanqie entropy if Æ(f) ≤ Æ(f + g), for any g ∈ K.

Problem 1. Is every bounded arithmetic function the sum of a minimal anqieentropy function and an element in K? Is the Mobius function μ a minimal anqieentropy function?

140 LIMING GE

For zero entropy functions, the above question has a positive answer. In general,we have the following theorem that describes the relation between zero entropyfunctions and elements in K.

Theorem 5.6. Let K be defined as above and E0 the C*-subalgebra of l∞(N)containing all zero anqie entropy functions (see Theorem 4.3). Then E0 + K is aC*-subalgebra of l∞(N).

Proof. Let K0 = K ∩ E0. Then K0 is a closed two-sided ideal in E0. ThusE0/K0 is a C*-algebra. Suppose B is the C*-algebra generated by E0 and K. Then

B/K is a C*-algebra. For any f ∈ E0/K0, f = f + K0, define ψ(f) = f + K in B.It is easy to check that ψ is a continuous, * preserving homomorphism. Clearly ψhas a trivial kernel and thus it is a *-isomorphism. It is also easy to see that ψ hasa dense range in B. Therefore ψ is onto. This proves that ‖f + K0‖ in C*-algebraE0/K0 is the same as ‖f + K‖ in B, which shows that all elements in B are givenby f +K for all f ∈ E0. �

From the above theorem, we see that E0 +K forms a C*-subalgebra of l∞(N).The completion of E0 (or E0 + K) in HE is denoted by H0. We have seen thatthe simplest zero (anqie) entropy functions are periodic ones. For the purpose ofapproximation, we introduce some generalized notions of periodicity.

An arithmetic function f in l∞(N) is said to be essentially periodic (or “e-

periodic”) if there is an n0 ≥ 1 such that f = An0f (or equivalently f = An0f inHE) and the smallest such n0 (≥ 1) is called an e-period of f .

It is not hard to check that e(√n) is an e-periodic function of e-period 1.

Arithmetic functions satisfying f(n) = f(n + 1) for all n must be constant ones.Thus e-periodic functions are far from periodic ones. In the following, we shallconstruct e-periodic 1 functions taking values only 0 and 1 but with an arbitraryvector norm between 0 and 1.

First we construct, for each 0 ≤ t ≤ 1, e-periodic functions ft of e-period 1such that Æ(ft) = 0 and E(ft) =

√t. This is easy when t is rational. Suppose

t is irrational and choose rational numbersnj

mj(for j ≥ 1 and nj ,mj ∈ N non

decreasing, respectively, with respect to j) so that limjnj

mj= t. We construct ft

successively: ft takes value 1 at the first n1 natural numbers in N, then followed by0’s at the next m1 − n1 numbers in N; repeat this process for n2 and m2 − n2, n3

and m3 − n3 and so on. Choose nj and mj (repeating themselves if necessary) sothat the ratio between

∑j≤n nj and

∑j≤n mj tends to t. One can show, in general,

that Æ(ft) = 0 and that ft also satisfies the requirement E(ft) =√t.

It is also easy to see that polynomials of e-periodic functions are again e-periodic. In general, e-periodic functions may not have zero entropy. But they are“nice” functions. The following definition is a generalization of e-periodicity.

Definition 5.7. A function f ∈ l∞(N) is called asymptotically periodic ifthere is a sequence of positive numbers nj ∈ N such that f − Anjf has limit zeroin HE (for any mean state E).

Asymptotically periodic functions exist naturally in many settings. Here welist some preliminary results, some of which might be well known. The detaileddiscussions and some of the proofs can be found in [4]. Again μ(n) is the Mobiusfunction.


Theorem 5.8. The following functions are asymptotically periodic:1. Uniform limits of e-periodic functions;2. For any real number θ, e(nθ);3. For any n1, . . . , nk ∈ N, An1(μ2) · · ·Ank(μ2).

It is not hard to show that uniform (or l∞-) limits of asymptotically peri-odic functions are again asymptotically periodic. Asymptotically periodic functionsthemselves may not have zero entropy, but they are weak (or l2-) limits of zero en-tropy functions (by certain averaging process). The orthogonality of arithmeticfunctions may be viewed as disjointness in number theory.

Theorem 5.9. All asymptotically periodic functions in l∞(N) are contained inH0, the closure (in HE) of all zero entropy functions in l∞(N). For any irrationalθ, the function f(n) = e(n2θ) (in E0) is perpendicular to all asymptotically periodicfunctions in l∞(N).

Thus measure-theoretically, asymptotically periodic functions are weak limits ofzero entropy functions. The proof of the above theorem is in [4]. A similar argumentwill show that the linear span of projections in E0 (or characteristic functions onsubsets of N with zero anqie entropy) is weakly dense in E0 (and therefore in H0)(see also [4] for details).

6. Arithmetics and topologies

Many problems in number theory are concerned with relations between arith-metic functions. Anqie independence of functions is a much stronger version oforthogonality (or disjointness). Interesting arithmetic functions span a wide rangein number theory. Some of them satisfy certain additive or multiplicative proper-ties. For example, e(nθ) can be viewed as a homomorphism from N (with respect toaddition) to S1 and thus is additive. On the other hand, the Mobius function μ(n)is a well known multiplicative function (with the property that μ(mn) = μ(m)μ(n)when (n,m) = 1). The disjointness between two functions f and g usually means

that limN1N

∑Nn=1 f(n)g(n) = 0. When f, g are complex-valued functions, we usu-

ally use g(n) instead of g(n) in the summation.In this section, we shall study the disjointness of arithmetic functions in the

sense of topology. Sarnak [10] (see also [8]) conjectured that the Mobius functionμ is disjoint from all continuous functions arising from any zero entropy dynamics.One can easily show that this conjecture is equivalent to that μ is orthogonal toE0(N) (orH0) inHE , the completion of all zero anqie entropy functions, for all meanstates E (see Section 5 for notations). The comments after Theorem 5.3 combinedwith Theorem 3.1 show that Sarnak’s Mobius disjointness conjecture can be furtherreduced to the disjointness of μ from zero-dimensional, zero entropy dynamicalsystems. From Theorem 5.3, we see that a positive answer to Sarnak’s disjointnessconjecture implies that the Mobius function is disjoint from many functions witha nonzero anqie entropy (or equivalently functions given by dynamics of nonzeroentropy). So one may ask: Is zero entropy essential for functions disjoint from theMobius function? Or is it the dynamics (or the topology of the system) that is moreimportant for the disjointness of the functions from μ? In [5], we have an examplewhere μ is embedded into C(S1) given by the dynamics (S1, T ), where T : z → z4

and x0 ∈ S1 is constructed depending on μ(n).

142 LIMING GE

Now, let us explore a little more the connections between topologies and arith-metics of N. Our “largest” possible transitive N-dynamics is the Stone-Cech β-compactification βN, also known as the maximal compactification of N. Similarproperty holds for E0-compactification of N, that is, E0(N) is the maximal zero(anqie) entropy compactification of N. Or, equivalently, we have the following:

Theorem 6.1. For any zero entropy N-dynamics (X,T ) and any continuous,dynamics preserving map ϕ : (βN, A) → (X,T ), there is a unique, continuous,dynamics preserving map ψ : (E0(N), A) → (X,T ) such that the following diagramcommutes:

(βN, A) −→ (E0(N), A)ϕ ↘ ↓ ψ

(X,T ).

It is probably known to specialists that any compact N-dynamics has a max-imal zero entropy factor (thanks to David Kerr for pointing it out). In our case,(E0(N), A) may be viewed as the maximal zero entropy factor of (βN, A). It is easyto show that βN is not homeomorphic to E0(N). But E0(N) seems much harder tounderstand. We list some properties in the following lemma.

Lemma 6.2. Let E0(N) be the E0-compactification of N. Then(1) E0(N) is uncountable;(2) Each n (in N) forms a closed and open subset of E0(N) and N embeds into

E0(N) as a dense (open) subset;(3) E0(N) is not extremally disconnected.

Similar results can be proven for Z and E0(Z). Clearly as dynamical systems,E0(Z) and E0(N) are quite different (the additive map A is invertible on E0(Z), butnot on E0(N)). Many questions may have easier answers for E0(Z) than that forE0(N). In this paper, we focus mostly on N. Some detailed discussions on E0(Z)will appear in [5].

We have seen that there are continuous maps from βN onto any (second count-able) compact spaces. From our examples, we also see that there are continuousmaps from E0(N) onto S1 or S1 × S1. From the fact that the entropy of any tran-sitive dynamics on [0, 1] has a positive lower bound, we know that the unit interval[0, 1] does not admit a transitive N-dynamics of zero entropy. This means thatthere is no surjective continuous map from E0(N) onto [0, 1]. Whether S2 has atransitive N-dynamics of zero entropy is still an open question. Thus far, we haveseen some obstructions in topological side in relations to arithmetic functions (oranqies) having zero entropy. The following definition seems natural.

Definition 6.3. A compact Hausdorff space X is called an E0-space if Xadmits a transitive N dynamics of zero entropy. For f ∈ l∞(N), we say that f istotally disjoint from a compact Hausdorff space X if for any anqie embeddingof C(X) into l∞(N), f is disjoint from C(X) (with respect to any mean state).

One can easily show that the Mobius function μ is totally disjoint from anyfinite set.

Problem 2. Is μ disjoint from minimal systems arising from [0, 1] or S1?

In the above, the entropy zero condition is removed. More topological restric-tions are imposed. Similar questions can be asked for S2, S1 × S1 etc.


Now let us return to zero entropy dynamics. Arithmetic functions arising fromany zero entropy N-dynamics belong to C(E0(N)) which is the same as E0(N) (theC*-algebra of all zero entropy bounded arithmetic functions). Thus to understandzero entropy functions, it is important to know E0(N).

It is easy to see that there are many infinite subsets of N having uncountableclosures in E0(N). To see this, we can choose an irrational θ and let A be the anqiegenerated by f(n) = e(nθ). Then there is an onto continuous map ψ from E0(N)to S1, the maximal ideal space of A. For any infinite subset S of N, if the closure off(S) in S1 is uncountable, then the closure of S in E0(N) is also uncountable. Forexample, this is the case when S = {n2 : n ∈ N}. A measure-theoretical argumentcan be applied to give the following theorem.

Theorem 6.4. The closure of any infinite subset S of N in E0(N) is alwaysuncountable. Thus E0(N) is not metrizable.

Finally we return to Sarnak’s disjointness conjecture. The following result onthe Mobius function μ generalizes a recent result in [9]. Its proof is similar to thatin [9].

Theorem 6.5. For any given k ∈ N∗, we have

limh→∞

limN→∞

1

h2N

N∑n=1

(h∑

j=1

μ(n+ jk))2 = 0.

The following corollary is an immediate consequence of the above theorem.

Corollary 6.1. The Mobius function μ is disjoint from the following func-tions:1. Arithmetic functions in the C*-algebra generated by all e-periodic functions inl∞(N);2. For any n1, . . . , nk in N, functions of the form An1(μ2) · · ·Ank(μ2), i.e., for anyn0, n1, . . . , nk in N,

limN→∞

1

N

N∑n=1

μ(n+ n0)μ2(n+ n1) · · ·μ2(n+ nk) = 0.

Although the above classes of arithmetic functions may not have zero entropy,in view of Theorem 5.3, the corollary may be considered as partial answers toSarnak’s disjointness conjecture. So after all, Sarnak’s disjointness conjecture mayreveal more than it seems.

References

[1] L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics,vol. 1513, Springer-Verlag, Berlin, 1992. MR1176513 (93g:58091)

[2] Liming Ge, Free probability, free entropy and applications to von Neumann algebras, Pro-ceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed.Press, Beijing, 2002, pp. 787–794. MR1957085 (2004a:46064)

[3] L. Ge, Numbers and figures (in Chinese), The Institute Lectures 2010, Science Press, 2012,Beijing, 1–12.

[4] L. Ge and F. Wei, On Sarnak’s Mobius disjointness conjecture, in preparation.[5] L. Ge, F. Wei and B. Xue, Entropy of arithmetic functions and arithmetic compactification

of natural numbers, in preparation.[6] L. K. Hua, “Introduction to Number Theory” (in Chinese), Science Press, 1957, Beijing.

144 LIMING GE

[7] R. Kadison and J. Ringrose, “Fundamentals of the Operator Algebras,” vols. I and II, Aca-demic Press, Orlando, 1983 and 1986.

[8] Jianya Liu and Peter Sarnak, The Mobius function and distal flows, Duke Math. J. 164(2015), no. 7, 1353–1399, DOI 10.1215/00127094-2916213. MR3347317

[9] K. Matomaki and M. Radziwill, Multiplicative functions in short intervals, preprint, arXiv1501.04585.

[10] P. Sarnak, “Three Lectures on the Mobius Function, Randomness and Dynamics”, IAS Lec-

ture Notes, 2009.[11] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79,

Springer-Verlag, New York-Berlin, 1982. MR648108 (84e:28017)

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Bei-

jing 100190, China – and – Department of Mathematics, University of New Hampshire,

Durham, New Hampshire 03824



Properness conditions for actions and coactions

S. Kaliszewski, Magnus B. Landstad, and John Quigg

Dedicated to R. V. Kadison — teacher and inspirator

Abstract. Three properness conditions for actions of locally compact groupson C∗-algebras are studied, as well as their dual analogues for coactions. Tomotivate the properness conditions for actions, the commutative cases (ac-tions on spaces) are surveyed; here the conditions are known: proper, locallyproper, and pointwise proper, although the latter property has not been sowell studied in the literature. The basic theory of these properness conditionsis summarized, with somewhat more attention paid to pointwise properness.C∗-characterizations of the properties are proved, and applications to C∗-dynamical systems are examined. This paper is partially expository, but someof the results are believed to be new.

1. Introduction

In our recent study of C∗-covariant systems (A,G, α) and crossed product al-gebras between the full crossed product A �α G and the regular crossed productA�α,r G, it turns out that various generalizations of the concept of proper actionsof G play an important role. We therefore start by taking a closer look at thisconcept, and it turns out that even for a classical action of G on a space X wemade what we believe to be new discoveries.

Classically (going back to Bourbaki [Bou60]), a G-space X is called proper ifthe map from G×X to X ×X given by

(s, x) �→ (x, sx)

is proper, i.e., inverse images of compact sets are compact.We call the action pointwise proper if the map from G to X given by

s �→ sx

is proper for each x ∈ X.There is also an intermediate property: X is locally proper if each point of X

has a G-invariant neighbourhood on which G acts properly.Apparently the above terminology is not completely standard. For a discrete

group, [DV97] uses the terms discontinuous, properly discontinuous, and stronglyproperly discontinuous instead of pointwise proper, locally proper, and proper, re-spectively. Palais uses Cartan instead of locally proper. And [Kos65] uses the

2000 Mathematics Subject Classification. Primary 46L55.Key words and phrases. Crossed product, action, proper action, coaction, Fourier-Stieltjes

algebra, exact sequence, Morita compatible.


145

146 S. KALISZEWSKI, MAGNUS B. LANDSTAD, AND JOHN QUIGG

terms P2, P1, and P , respectively. A characteristic property of properness (seeLemma 2.3 below) is sometimes referred to as “compact sets are wandering”.

It is folklore that for proper G-spaces X the full crossed product C0(X)�α Gis isomorphic to the reduced crossed product C0(X) �α,r G (see [Phi89] for thesecond countable case). In Proposition 6.12 (perhaps also folklore) we show thatthis carries over to locally proper actions. We will show in Theorem 6.2 (believed tobe new) that this is true also if X is first countable, but the action is only assumedto be pointwise proper.

We propose the following as natural generalizations of properness to a generalC∗-covariant system (A,G, α):

Definition.

• (A,G, α) is s-proper if for all a, b ∈ A the map

g �→ αg(a)b is in C0(G,A).

• (A,G, α) is w-proper if for all a ∈ A, φ ∈ A∗ the map

g �→ φ(αg(a)) is in C0(G).

This is consistent with the classical case: for A = C0(X) we have

(X,G) is proper ⇐⇒ (C0(X), G) is s-proper

(X,G) is pointwise proper ⇐⇒ (C0(X), G) is w-proper.

One indication that w-properness is an interesting property is the following:

Proposition. Suppose (A,G, α) is w-proper, π a nondegenerate representationof A, and s �→ Us a continuous map into the unitaries (but not necessarily ahomomorphism) such that π(αs(a)) = Usπ(a)U

∗s . Then for all ξ, η in the Hilbert

space the coefficient function s �→ 〈Usξ, η〉 is in C0(G).

We treat the classical situation of a G-space X in Sections 2 and 3, and discussgeneral C∗-covariant systems in Section 4.

For a C∗-covariant system (A,G, α), there are various definitions of properness(by Rieffel and others) involving some integrability properties. We show in Sec-tion 5 that they imply s- or w-properness. The main purpose of these integrabilityproperties is to define a suitable fixed point algebra in M(A), so our propernessdefinitions are too general for this purpose.

The natural dual concept of a C∗-covariant system is that of a coaction. As webriefly describe in Section 7, it turns out that s- and w-properness can be definedin a similar way for coactions, and we describe some of the relevant results.

In Section 8 we describe a general construction of crossed product algebrasbetween A �α G and A �α,r G. We claim that the interesting ones are obtainedby first taking as our group C∗-algebra C∗(G)/I where I is a small ideal of C∗(G)(i.e. I is δG-invariant and contained in the kernel of the regular representation λof C∗(G)). We showed in [KLQ13] that I is a small ideal of C∗(G) if and only ifthe annihilator E = I⊥ in B(G) is a large ideal, in the sense that it is a nonzero,weak* closed, and G-invariant ideal of the Fourier-Stieltjes algebra B(G). Thereare various interesting examples (see [BG] and [KLQ13]).

Now to a C∗-covariant system (B,G, α) and E as above one can define an E-crossed product B �α,E G between the full and the reduced crossed product. In[KLQ13] we show that if the coaction is w-proper then there is a Galois theorydescribing these crossed products.

PROPERNESS CONDITIONS FOR ACTIONS AND COACTIONS 147

Finally we mention the work by Baum, Guentner, and Willet [BGW] on theBaum-Connes conjecture. They have shown that there is a unique minimal exactand Morita compatible functor that assigns to a C∗-covariant system (A,G, α) aC∗-algebra between A�α G and A�α,r G. At least one of the authors doubts thatthis minimal functor is an E-crossed product for some large ideal E, although thisremains an open problem.

In Sections 2–6 we give a fairly detailed exposition, in particular proofs ofresults we believe to be new. Sections 7–8 will be more descriptive, referring to theliterature for details and proofs.

2. Actions on spaces

Throughout, G will be a locally compact group, A will be a C∗-algebra, andX will be a locally compact Hausdorff space. We will be concerned with actionsα of G on A, and we just say (A,α) is an action since the group G will typicallybe fixed. If G acts on X then we sometimes call X a G-space, and the associatedaction (C0(X), α) is defined by

αs(f)(x) = f(s−1x) for s ∈ G, f ∈ C0(X), x ∈ X.

Recall that, since the map (s, x) �→ sx from G×X toX is continuous, the associatedaction α is strongly continuous in the sense that for all f ∈ C0(X) the map s �→αs(f) from G to C0(X) is continuous for the uniform norm.

The following notation is borrowed from Palais [Pal61]:

Notation 2.1. If G acts on X, then for two subsets U, V ⊂ X we define

((U, V )) = {s ∈ G : sU ∩ V �= ∅}.

Note that if U and V are compact then ((U, V )) is closed in G.Much of the following discussion of actions on spaces is well-known; we present

it in a formal way for convenience. We make no attempt at completeness, butat the same time we include many proofs to make this exposition self-contained.When a result can be explicitly found in [Pal61], we give a precise reference, butlack of such a reference should not be taken as any claim of originality. In muchof the literature on proper actions the spaces are only required to be Hausdorff, orcompletely regular; in the proofs we will take full advantage of our assumption thatour spaces are locally compact Hausdorff.

Definition 2.2. A G-space X is proper if the map φ : X×G → X×X definedby φ(x, s) = (x, sx) is proper, i.e., inverse images of compact sets are compact.

The following is routine, and explains why properness is sometimes referred toas “compact sets are wandering” (e.g., [Rie82, Situation 2]):

Lemma 2.3. A G-space X is proper if and only if for every compact K ⊂ X theset ((K,K)) is compact, equivalently for every compact K,L ⊂ X the set ((K,L))is compact.

Example 2.4. If H is a closed subgroup of G, then it is an easy exercise thatthe action of G on the homogeneous space G/H by translation is proper if and onlyif H is compact.

The following result is contained in [Pal61, Theorem 1.2.9].


Proposition 2.5. A G-space X is proper if and only if for all x, y ∈ X thereare neighborhoods U of x and V of y such that ((U, V )) is relatively compact.

Proof. One direction is obvious, since if the action is proper we only need tochoose the neighborhoods U and V to be compact.

Conversely, assume the condition involving pairs of points x, y, and let K ⊂ Xbe compact. To show that ((K,K)) is compact, we will prove that any net {si}in ((K,K)) has a convergent subnet. For every i we can choose xi ∈ K such thatsixi ∈ K. Passing to subnets and relabeling, we can assume that xi → x andsixi → y for some x, y ∈ K. By assumption we can choose compact neighborhoodsU of x and V of y such that ((U, V )) is compact. Without loss of generality, for all iwe have xi ∈ U and sixi ∈ V , and hence si ∈ ((U, V )). Thus {si} has a convergentsubnet by compactness. �

Definition 2.6. A G-space X is locally proper if it is a union of open G-invariant sets on which G acts properly.

Palais uses the term Cartan instead of locally proper. The forward directionof the following result is [Pal61, Proposition 1.2.4].

Lemma 2.7. A G-space X is locally proper if and only if every x ∈ X has aneighborhood U such that ((U,U)) is compact.

Proof. First assume that the action is locally proper, and let x ∈ X. Choosean open G-invariant set V containing x on which G acts properly. Then choosea compact neighborhood U of x contained in V . Then ((U,U)) is compact byproperness.

Conversely, assume the condition involving compact sets ((U,U)). Choose anopen neighborhood V of x such that ((V, V )) is relatively compact, and let U = GV .We will show that the action of G on U is proper. Let y, z ∈ U . Choose s, t ∈ Gsuch that y ∈ sV and z ∈ tV . Then we have neighborhoods sV of y and tV of z,and

((sV, tV )) = t((V, V ))s−1

is relatively compact. �

The following result displays a kind of semicontinuity of the sets ((V, V )), andalso of the stability subgroups. The forward direction is [Pal61, Proposition 1.1.6].

Proposition 2.8. A G-space X is locally proper if and only if for all x ∈ X,the isotropy subgroup Gx is compact and for every neighborhood U of Gx there is aneighborhood V of x such that ((V, V )) ⊂ U .

Proof. First assume that the action is locally proper. We argue by contra-diction. Suppose we have x ∈ X and a neighborhood U of Gx such that for everyneighborhood V of x there exists s ∈ ((V, V )) such that s /∈ U . Fix a neighborhoodR of x such that ((R,R)) is compact. Restricting to neighborhoods V of x withV ⊂ R, we see that we can find nets {si} in the complement Uc and {yi} in R suchthat

• siyi ∈ R for all i,• yi → x, and• siyi → x.


Then si ∈ ((R,R)) for all i, so passing to subnets and relabeling we can assumethat si → s for some s ∈ G. Then siyi → sx, so sx = x. Thus s ∈ Gx. But theneventually si ∈ U , which is a contradiction.

Conversely, assume the condition regarding isotropy groups and neighborhoodsthereof, and let x ∈ X. SinceGx is compact, we can choose a compact neighborhoodU of Gx, and then we can choose a neighborhood V of x such that ((V, V )) ⊂ U .Then ((V, V )) is relatively compact, and we have shown that the action is locallyproper. �

The following result is contained in [Pal61, Theorem 1.2.9].

Proposition 2.9. A G-space X is proper if and only if it is locally proper andG\X is Hausdorff.

Proof. First assume that the action is proper. Then it is locally proper, andto show that G\X is Hausdorff, we will prove that if a net {Gxi} in G\X convergesto both Gx and Gy then Gx = Gy. Since the quotient map X → G\X is open, wecan pass to a subnet and relabel so that without loss of generality xi → x. Thenagain passing to a subnet and relabeling we can find si ∈ G such that sixi → y.Choose compact neighborhoods U of x and V of y, so that ((U, V )) is compactby properness. Without loss of generality xi ∈ U and sixi ∈ V for all i. Thensi ∈ ((U, V )) for all i, so by compactness we can pass to subnets and relabel so that{si} converges to some s ∈ G. Then sixi → sx, so sx = y, and hence Gx = Gy.

Conversely, assume that the action is locally proper and G\X is Hausdorff. Letx, y ∈ X. By assumption we can choose a compact neighborhood U of x such that((U,U)) is compact. Now choose any compact neighborhood V of y. To show thatthe action is proper, we will prove that ((U, V )) is compact. Let {si} be any netin ((U, V )). For each i choose xi ∈ U such that sixi ∈ V . By compactness wecan pass to subnets and relabel so that xi → z and sixi → w for some z ∈ U andw ∈ V . Then by Hausdorffness we can write

Gz = limGxi = limGsixi = Gw,

so we can choose s ∈ G such that w = sz. Then sixi → sz, so

s−1sixi → z.

Without loss of generality, for all i we can assume that s−1sixi ∈ U , so thats−1si ∈ ((U,U)). By compactness we can pass to subnets and relabel so thats−1si → t for some t ∈ G. Thus si → st, and we have found a convergent subnetof {si}. Thus ((U, V )) is compact. �

Example 2.10. It is a well-known fact in topological dynamics that there areactions that are locally proper but not proper, e.g., the action of Z on

[0,∞)× [0,∞) \ {(0, 0)}generated by the homeomorphism (x, y) �→ (2x, y/2), where any compact neigh-borhood of {(1, 0), (0, 1)} meets itself infinitely often. This action is locally properbecause its restriction to each of the open sets [0,∞)× (0,∞) and (0,∞)× [0,∞),which cover the space, are proper. A closely related example is given by letting Ract on the same space by s(x, y) = (esx, e−sy).

Definition 2.11. A G-spaceX is pointwise proper if for all x ∈ X and compactK ⊂ X, the set ((x,K)) is compact.


The above properness condition does not seem to be very often studied in thedynamics literature, and the term we use is not standard, as far as we have beenable to determine.

It is obvious that the above definition can be reformulated as follows:

Lemma 2.12. A G-space X is pointwise proper if and only if for every x ∈ Xthe map s �→ sx from G to X is proper.

Proposition 2.13. If a G-space X is pointwise proper then orbits are closed,and hence G\X is T1.

Proof. Let x ∈ X, and suppose we have a net {six} in the orbit Gx convergingto y ∈ X. Choose a compact neighborhood U of y. Without loss of generality, forall i we have six ∈ U , and hence si ∈ ((x, U)). This set is compact by pointwiseproperness, so passing to a subnet and relabeling we can assume that si → s forsome s ∈ G. Then six → sx, so y = sx ∈ Gx. �

Notation 2.14. For x ∈ X let Gx denote the isotropy subgroup.

Proposition 2.15. A G-space X is pointwise proper if and only if for all x ∈ Xthe isotropy subgroup Gx is compact and the map s �→ sx from G to Gx is relativelyopen, equivalently, the action of G on the orbit Gx is conjugate to the action onthe homogeneous space G/Gx.

Proof. First assume that the action is pointwise proper, and let x ∈ X. ThenGx is trivially compact. By homogeneity it suffices to show that the map s �→ sxfrom G to Gx is relatively open at e. Let W be a neighborhood of e. Suppose thatWx is not a relative neighborhood of x in the orbit Gx. Then we can choose a net{si} in G such that six /∈ Wx and six → x. Let U be a compact neighborhood ofx. Then ((x, U)) is compact. Without loss of generality, for all i we have six ∈ U ,and so si ∈ ((x, U)). By compactness we can pass to a subnet and relabel so thatsi → s for some s ∈ G. Then six → sx. Thus sx = x, and so s ∈ Gx. But theneventually si ∈ WGx, which is a contradiction because WGxx = Wx.

The converse is obvious, since if Gx is compact the action of G on G/Gx isproper. �

We will show that pointwise properness is weaker than local properness, butfor this we need a version of Proposition 2.13 for local properness. The followingresult is contained in [Pal61, Proposition 1.1.4].

Lemma 2.16. If a G-space X is locally proper then orbits are closed.

Proof. Let x ∈ X, and suppose we have a net {si} in G such that six → y.Choose an open G-invariant subset U containing y on which G acts properly. Thenthe action of G on U is pointwise proper, so the orbit Gx is closed in U , and hencey ∈ Gx. �

Corollary 2.17. If a G-space X is locally proper then it is pointwise proper.

Proof. Let x ∈ X. Choose an open G-invariant neighborhood U of x suchthat the action of G on U is proper. Let K ⊂ X be compact, and put L = K ∩Gx.Then L is compact because Gx is closed, and L ⊂ U . Thus ((x,K)) = ((x, L))is compact because {x} and L are compact subsets of U and G acts properly onU . �


Example 2.18. This example is taken from [DV97, Example 5 in Section 2].Recall that in Example 2.10 we had an action of Z on the space

X =([0,∞)× [0,∞)

)\ {(0, 0}

generated by the homeomorphism (x, y) �→ (2x, y/2). We form the quotient of Xby identifying {0} × (0,∞) with (0,∞)× {0} via

(0, y) ∼ (1/y, 0).

Then the action descends to the identification space, and the quotient action ispointwise proper but not locally proper.

With suitable countability assumptions, there is a surprise:

Corollary 2.19 (Glimm). Let G act on X, and assume that G and X aresecond countable, and that every isotropy subgroup is compact. Then the followingare equivalent:

(1) the action is pointwise proper;(2) for all x ∈ X the map sGx �→ sx from G/Gx to Gx is a homeomorphism;(3) G\X is T0;(4) G\X is T1;(5) every orbit is locally compact in the relative topology from X;(6) every orbit is closed in X.

Proof. Because we assume that the isotropy groups are compact, we know(1) ⇐⇒ (2). Glimm [Gli61, Theorem 1] proves that, in the second countablecase, (2) ⇐⇒ (3) ⇐⇒ (5). We also know (1) ⇒ (6) ⇒ (4). Finally, (4) ⇒ (3)trivially. �

3. C∗-ramifications

Let X be a G-space, and let α be the associated action of G on C0(X). Inthis section we examine the ramifications for the action α of the various propernessconditions covered in Section 2. For the state of the art in the case of properactions, see [EE11].

Notation 3.1. If ψ : X → Y is a continuous map between locally compactHausdorff spaces, define ψ∗ : C0(Y ) → Cb(X) by ψ∗(f) = f ◦ ψ.

It is an easy exercise to show:

Lemma 3.2. For a continuous map ψ : X → Y between locally compact Haus-dorff spaces, the following are equivalent:

(1) ψ is proper(2) ψ∗ maps C0(Y ) into C0(X)(3) ψ∗ maps Cc(Y ) into Cc(X).

Proposition 3.3. The G-space X is proper if and only if for all f, g ∈ C0(X)the map s �→ αs(f)g from G to C0(X) vanishes at infinity.

Proof. First assume that the action is proper. Since Cc(X) is dense in C0(X),by continuity it suffices to show that for all f, g ∈ Cc(X) the continuous maps �→ αs(f)g from G to C0(X) has compact support. Define f × g ∈ Cc(X ×X) by

f × g(x, y) = f(x)g(y).


Since the map φ : G ×X → X × X given by φ(s, x) = (sx, x) is proper, we haveφ∗(f × g) ∈ Cc(G×X), so there exist compact sets K ⊂ G and L ⊂ X such thatfor all (s, x) /∈ K × L we have

0 = φ∗(f × g)(s, x) = f × g(sx, x) = f(sx)g(x) =(αs−1(f)g

)(x).

Since s /∈ K implies (s, x) /∈ K × L, we see that the map s �→ αs(f)g is supportedin the compact set K−1.

Conversely, assume the condition regarding αs(f)g. To show that the action isproper, we will show that the map φ is proper, and by Lemma 3.2 it suffices to showthat if h ∈ Cc(X ×X) then φ∗(h) ∈ Cc(G×X). The support of h is contained in aproduct M ×N for some compact sets M,N ⊂ X, and we can choose f, g ∈ Cc(X)with f = 1 on M and g = 1 on N . Then h(f × g) = h, so it suffices to showthat φ∗(f × g) has compact support. By assumption the support K of s �→ αs(f)gis compact, and letting L be the support of g we see that for all (s, x) not in thecompact set K−1 × L we have

φ∗(f × g)(s, x) =(αs−1(f)g

)(x) = 0. �

Proposition 3.4. The G-space X is pointwise proper if and only if for allf ∈ C0(X) and μ ∈ M(X) = C0(X)∗ the map

g(s) =

∫X

f(sx) dμ(x)

is in C0(G).

Proof. First assume that the action is pointwise proper. Let f ∈ C0(X) andμ ∈ M(X), and define g as above. Note that g is continuous since the associatedaction (C0(X), α) is strongly continuous. Suppose that g does not vanish at ∞,and pick ε > 0 such that the closed set

S := {s ∈ G : |g(s)| ≥ ε}is not compact. It is a routine exercise to verify that we can find a sequence {sn}in S and a compact neighborhood V of e such that the sets {snV } are pairwisedisjoint. Then for each x ∈ X we have limn→∞ f(snx) = 0, because for fixed x andany δ > 0 it is an easy exercise to see that the compact set {s ∈ G : |f(sx)| ≥ δ} canonly intersect finitely many of the sets {snV }. Thus by the Dominated Convergencetheorem limn→∞ g(xn) = 0, contradicting sn ∈ S for all n.

The converse follows immediately by taking μ to be a Dirac measure and ap-plying Lemma 2.12. �

Proposition 3.5 below is the first time we need vector-valued integration. Thereare numerous references dealing with this topic. We are interested in integratingfunctions f : Ω → B, where Ω is a locally compact Hausdorff space equipped with aRadon measure μ (sometimes complex, but other times positive, and then frequentlyinfinite), and B is a Banach space. Rieffel [Rie04, Section 1] handles continuousbounded functions to a C∗-algebra using C∗-valued weights. Exel [Exe99, Sec-tion 2] develops a theory of unconditionally integrable functions with values in aBanach space, involving convergence of the integrals over relatively compact subsetsof G. Williams [Wil07, Appendix B.1] gives an exposition of the general theory ofL1(Ω, B), that in some sense unifies the treatments in [DS88, Chapter 3], [Bou63],[FD88, Chapter II], and [HP74, part I, Section III.1]. However, Williams uses apositive measure throughout, and we occasionally need complex measures; this


poses no problem, since the theory of [Wil07] can be applied to the positive andnegative variations of the real and imaginary parts of a complex measure. We pre-fer to use [Wil07] as our reference for vector-valued integration, mainly becauseit entails absolute integrability rather than unconditional integrability (see the firstitem in the following list). Here are the main properties of L1(Ω, B) that we need:

• The map f �→∫Ωf dμ from L1(Ω, B) to B is bounded and linear, where

‖f‖1 =∫Ω‖f(x)‖ d|μ|(x).

• If f ∈ L1(Ω, B) and ω is a bounded linear functional on B, then ω ◦ f ∈L1(Ω) and

ω

(∫Ω

f(x) dμ(x)

)=

∫Ω

ω(f(x)) dμ(x).

• If f ∈ L1(Ω) and b ∈ B then∫Ω

(f ⊗ b) dμ =

(∫Ω

f dμ

)b,

where (f ⊗ b)(x) = f(x)b.• Every continuous bounded function from Ω to B is measurable, and is

also essentially-separably valued on compact sets, and so is integrablewith respect to any complex measure.

Of course, we refer to the elements of L1(Ω, B) as the integrable functions from Ωto B.

If X is a G-space, then C0(X) gets a Banach-module structure over M(G) =C0(G)∗ by

μ ∗ f(x) =∫G

f(sx) dμ(s) for μ ∈ M(G), f ∈ C0(X), x ∈ X.

Here we are integrating the continuous bounded function s �→ αs−1(f) with respectto the complex measure μ.

The following is a special case of Proposition 4.6 below.

Proposition 3.5. The action on X is pointwise proper if and only if for eachf the map μ �→ μ ∗ f is weak*-to-weakly continuous.

4. Properness conditions for actions on C∗-algebras

Propositions 3.3 and 3.4 motivate the following:

Definition 4.1. An action (A,α) is s-proper if for all a, b ∈ A the map s �→αs(a)b from G to A vanishes at infinity.

Taking adjoints, we see that the above map could equally well be replaced bys �→ aαs(b).

Definition 4.2. An action (A,α) is w-proper if for all a ∈ A and all ω ∈ A∗

the mapg(s) = ω

(αs(a)

)is in C0(G).

We use the admittedly nondescriptive terminology s-proper and w-proper toavoid confusion with the myriad other uses of the word “proper” for actions onC∗-algebras.


Remark 4.3. It is almost obvious that a G-space X is locally proper if andonly if there is a family of α-invariant closed ideals of C0(X) that densely spanC0(X) and on each of which α has the property in Proposition 3.3. In fact, we willuse this in the proof of Proposition 6.12. This could be generalized in various waysto actions on arbitrary C∗-algebras, but since we have no applications of this wewill not pursue it here.

Propositions 3.3 and 3.4 can be rephrased as follows:

Corollary 4.4. A G-space X is proper if and only if the associated action(C0(X), α) is s-proper, and is pointwise proper if and only if α is w-proper.

Remark 4.5. If an action (A,α) is s-proper then it is w-proper, since by theCohen-Hewitt factorization theorem every functional in A∗ can be expressed in theform ω · a, where

ω · a(b) = ω(ab) for ω ∈ A∗, a, b ∈ A.

On the other hand, Example 2.10 implies that α can be w-proper but not s-proper.

If (A,α) is an action then A gets a Banach module structure over M(G) by

μ ∗ a =

∫G

αs(a) dμ(s) for μ ∈ M(G), a ∈ A.

Proposition 3.5 is the commutative version of the following:

Proposition 4.6. An action (A,α) is w-proper if and only if for each a ∈ Athe map μ �→ μ ∗ a is weak*-to-weakly continuous.

Proof. First assume that α is w-proper, and let a ∈ A. Let μi → 0 weak* inM(G), and let ω ∈ A∗. Then

ω(μi ∗ a) = ω

(∫G

αs(a) dμi(s)

)=

∫ω(αs(a)) dμi(s) → 0,

because the map s �→ ω(αs(a)) is in C0(G).Conversely, assume the weak*-weak continuity, and let a ∈ A and ω ∈ A∗. If

μi → 0 weak* in M(G), then∫G

ω(αs(a)) dμi(s) = ω(μi ∗ a) → 0

by continuity. By the well-known Lemma 4.7 below, the element s �→ ω(αs(a)) ofCb(G) lies in C0(G). �

In the above proof we appealed to the following well-known fact:

Lemma 4.7. Let f ∈ Cb(G). Then f ∈ C0(G) if and only if for every net {μi}in M(G) converging weak* to 0 we have∫

f dμi → 0.

The properties of s-properness and w-properness are both preserved by mor-phisms:

Proposition 4.8. Let φ : A → M(B) be a nondegenerate homomorphism thatis equivariant for actions α and β, respectively. If α is s-proper or w-proper, thenβ has the same property.


Proof. First assume that α is s-proper. Let c, d ∈ B. By the Cohen-HewittFactorization theorem, c = c′φ(a) and d = φ(b)d′ for some a, b ∈ A and c′, d′ ∈ B.Then

βs(c)d = βs(c′φ(a))φ(b)d′

= βs(c′)φ

(αs(a)b

)d′,

which vanishes at infinity because s �→ αs(a)b does and s �→ βs(c′) is bounded.

Now assume that α is w-proper. Let b ∈ B and ω ∈ B∗. We must show thatthe function s �→ ω ◦ βs(b) vanishes at ∞, and it suffices to do this for ω positive.By the Cohen-Hewitt Factorization theorem we can assume that b = φ(a∗)c witha ∈ A and c ∈ B. By the Cauchy-Schwarz inequality for positive functionals onC∗-algebras, we have ∣∣ω ◦ βs(b)

∣∣2 =∣∣∣ω(φ(αs(a

∗))βs(c))∣∣∣2

≤ ω ◦ φ(αs(a∗a))ω(βs(c

∗c)),

which vanishes at ∞ since s �→ ω ◦φ(αs(a∗a)) does and s �→ ω(βs(c

∗c)) is bounded.�

In Section 7 we will discuss properness for coactions, the dualization of actions.Here we record an easy corollary of Proposition 4.8 that involves coactions, becauseit gives a rich supply of s-proper actions. For now we just need to recall that if(A, δ) is a coaction of G, with crossed product C∗-algebra A�δ G, then there is apair of nondegenerate homomorphisms

AjA �� M(A�δ G) C0(G)

jG��

such that (jA, jG) is a universal covariant homomorphism. The dual action δ of Gon A�δ G is characterized by

δs ◦ jA = jA

δs ◦ jG = jG ◦ rts,where rt is the action of G on C0(G) by right translation.

Corollary 4.9. Every dual action is s-proper.

Proof. If δ is a coaction of G on A, then the canonical nondegenerate homo-

morphism jG : C0(G) → M(A�δ G) is rt− δ equivariant. Thus δ is s-proper sincert is. �

[BG12, Corollary 5.9] says that if an action of a discrete group G on a compactHausdorff space X is a-T-menable in the sense of [BG12, Definition 5.5], then everycovariant representation of the associated action (C(X), α) is weakly contained ina representation (π, U), on a Hilbert space H, such that for all ξ, η in a densesubspace of H the function s �→< 〈Usξ, η〉 is in c0(G). The following propositionshows that w-proper actions on arbitrary C∗-algebras have a quite similar property:

Proposition 4.10. Let (A,α) be a w-proper action, let π be a representationof A on a Hilbert space H, and for each s ∈ G suppose we have a unitary operatorUs on H such that AdUs ◦ π = π ◦ αs. Then for all ξ, η ∈ H the function

s �→ 〈Usξ, η〉


vanishes at infinity.

Proof. We can assume that π is nondegenerate. Then we can factor ξ = π(a)ξ′

for some a ∈ A, ξ′ ∈ H, and we have

|〈Usπ(a)ξ′, η〉| = |〈Usξ

′, π(αs(a∗))η〉|

≤ ‖ξ′‖〈π(αs(aa∗)η, η〉1/2,

so we can appeal to w-properness with ω ∈ A∗ defined by

ω(b) = 〈π(b)η, η〉. �Remark 4.11. Note that in the above proposition we do not require U to be

a homomorphism; it could be a projective representation.

Remark 4.12. Thus it would be interesting to study the relation between a-T-menable actions in the sense of [BG12] and pointwise proper actions. As itstands, the connection would be subtle, because an infinite discrete group cannotact pointwise properly on a compact space.

Action on the compacts. The following gives a strengthening of a specialcase of Proposition 4.10:

Proposition 4.13. Let H be a Hilbert space, and let α be an action of G onK(H). For each s ∈ G choose a unitary operator Us such that αs = AdUs. Thefollowing are equivalent:

(1) α is s-proper;(2) α is w-proper;(3) s �→ 〈Usξ, ξ〉 vanishes at infinity for all ξ ∈ H.(4) s �→ 〈Usξ, η〉 vanishes at infinity for all ξ, η ∈ H.

Proof. We know (1) ⇒ (2) ⇒ (3) by Remark 4.5 and Proposition 4.10, and(3) ⇒ (4) by polarization.

Assume (4). Let E(ξ, η) be the rank-1 operator given by ζ �→ 〈ζ, η〉ξ. Forξ, η, γ, κ ∈ H, a routine computation shows

E(ξ, η)αs(E(γ, κ)) = 〈Usγ, η〉E(ξ, κ)U∗s ,

so ∥∥E(ξ, η)αs(E(γ, κ))∥∥ ≤

∣∣〈Usγ, η〉∣∣‖E(ξ, κ)‖,

which vanishes at infinity. Thus s �→ aαs(b) is in C0(G,K(H)) whenever a and bare rank-1, and by linearity and density it follows that α is s-proper. �

In Proposition 4.13, when U can be chosen to be a representation of G, we havethe following:

Corollary 4.14. Let U be a representation of G on a Hilbert space H, and letα = AdU be the associated action of G on K(H). Suppose that ξ is a cyclic vectorfor the representation U . If s �→ 〈Usξ, ξ〉 vanishes at infinity, then α is s-proper.

Proof. As in [BG12, Remark 2.7], it is easy to see that for all η, κ in thedense subspace of H spanned by {Usξ : s ∈ G} the function s �→ 〈Usη, κ〉 vanishesat infinity. Then for all η, κ ∈ H we can find sequences {ηn}, {κn} such that‖ηn − η‖ → 0, ‖κn − κ‖ → 0, and for all n the function s �→ 〈Usηn, κn〉 vanishesat infinity. Then a routine estimation shows that the functions s �→ 〈Usηn, κn〉converge uniformly to the function s �→ 〈Usη, κ〉, and hence this latter functionvanishes at infinity. The result now follows from Proposition 4.13. �


5. Rieffel properness

We will show that if an action (A,α) is proper in Rieffel’s sense [Rie90, Defini-tion 1.2] (see also [Rie04, Definition 4.5] then it is s-proper. Rieffel’s definitions ofproper action in both of the above papers involve integration of A-valued functionson G, and we have recorded our conventions regarding vector-valued integration inthe discussion preceding Proposition 3.5. In [Rie90], Rieffel defined an action (A,α)to be proper (and we follow [BE] in using the term Rieffel proper) if s �→ αs(a)b isintegrable for all a, b in some dense subalgebra, plus other conditions that we willnot need.

Corollary 5.1. Let (A,α) be an action.

(1) Suppose that there is a dense α-invariant subset A0 of A such that for alla, b ∈ A0 the function

(5.1) s �→ αs(a)b

is integrable. Then α is s-proper in the sense of Definition 4.1.(2) Suppose that there is a dense α-invariant subset A0 of A such that for all

a ∈ A0 and all ω ∈ A∗ the function

s �→ ω(αs(a))

is integrable. Then α is w-proper in the sense of Definition 4.2.

Proof. (1) Since the functions (5.1) are uniformly continuous in norm, itfollows immediately from the elementary lemma Lemma 5.2 below that s �→ αs(a)bis in C0(G,A) for all a, b ∈ A0, and then (1) follows by density.

(2) This can be proved similarly to (1), except now the functions are scalar-valued. �

In the above proof we referred to the following:

Lemma 5.2. Let B be a Banach space, and let f : G → B be uniformly contin-uous and integrable. Then f vanishes at infinity.

Proof. Since the composition of f with the norm on B is uniformly continu-ous, and ‖f‖1 =

∫G‖f(s)‖ ds < ∞ by hypothesis, this follows immediately from the

scalar-valued case (for which, see [Car96, Theorem 1]), and which itself is a rou-tine adaptation of a classical result about scalar-valued functions on R, sometimesreferred to as Barbalat’s Lemma. �

In the commutative case, Corollary 5.1 (1) has a converse. First, following[BE], we will call an action (A,α) Rieffel proper if it satisfies the conditions of[Rie90, Definition 1.2].

Proposition 5.3. If A = C0(X) is commutative, then an action (A,α) iss-proper if and only if it is Rieffel proper.

Proof. First assume that α is s-proper. Then by Theorem 4.4 the G-space Xis proper, and then it follows from [Rie04, Theorem 4.7 and and its proof] that αis Rieffel proper.

Conversely, if α is Rieffel proper, then in particular it satisfies the hypothesisof Corollary 5.1 (1), so α is s-proper. �


Remark 5.4. Thus, if the G-space X is proper, then by [Rie90, Theorem 1.5](for the case of free action, see also [Rie82, Situation 2], which refers to [Gre77])there is an ideal of C0(X)�r G (which is known to equal C0(X)�G in this case —see Proposition 6.12 below) that is Morita equivalent to C0(G\X). This uses thefollowing: for f ∈ Cc(X) the integral

f(Gx) :=

∫G

f(sx) ds

defines f ∈ Cc(G\X). If the action on X is just pointwise proper, the integral∫Gf(sx) ds still makes sense for f ∈ Cc(X). It would be interesting to know what

properties persist in this case.

Example 5.5. Proposition 5.3 is not true for arbitrary actions (A,α). Forexample, let G be the free group Fn with n > 1, and let l be the length function.Haagerup proves in [Haa79] that for any a > 0 the function s �→ e−al(s) is positivedefinite.

For k ∈ N define hk(s) = e−l(s)/k, and let Uk be the associated cyclic represen-tation on a Hilbert space Hk, so that we have a cyclic vector ξk for Uk with

〈Uk(s)ξk, ξk〉 = hk(s).

For each k, since hk vanishes at infinity the associated inner action αk = AdUk ofG on K(Hk) is s-proper, by Corollary 4.14.

We claim that not all these actions αk can be Rieffel proper. Rieffel showsin [Rie04, Theorem 7.9] that the action α is proper in the sense of [Rie04, Def-inition 4.5] if and only if the representation U is square-integrable in the sense of[Rie04, Definition 7.8]. This latter definition is somewhat nonstandard, in thatit uses concepts from the theory of left Hilbert algebras. Also, Rieffel’s definitionof proper action in [Rie04] is somewhat complicated in that it involves C∗-valuedweights. In this paper we prefer to deal with the more accessible definition ofRieffel-proper action in [Rie90, Definition 1.2], which Rieffel shows implies theproperness condition [Rie04, Definition 4.5]. Actually, we need not concern our-selves here with Rieffel’s definition of square-integrable representations, rather allwe need is his reassurance (see [Rie04, Corollary 7.12 and Theorem 7.14]) that acyclic representation of G is square-integrable in his sense if and only if it is con-tained in the regular representation of G — so his notion of square integrabilityis equivalent to the more usual one (as he assures us in his comment following[Rie04, Definition 7.8]).

Suppose that for every k ∈ N the action αk of G on K(Hk) is Rieffel proper.Then, as noted above, αk is also proper in the sense of [Rie04, Definition 4.5], andso the representation Uk is contained in the regular presentation λ. Now we argueexactly as in [BG12, proof of Proposition 4.4]: since the functions hk converge to1 pointwise on the discrete group G, for all s ∈ G we have

〈Uk(s)ξk, ξk〉 → 1,

and hence

‖Uk(s)ξk − ξk‖ → 0.

Thus the direct sum representation⊕

k Uk weakly contains the trivial representa-tion. But since each Uk is contained in λ, the direct sum is weakly contained in λ.This gives a contradiction, since G = Fn is nonamenable.


6. Full equals reduced

Definition 6.1. Let (A,α) be an action. We say the full and reduced crossedproducts of (A,α) are equal if the regular representation

Λ : A�α G → A�α,r G

is an isomorphism.

It is an old theorem [Phi89] that if X is a second countable proper G-spacethen the associated action (C0(X), α) has full and reduced crossed products equal.It is folklore that the second-countability hypothesis can be removed — see theproof of Proposition 6.12 and Remark 6.14. We extend this to pointwise properactions and weaken the countability hypothesis:

Theorem 6.2. If X is a first countable pointwise proper G-space, then the fulland reduced crossed products of the associated action (C0(X), α) are equal.

We need some properties of the “full = reduced” phenomenon for actions. First,it is frequently inherited by invariant subalgebras:

Lemma 6.3. Let (A,α) and (B, β) be actions, and let φ : A → M(B) bean injective α − β equivariant homomorphism. Suppose that the crossed-producthomomorphism

φ�G : A�α G → M(B �β G)

is faithful. If the full and reduced crossed products of β are equal, then the full andreduced crossed products of α are equal.

Proof. We have a commutative diagram

A�α Gφ�G

��

Λα

��

M(B �β G)

Λβ

��

A�α,r Gφ�rG

�� M(B �β,r G),

and the composition Λβ ◦ (φ�G) is faithful, and therefore Λα is faithful. �

Next, “full = reduced” is preserved by extensions:

Lemma 6.4. Let (A,α) be an action, and let J be a closed invariant ideal of A.If the actions of G on J and on A/J both have full and reduced crossed productsequal, then the full and reduced crossed products of α are equal.

Proof. Let φ : J ↪→ A be the inclusion map, and let ψ : A → A/J be thequotient map. We have a commutative diagram

J �Gφ�G

��

ΛJ

��

A�Gψ�G

��

ΛA

��

A/J �G

ΛA/J

��

J �r Gφ�rG

�� A�r Gψ�rG

�� A/J �r G.

The argument is a routine diagram-chase. The vertical maps are the regular rep-resentations, which are surjective, and moreover ΛJ and ΛA/J are injective byhypothesis. Since J is an ideal, the map φ�G is an isomorphism onto the kernel of


ψ�G [Gre78, Proposition 12]. Further, since J is an invariant subalgebra, φ�r Gis injective. Let x be in the kernel of ΛA. Then

0 = (ψ �r G) ◦ ΛA(x) = ΛA/J ◦ (ψ �G)(x),

so x is in the kernel of ψ �G. Thus x ∈ J �G, and

0 = ΛA ◦ (φ�G)(x) = (φ�r G) ◦ ΛJ (x),

so x = 0. �

Next we show that “full = reduced” is preserved by direct sums:

Lemma 6.5. Let {(Ai, αi)}i∈I be a family of actions, and assume that the fulland reduced crossed products are equal for every αi. Then the direct sum action(⊕

i∈I

Ai,⊕i∈I

αi

)also has full and reduced crossed products equal.

Proof. By Lemma 6.4, the conclusion holds if I has cardinality 2, and by in-duction it holds if I is finite. By [Gre78, Proposition 12], we can regard (

⊕i∈I Ai)�

G as the inductive limit of the ideals (⊕

i∈F Ai) � G for finite F ⊂ I. Similarly(but not requiring the reference to [Gre78]), we can regard (

⊕i∈I Ai)�r G as the

inductive limit of the ideals (⊕

i∈F Ai) �r G. For every finite F ⊂ I we have acommutative diagram(⊕

i∈F Ai

)�G � � ��

ΛF ��

(⊕i∈I Ai

)�G

ΛI

��(⊕i∈F Ai

)�r G

� � ��(⊕

i∈I Ai

)�r G,

where the vertical arrows are the regular representations. Thus ΛI must be anisomorphism, by properties of inductive limits. �

Corollary 6.6. Let (A,α) be an action, let {(Ai, αi)}i∈I be a family of actionsfor which the full and reduced crossed products are equal, and for each i let φi : A →M(Ai) be an α− αi equivariant homomorphism. Let

φ : A → M

(⊕i∈I

Ai

)be the associated equivariant homomorphism. Suppose that

⋂i∈I kerφi = {0}, and

that the crossed-product homomorphism

A�α G → M

((⊕i∈I

Ai

)�αi

G

)is faithful. Then α also has full and reduced crossed products equal.

Proof. This follows immediately from Lemmas 6.3 and 6.5. �

We are almost ready for the proof of Theorem 6.2, but first we need to recallthe notion of quasi-regularity, and we only need this in the special case of closedorbits:


Definition 6.7 (special case of [Gre78, Page 221]). Let G act on X, andassume that all orbits are closed. Then the associated action of G on C0(X) isquasi-regular if for every irreducible covariant representation (π, U) of (C0(X), G)there is an orbit G · x such that

kerπ = {f ∈ C0(X) : f |G·x = 0}.In this case, π factors through a faithful representation ρ of C0(G ·x) such that

the covariant pair (ρ, U) is an irreducible representation of the restricted action(C0(G · x), α). By [Gre78, Corollary 19], the action is quasi-regular if the orbitspace G\X is second countable or almost Hausdorff in the sense that every closedsubset contains a dense relative open Hausdorff subset. Here we will prove a variantof this result:

Proposition 6.8. If a G-space X is pointwise proper and first countable, thenthe associated action of G on C0(X) is quasi-regular.

We first need a topological property of pointwise proper actions on first count-able spaces:

Lemma 6.9. If a G-space X is pointwise proper and first countable, then eachorbit is a countable decreasing intersection of open G-invariant sets.

Proof. Since orbits are closed, the quotient space G\X is T1. Since the quo-tient map is continuous and open, G\X is first countable. In particular, every pointis a countable decreasing intersection of open sets, and the result follows. �

Remark 6.10. In Lemma 6.9 the first countability assumption could be weak-ened to: every point in X is a Gδ.

It seems to us that the proof of Proposition 6.8 is clearer if we separate out aspecial case:

Lemma 6.11. If a G-space X is pointwise proper and first countable, and ifthere is an irreducible covariant representation (π, U) of (C0(X), G) such that π isfaithful, then X consists of a single orbit.

Proof. We can extend π to a representation of the algebra of bounded Borelfunctions on X, and we let P be the associated spectral measure (see, e.g., [Mur90,Theorem 2.5.5] for a version of the relevant theorem in the nonsecond-countablecase; Murphy states the theorem for compact Hausdorff spaces, but it appliesequally well to locally compact spaces by passing to the one-point compactifica-tion). Since (π, U) is irreducible, for every G-invariant Borel set E we have P (E)= 0 or 1. In particular each orbit has spectral measure 0 or 1, and there can be atmost one orbit with measure 1.

Claim: every nonempty G-invariant open subset O of X has spectral measure1. It suffices to show that P (O) �= 0. Since O �= ∅, we can choose a nonzerof ∈ C0(X) supported in O. Then

0 �= π(f) = π(fχO) = π(f)P (O),

so P (O) �= 0.Let x ∈ X. We will show that X = G · x. By Lemma 6.9 we can choose a

decreasing sequence {On} of open G-invariant sets with⋂∞

1 On = G · x. By theproperties of spectral measures, we have

P (G · x) = limn

P (On) = 1.


Thus every orbit has spectral measure 1, so there can be only one orbit. �

Proof of Proposition 6.8. Let (π, U) be an irreducible covariant represen-tation of (C0(X), G) on a Hilbert space H. Then kerπ is a G-invariant ideal ofC0(X), so there is a closed G-invariant subset Y of X such that

kerπ = {f ∈ C0(X) : f |Y = 0}.

We will show that Y consists of a single orbit. The restriction map f �→ f |Yis a G-equivariant homomorphism of C0(X) to C0(Y ), and kerπ = C0(X \ Y ),so π factors through a faithful representation ρ of C0(Y ) such that (ρ, U) is anirreducible covariant representation of (C0(Y ), G). Then Y is a single orbit, byLemma 6.11. �

Proof of Theorem 6.2. For each x ∈ X, the orbit G·x is closed, the isotropysubgroup Gx is compact, and the canonical bijection G/Gx → G·x is an equivarianthomeomorphism. Thus Gx is in particular amenable, so it follows from the aboveand [QS92, Corollary 4.3] (see also [Kas88, Theorem 3.15]) the associated actionof G on C0(G · x) has full and reduced crossed products equal. The restrictionmap φx : C0(X) → C0(G · x) is equivariant, and we get an equivariant injectivehomomorphism

φ : C0(X) → M

(⊕x∈X

C0(G · x)).

By Proposition 6.8 the action of G on C0(X) is quasi-regular, so every irre-ducible covariant representation of (C0(X), G) factors through a representation of(C0(G · x), G) for some orbit G · x. It follows that the crossed-product homomor-phism

φ�G : C0(X)�G → M

((⊕x∈X

C0(G · x))�G

)is faithful. Therefore the theorem follows from Corollary 6.6. �

The above strategy can also be used to prove the following folklore result, whichis a mild extension of Phillips’ full-equals-reduced theorem. Actually, we could notfind the following result explicitly recorded in the literature, but it seems to us thatit must have been noticed before.

Proposition 6.12. If a G-space X is locally proper then the associated action(C0(X), α) has full and reduced crossed products equal.

Note that there is no countability hypothesis on X.We need the following, which will play a role similar to that of Corollary 6.6 in

the pointwise proper case:

Corollary 6.13. Let (A,α) be an action, and let {Ji}i∈I be a family of G-invariant ideals that densely span A. If for every i the restriction of the action toJi has full and reduced crossed products equal, then the action on A has the sameproperty.

Proof. For each i let αi = α|Ji, let φi : A → M(Ji) be the α − αi equivari-

ant homomorphism induced by the A-bimodule structure on Ji, and let φ : A →


M(⊕

i∈I Ji) be the associated equivariant homomorphism, since A = spani∈I Ji,we have

⋂i∈I kerφi = {0}. Thus, by Corollary 6.6 we only need to show that

φ�G : A�α G → M

((⊕i∈I

Ji

)�αi

G

)is faithful. Suppose that ker(φ � G) �= {0}. The ideals Ji �αi

G densely spanA�α G, since the Ji’s densely span A. Thus we can find i ∈ J such that

{0} �= ker(φ�G) ∩ (Ji �αiG) = ker(φ|Ji

�G).

But φ|Ji� G is faithful since φ|Ji

is faithful and Ji is a G-invariant ideal, so wehave a contradiction. �

Proof of Proposition 6.12. First, if the G-space X is actually proper, thenG\X is Hausdorff, so by [Gre78, Corollary 19] the action of G on C0(X) is quasi-regular, so the conclusion follows as in the proof of Proposition 6.2. In the generalcase, X is a union of open G-invariant subsets Ui, on each of which G acts prop-erly. Then C0(X) is densely spanned by the ideals C0(Ui), so by properness theassociated actions αi have full and reduced crossed products equal, and hence theconclusion follows from Lemma 6.13. �

Remark 6.14. In the above proof we appealed to [Gre78, Corollary 19], whoseproof involved dense points in irreducible closed sets. In the spirit of the techniquesof the current paper, we offer an alternative argument: assume that X is a properG-space. To see that the action is quasi-regular, as in the proof of Proposition 6.8we can assume without loss of generality that there is an irreducible covariantrepresentation (π, U) of (C0(X), G) such that π is faithful. We must show thatX consists of a single G-orbit. Suppose G · x and G · y are distinct orbits in X.By properness, the quotient space G\X is Hausdorff, so we can find disjoint openneighborhoods of G · x and G · y in G\X, and hence nonempty disjoint open G-invariant sets U and V in X. But, as in the proof of Lemma 6.11, letting P denotethe spectral measure associated to the representation π of C0(X), every nonemptyG-invariant open subset O of X has P (O) = 1. Since we cannot have two disjointopen sets with spectral measure 1, we have a contradiction.

The above methods quickly lead to another property of the crossed product.Recall that a C∗-algebra is called CCR, or liminal [Dix77, Definition 4.2.1], ifevery irreducible representation is by compacts. In the second countable case, thefollowing result is contained in [Wil07, Proposition 7.31].

Proposition 6.15. Let X be a G-space. In either of the following two situa-tions, the crossed product C0(X)�G is CCR:

(1) the action of G is locally proper;(2) the action is pointwise proper and X is first countable.

Proof. (1) If the G-space X is actually proper, then this is well-known. Toillustrate how the above methods apply, we give the following argument. We haveseen above that the action is quasi-regular, and hence every irreducible covariantrepresentation (π, U) of (C0(X), G) factors through an irreducible representationof the restriction of the action to (C0(G · x), G) for some x ∈ X. The G-spacesG ·x and G/Gx are isomorphic, and C0(G/Gx)�G is Morita equivalent to C∗(Gx)by Rieffel’s version of Mackey’s Imprimitivity Theorem [Rie74, Section 7]. Since


the isotropy subgroup Gx is compact, C∗(Gx) is CCR, and hence the image of theintegrated form ρ× U , which equals the image of π × U , is the algebra of compactoperators.

In the general case, X is a union of open G-invariant proper G-spaces Ui, soC0(X)�G is the closed span of the CCR ideals C0(Ui)�G. Since every C∗-algebrahas a largest CCR ideal [Dix77, Proposition 4.2.6], C0(X)�G must be CCR.

(2) By Proposition 6.8 the action is quasi-regular, and it follows as in part (1)that C0(X)�G is CCR. �

Remark 6.16. As remarked in [AD02, Example 2.7 (3)], it follows from[ADR00, Corollary 2.1.17] that if an action of G on X is proper then the actionis amenable (a condition involving approximation by positive-definite functions).By [AD02, Theorem 5.3], if a G-space X is amenable then the associated actionα on C0(X) has full and reduced crossed products equal. This raises a question: isevery pointwise proper action amenable? It seems that amenability of the G-spaceis closely related to equality of full and reduced crossed products: by [Mat14, The-orem 3.3], for an action of a discrete exact group G on a compact space X, if αhas full and reduced crossed products equal then the action is amenable. Unfortu-nately, this is of no help for our question, because a noncompact group cannot actpointwise properly on a compact space.

7. Properness conditions for coactions

We will now dualize the properness properties of Definitions 4.1 and 4.2. Tomotivate how this will go, we pause to recall some basic facts regarding C∗-tensorproducts, commutative C∗-algebras, and actions.

For locally compact Hausdorff spaces X,Y we have the standard identifications

C0(X × Y ) = C0(X)⊗ C0(Y )

and

Cb(X) = M(C0(X)).

For a C∗-algebra A we have

A⊗ C0(G) = C0(G,A)

and

M(A⊗ C0(G)) = Cb(G,Mβ(A)),

where Mβ(A) denotes the multiplier algebra M(A) with the strict topology.For an action (A,α) we have a homomorphism

α : A → M(A⊗ C0(G))

given by

α(f)(s, x) = α(f)(s)(x) = f(sx) = αs−1(f)(x).

In fact, the image of α lies in the C∗-subalgebra M(A ⊗ C0(G)), where for anyC∗-algebras A and D

M(A⊗D) := {m ∈ M(A⊗D) : m(1⊗D) ∪ (1⊗D)m ⊂ A⊗D}.

Using the above facts, Corollary 4.4 can be restated as follows:


Lemma 7.1. An action (A,α) is s-proper if and only if

α(A)(A⊗ 1M(C0(G))) ⊂ A⊗ C0(G),

and is w-proper if and only if for all ω ∈ A∗,

(ω ⊗ id) ◦ α(A) ⊂ C0(G).

Now consider a coaction (A, δ) of G. The main difference from actions is thatthe commutative C∗-algebra C0(G) is replaced by C∗(G). Here we will use thestandard conventions for tensor products and coactions (see, e.g., [EKQR06, Ap-pendix A], in particular, the coaction is a homomorphism

δ : A → M(A⊗ C∗(G)).

Definition 7.2. A coaction (A, δ) is s-proper if

δ(A)(A⊗ 1M(C∗(G))

)⊂ A⊗ C∗(G),

and is w-proper if for all ω ∈ A∗ we have

(ω ⊗ id) ◦ δ(A) ⊂ C∗(G).

Remark 7.3. In [KLQ, Definition 5.1] we introduced the above propernessconditions, but in that paper we used the term proper coaction for the above s-proper coaction, and slice proper coaction for the above w-proper coaction (be-cause it involves the slice map ω ⊗ id). After we submitted [KLQ], we learnedthat Ellwood had defined properness more generally for coactions of Hopf C∗-algebras [Ell00, Definition 2.4]. Indeed, Proposition 3.3 is essentially [Ell00, The-orem 2.9(b)]. Definition 7.2 should also be compared with Condition (A1) in[GK03, Section 4.1], which concerns discrete quantum groups and involves thealgebraic tensor product.

Remark 7.4. An action on C0(X) can be w-proper without being s-proper, anda fortiori a coaction can be w-proper without being s-proper, even for G abelian.

Remark 7.5. (1) Just as every action of a compact group is s-proper, everycoaction of a discrete group is s-proper, because then we in fact have δ(A) ⊂A⊗ C∗(G).

(2) For any locally compact group G the canonical coaction δG on C∗(G) givenby the comultiplication is s-proper, because it is symmetric in the sense that

δG = Σ ◦ δG,where Σ is the flip automorphism on C∗(G)⊗ C∗(G).

If (A, δ) is a coaction, then A gets a Banach module structure over the Fourier-Stieltjes algebra B(G) = C∗(G)∗ by

f · a = (id⊗ f) ◦ δ(a) for f ∈ B(G), a ∈ A.

In [KLQ, Lemma 5.2] we proved the following dual analogue of Lemma 4.6

Lemma 7.6. A coaction (A, δ) is w-proper if and only if for all a ∈ A the mapf �→ f · a is weak*-to-weakly continuous.

Both s-properness and w-properness are preserved by morphisms. For w-properness this is proved in [KLQ, Proposition 5.3], and here it is for s-properness:


Proposition 7.7. Let φ : A → M(B) be a nondegenerate homomorphism thatis equivariant for coactions δ and ε, respectively. If δ is s-proper then ε has thesame property.

Proof. We have

(B ⊗ 1)ε(B) = (Bφ(A)⊗ 1)(φ⊗ id)(δ(A))ε(B)

= (B ⊗ 1)(φ(A)⊗ 1)(φ⊗ id)(δ(A))ε(B)

= (B ⊗ 1)(φ⊗ id)((A⊗ 1)δ(A)

)ε(B)

⊂ (B ⊗ 1)(φ⊗ id)(A⊗ C∗(G))ε(B)

= (B ⊗ C∗(G))ε(B)

⊂ B ⊗ C∗(G)). �

Corollary 7.8. Every dual coaction is s-proper.

Proof. If (A,α) is an action, then the canonical nondegenerate homomor-phism iG : C∗(G) → M(A �α G) is δG − α equivariant, where δG is the canonicalcoaction on C∗(G) given by the comultiplication. Thus α is s-proper since δG is. �

Recall that if (A, δ) is a coaction then the spectral subspaces {As}s∈G are givenby

As = {a ∈ M(A) : δ(a) = a⊗ s},and the fixed-point algebra is Aδ = Ae.

Proposition 7.9. Suppose A ∩ Aδ �= {0}. Then the following are equivalent:

(1) δ is s-proper;(2) δ is w-proper;(3) G is discrete.

Proof. We know (1) implies (2) and (3) implies (1). Assume (2), and letae ∈ A ∩ Aδ be nonzero. Then

f �→f · ae = (id⊗ f) ◦ δ(ae) = (id⊗ f)(ae ⊗ 1) = f(e)ae

is weak*-weak continuous from B(G) to A, so f �→ f(e) is a weak* continuouslinear functional on B(G), which implies e ∈ C∗(G), and hence G is discrete. �

Remark 7.10. Of course, the above proposition applies if A is unital. Alsonote that when G is nondiscrete a coaction (A, δ) can be s-proper and still havenonzero spectral subspaces As (and hence nontrivial fixed-point algebra Aδ, butthese will be subspaces in M(A) that intersect A trivially.

For the next lemma, recall that if (A, δ) is a coaction, then a projection p ∈M(A) is called δ-invariant if p ∈ Aδ, and in this case δ restricts to a coaction δpon the corner pAp:

δp(pap) = (p⊗ 1)δ(a)(p⊗ 1) ∈ M(pAp⊗ C∗(G)) for a ∈ A.

Lemma 7.11. Let (A, δ) be a coaction, and let p be a δ-invariant projection inM(A). If (A, δ) is s-proper, then so is the corner coaction (pAp, δp) defined above.


Proof. This is a routine computation:

δp(pAp)(pAp⊗ 1) ⊂ (p⊗ 1)δ(A)(A⊗ 1)(p⊗ 1)

⊂ (p⊗ 1)(A⊗ C∗(G))(p⊗ 1)

= pAp⊗ C∗(G). �

For the definitions of normalization and maximalization, we refer to [EKQR06,Appendix A.7] and [EKQ04]. Normalizations and maximalizations always exist,and are unique up to equivariant isomorphism.

Proposition 7.12. For any coaction (A, δ), the following are equivalent:

(1) (A, δ) is s-proper;(2) The normalization (An, δn) is s-proper;(3) The maximalization (Am, δm) is s-proper.

Proof. It follows from Proposition 4.8 that (1) implies (2) and (3) implies (1),and a careful examination of the construction of the maximalization in [EKQ04](particularly Lemma 3.6 and the proof of Theorem 3.3 in that paper) shows that(2) implies (3). �

Remark 7.13. In case the above proof seems overly fussy, note that it would

not be enough to observe that the double-dual coactionδ is automatically s-proper

and the maximalization δm is Morita equivalent toδ, because s-properness is not

preserved by Morita equivalence — otherwise every coaction of an amenable groupwould be s-proper!

Recall from [KMQW10, Proposition 3.1] that if A → G is a Fell bundle thenthere is a coaction δA of G on the (full) bundle algebra C∗(A). (That result wasstated for separable Fell bundles, but the proof did not require separability.)

Proposition 7.14. Let A → G be a Fell bundle. Then the coaction (C∗(A), δA)is s-proper.

Proof. We must show that for all a, b ∈ C∗(A) we have δ(a)(b⊗1) ∈ C∗(A)⊗C∗(G), and by density and nondegeneracy it suffices to take a ∈ Γc(A) and b of theform f · b for f ∈ A(G) ∩ Cc(G):

δ(a)(f · b⊗ 1) =

∫G

(a(t)f · b⊗ t

)dt

=

∫G

(a(t)b⊗ tf

)dt (justified below)

∈ C∗(A)⊗ C∗(G),

because the integrandt �→ a(t)b⊗ tf

is in Cc(G,C∗(A)⊗ C∗(G)). In the above computation we used the equality

a(t)f · b⊗ t = a(t)b⊗ tf for all t ∈ G,

which we justify as follows: computing inside M(C∗(A)⊗ C∗(G)), we have

a(t)f · b⊗ t =(a(t)⊗ t

)(f · b⊗ 1

)=(a(t)⊗ t

)(b⊗ f

)(justified below)

= a(t)b⊗ tf,


where we must now justify the equality f · b⊗1 = b⊗f : both sides can be regardedas compactly supported strictly continuous functions from G toM(C∗(A)⊗C∗(G)),and for all s ∈ G we have

(f · b⊗ 1)(s) = (f · b)(s)⊗ 1

= f(s)b(s)⊗ 1

= b(s)⊗ f(s) (since f(s) ∈ C)

= (b⊗ f)(s). �

Remark 7.15. Let A be a Fell bundle over G, and let

δrA = (id⊗ λ) ◦ δA : C∗(A) → M(C∗(A)⊗ C∗r (G))

be the reduction of the coaction δA. [Bus10, Theorem 3.10] shows that δrA isintegrable in the sense that the set of positive elements a in A for which δrA(a) isin the domain of the operator-valued weight id⊗ ϕ is dense in A+, where ϕ is thePlancherel weight on C∗

r (G).

Corollary 7.16 below is a dual analogue of Corollary 5.1 (1). To explain theterminology, we recall a few things from Buss’ thesis [Bus07]. Buss worked withreduced coactions, but as he points out in [Bus07, Remark 2.6.1 (4)], the the-ory carries over to full coactions by considering the reductions of the coactions.Throughout, (A, δ) is a coaction of G.

Let ϕ be the Plancherel weight on C∗(G), let M+ϕ = {c ∈ C∗(G)+ : ϕ(c) < ∞},

Nϕ = {c ∈ C∗(G) : c∗c ∈ Mϕ}, and Mϕ = spanM+ϕ , so that M+

ϕ is a hereditary

cone in C∗(G), and coincides with both Mϕ ∩ C∗(G)+ and spanN ∗ϕNϕ, and ϕ

extends uniquely to a linear functional on Mϕ.Let id ⊗ ϕ denote the associated M(A)-valued weight on A ⊗ C∗(G), with

associated objects M+id⊗ϕ, Nid⊗ϕ, and Mid⊗ϕ, and characterized as follows: for

x ∈ (A⊗ C∗(G))+ we have x ∈ M+id⊗ϕ if and only if there exists a ∈ M(A)+ such

thatθ(a) = (id⊗ ϕ)

((θ ⊗ id)(x)

)for all θ ∈ A∗+,

in which case (id ⊗ ϕ)(x) = a. We have (id ⊗ ϕ)(a⊗ c) = ϕ(c)a for all a ∈ A andc ∈ Mϕ.

Let Λ : Nϕ → L2(G) be the canonical embedding associated to the GNSconstruction for ϕ, so that Λ(bc) = λ(b)Λ(c) for all b ∈ C∗(G) and c ∈ Nϕ.

Let id⊗Λ : Nid⊗ϕ → M(A⊗L2(G)) = L(A,A⊗L2(G)) be the map associatedto the KSGNS construction for id⊗ ϕ, characterized by

(id⊗ Λ)(x)∗(a⊗ Λ(c)

)= (id⊗ ϕ)

(x∗(a⊗ c)

)for all x ∈ Nid⊗ϕ, a ∈ A, and c ∈ Nϕ. We have (id ⊗ Λ)(a ⊗ c) = a ⊗ Λ(c) for alla ∈ A and c ∈ Nϕ, and

(id⊗ Λ)(xy) = (id⊗ λ)(x)(id⊗ Λ)(y)

for all x ∈ M(A⊗ C∗(G)) and y ∈ Nid⊗Λ.The weight ϕ extends canonically to M(C∗(G)), and the associated objects are

denoted by M+

ϕ , Nϕ, and Mϕ. Similarly for the canonical extension of id ⊗ ϕ to

M(A⊗ C∗(G)), M+

id⊗ϕ, etc.Let

Asi = {a ∈ A : δ(aa∗) ∈ M+

id⊗ϕ}.


Then the coaction δ is square-integrable if Asi is dense in A. For a ∈ Asi define

〈〈a| ∈ M(A⊗ L2(G)) = L(A,A⊗ L2(G))

by

〈〈a|(b) = (id⊗ Λ)(δ(a)∗(b⊗ 1)

),

then define |a〉〉 = 〈〈a|∗ ∈ L(A⊗ L2(G), A), and for a, b ∈ Asi define

〈〈a|b〉〉 = 〈〈a| ◦ |b〉〉 ∈ L(A⊗ L2(G)).

Then (A, δ) is continuously square-integrable if there is a dense subspace R ⊂ Asi

such that

〈〈a|b〉〉 ∈ A�δ G ⊂ L(A⊗ L2(G)) for all a, b ∈ Asi.

Corollary 7.16. Every continuously square-integrable coaction is s-proper.

Proof. Let (A, δ) be a continuously square-integrable coaction. [Bus07, Sec-tion 6.8 and Proposition 6.9.4] gives a Fell bundle A over G and a δA−δ equivariantsurjective homomorphism κ : C∗(A) → A. By Proposition 4.8, every quotient ofan s-proper coaction is s-proper, so the corollary follows from Proposition 7.14. �

8. E-crossed products

To every action (B,α) one can associate the full crossed product B�αG and thereduced crossed product B�α,r G. But there are frequently many “exotic” crossedproducts in between, i.e., quotients (B�αG)/J where J is a nonzero ideal properlycontained in the kernel of the regular representation Λ. In [KLQ13], inspired bywork of Brown and Guentner [BG12], we introduced a tool that produces many(but not all) of these exotica. Our strategy is to base everything on “interesting”C∗-algebras C∗(G)/I between C∗(G) and C∗

r (G). We call a closed ideal I of C∗(G)small if it is contained in the kernel of the regular representation λ and is δG-invariant, i.e., the coaction δG descends to a coaction on C∗(G)/I. In [KLQ13,Corollary 3.13] we proved that I is small if and only if the annihilator E = I⊥ inB(G) is an ideal, which will then be large in the sense that it is nonzero, weak*closed, and G-invariant, where B(G) is given the G-bimodule structure

(s · f · t)(u) = f(tus) for f ∈ B(G), s, t, u ∈ G.

Large ideals automatically contain the reduced Fourier-Stieltjes algebra Br(G) =C∗

r (G)∗ [KLQ13, Lemma 3.14], and the map E �→ ⊥E gives a bijection betweenthe large ideals of B(G) and the small ideals I of C∗(G). For a large ideal E thequotient map

qE : C∗(G) → C∗E(G) := C∗(G)/⊥E

is equivariant for δG and a coaction δEG .

Example 8.1. E = B(G) ∩ C0(G) is a large ideal, and if G is discrete then Ghas the Haagerup property if and only if E = B(G) [BG12, Corollary 3.5].

Example 8.2. For 1 ≤ p ≤ ∞, Ep := B(G) ∩ Lp(G) is a large ideal. Ofcourse E∞ = B(G). For p ≤ 2 we have Ep = Br(G) [KLQ13, Proposition 4.2](and [BG12, Proposition 2.11] for discrete G). If G = Fn for n > 1, it hasbeen attributed to Okayasu [Oka] and (independently) to Higson and Ozawa (see[BG12, Remark 4.5]) that for 2 ≤ p < ∞ the ideals Ep are all distinct.


Given an action (B,α), we use large ideals to produce exotic crossed productsby involving the dual coaction α on B �α G. As in [KLQ], the process is mostcleanly expressed in terms of an abstract coaction (A, δ). An ideal J of A is calledδ-invariant if δ descends to a coaction on the quotient A/J . We call an ideal Jsmall if it is invariant and contained in the kernel of jA, where (jA, jG) is thecanonical covariant homomorphism of (A,C0(G)) into the multiplier algebra of thecrossed product A �δ G. For the coaction (C∗(G), δG), this is consistent with theabove notion of small ideals of C∗(G).

Recall that A gets a B(G)-module structure by

f · a = (id⊗ f) ◦ δ(a) for f ∈ B(G), a ∈ A.

For any large ideal E of B(G),

J (E) = {a ∈ A : f · a = 0 for all f ∈ E}is a small ideal of A [KLQ, Observation 3.10]. For a dual coaction (B�αG, α), wecall the quotient

B �α,E G := (B �α G)/J (E)

an E-crossed product.In the other direction, for any small ideal J of A,

E(J) = {f ∈ B(G) : (s · f · t) · a = 0 for all a ∈ J, s, t ∈ G}is an ideal of B(G), which is G-invariant by construction, and which will be weak*-closed if the coaction is w-proper. The following is [KLQ, Lemma 6.4]:

Lemma 8.3. For any w-proper coaction (A, δ), the above maps J and E forma Galois correspondence between the large ideals of B(G) and the small ideals of A.

By Galois correspondence we mean that J and E reverse inclusions, E ⊂E(J (E)) for every large ideal E of B(G), and J ⊂ J (E(J)) for every small ideal Jof A.

Since every dual coaction is s-proper, and hence w-proper, Lemma 8.3 is ap-plicable to (B �α G, α) for any action (B,α). In [KLQ, Theorem 6.10] we usedthis Galois correspondence to exhibit examples of small ideals J that are not ofthe form J (E) for any large ideal E. Buss and Echterhoff [BE13, Example 5.3]have given examples that are better in the sense that the coaction (A, δ) is of theform (B �α G, α). Consequently, there are exotic crossed products that are notE-crossed products for any large ideal E.

However, the real goal is not to look at exotic crossed products one at a time,but rather all at once: In [BGW], Baum, Guentner, and Willett define a crossed-product as a functor (B,α) �→ B�α,τG, from the category of actions to the categoryof C∗-algebras, equipped with natural transformations

B �α G ��

��

B �α,τ G

��

��

B �α,r G,

where the vertical arrow is the regular representation, such that the horizontalarrow is surjective.

For a large ideal E of B(G), the E-crossed product

(B,α) �→ B �α,E G


gives a crossed-product functor in the sense of [BGW].[BGW] defines a crossed-product functor τ to be exact if for every short exact

sequence

0 → (B1, α1) → (B2, α2) → (B3, α3) → 0

of actions the corresponding sequence of C∗-algebras

0 → B1 �α1,τ G → B2 �α2,τ G → B3 �α3,τ G → 0

is exact, and Morita compatible if for every action (B,α) the canonical untwistingisomorphism

(B ⊗KG)�G " (A�G)⊗KG,

where KG denotes the compact operators on⊕∞

n=1 L2(G), descends to an isomor-

phism

(B ⊗KG)�τ G " (A�τ G)⊗KG

of τ -crossed products. [BGW, Theorem 3.8] (with an assist from Kirchberg) showsthat there is a unique minimal exact and Morita compatible crossed product, and[BGW] uses this to give a promising reformulation of the Baum-Connes conjecture.

If E is any large ideal of B(G), the E-crossed product

(B,α) �→ B �α,E G

is a crossed-product functor in the sense of [BGW], and it is automatically Moritacompatible [BGW, Lemma A.5].

It is an open problem whether the minimal functor of [BGW] is an E-crossedproduct for some large ideal E. The counterexamples of [BE13] do not necessarilygive a negative answer, because it is unknown whether they fit into a crossed-product functor. The state of the art regarding E-crossed products is depressinglymeager at this early stage — we do not even know any examples other than B(G)itself of large ideals E for which the E-crossed-product functor is exact for all G!Of course, by definition the Br(G)-crossed product is exact for an exact groupG (where Br(G) = C∗

r (G)∗ denotes the reduced Fourier-Stieltjes algebra). Butnonexact groups are quite mysterious.

References

[BGW] P. Baum, E. Guentner, and R. Willett, Expanders, exact crossed products, and theBaum-Connes conjecture, Annals of K-Theory 1 (2016), no. 2, 155–208.

[Bou60] N. Bourbaki, Elements de mathematique. Premiere partie. (Fascicule III.) Livre III;Topologie generale. Chap. 3: Groupes topologiques. Chap. 4: Nombres reels (French),Troisieme edition revue et augmentee, Actualites Sci. Indust., No. 1143. Hermann,Paris, 1960. MR0140603 (25 #4021)

[Bou63] N. Bourbaki, Elements de mathematique. Fascicule XXIX. Livre VI: Integration.Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et representations, ActualitesScientifiques et Industrielles, No. 1306, Hermann, Paris, 1963.

[BG12] N. P. Brown and E. P. Guentner, New C∗-completions of discrete groupsand related spaces, Bull. Lond. Math. Soc. 45 (2013), no. 6, 1181–1193, DOI10.1112/blms/bdt044. MR3138486

[Bus07] A. Buss, Generalized fixed point algebras for coactions of locally compact quantumgroups, Ph.D. thesis, Westfalische Wilhelms-Universitat-Munster, 2007.

[Bus10] A. Buss, Integrability of dual coactions on Fell bundle C∗-algebras, Bull. Braz. Math.Soc. (N.S.) 41 (2010), no. 4, 607–641, DOI 10.1007/s00574-010-0028-6. MR2737319(2012e:46145)

[BE] A. Buss and S. Echterhoff, Rieffel proper actions, preprint, arXiv:1409.3977[math.OA].


[BE13] A. Buss and S. Echterhoff, Universal and exotic generalized fixed-point algebras forweakly proper actions and duality, Indiana Univ. Math. J. 63 (2014), no. 6, 1659–1701, DOI 10.1512/iumj.2014.63.5405. MR3298718

[AD02] C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems andtheir C∗-algebras, Trans. Amer. Math. Soc. 354 (2002), no. 10, 4153–4178 (elec-tronic), DOI 10.1090/S0002-9947-02-02978-1. MR1926869 (2004e:46082)

[ADR00] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies

de L’Enseignement Mathematique [Monographs of L’Enseignement Mathematique],vol. 36, L’Enseignement Mathematique, Geneva, 2000. With a foreword by GeorgesSkandalis and Appendix B by E. Germain. MR1799683 (2001m:22005)

[Car96] G. Carcano, Left and right on locally compact groups, Collect. Math. 47 (1996), no. 2,179–186. MR1402068 (97f:22008)

[DV97] S. Deo and K. Varadarajan, Discrete groups and discontinuous actions, Rocky Moun-tain J. Math. 27 (1997), no. 2, 559–583, DOI 10.1216/rmjm/1181071925. MR1466157(98i:57069)

[Dix77] J. Dixmier, C∗-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathe-matical Library, Vol. 15. MR0458185 (56 #16388)

[DS88] N. Dunford and J. T. Schwartz, Linear operators. Part I, Wiley Classics Library, JohnWiley & Sons, Inc., New York, 1988. General theory; With the assistance of WilliamG. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-IntersciencePublication. MR1009162 (90g:47001a)

[EE11] S. Echterhoff and H. Emerson, Structure and K-theory of crossed products by properactions, Expo. Math. 29 (2011), no. 3, 300–344, DOI 10.1016/j.exmath.2011.05.001.MR2820377 (2012f:19014)

[EKQ04] S. Echterhoff, S. Kaliszewski, and J. Quigg, Maximal coactions, Internat. J. Math. 15(2004), no. 1, 47–61, DOI 10.1142/S0129167X04002107. MR2039211 (2004j:46087)

[EKQR06] S. Echterhoff, S. Kaliszewski, J. Quigg, and I. Raeburn, A categorical approach to im-primitivity theorems for C∗-dynamical systems, Mem. Amer. Math. Soc. 180 (2006),no. 850, viii+169, DOI 10.1090/memo/0850. MR2203930 (2007m:46107)

[Ell00] D. A. Ellwood, A new characterisation of principal actions, J. Funct. Anal. 173(2000), no. 1, 49–60, DOI 10.1006/jfan.2000.3561. MR1760277 (2001c:46126)

[Exe99] R. Exel, Unconditional integrability for dual actions, Bol. Soc. Brasil. Mat. (N.S.) 30(1999), no. 1, 99–124, DOI 10.1007/BF01235677. MR1686980 (2000f:46071)

[FD88] J. M. G. Fell and R. S. Doran, Representations of ∗-algebras, locally compact groups,and Banach ∗-algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125,Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups andalgebras. MR936628 (90c:46001)

[Gli61] J. Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101(1961), 124–138. MR0136681 (25 #146)

[GK03] D. Goswami and A. O. Kuku, A complete formulation of the Baum-Connes conjecturefor the action of discrete quantum groups, K-Theory 30 (2003), no. 4, 341–363, DOI10.1023/B:KTHE.0000021930.34846.51. Special issue in honor of Hyman Bass on hisseventieth birthday. Part IV. MR2064244 (2005h:46097)

[Gre77] P. Green, C∗-algebras of transformation groups with smooth orbit space, Pacific J.Math. 72 (1977), no. 1, 71–97. MR0453917 (56 #12170)

[Gre78] P. Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978),no. 3-4, 191–250. MR0493349 (58 #12376)

[Haa79] U. Haagerup, An example of a nonnuclear C∗-algebra, which has the met-ric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293, DOI10.1007/BF01410082. MR520930 (80j:46094)

[HP74] E. Hille and R. S. Phillips, Functional analysis and semi-groups, American Mathe-

matical Society, Providence, R. I., 1974. Third printing of the revised edition of 1957;American Mathematical Society Colloquium Publications, Vol. XXXI. MR0423094(54 #11077)

[KLQ] S. Kaliszewski, M. B. Landstad, and J. Quigg, Exotic coactions, Proc. EdinburghMath. Soc. 59 (2016), 411–434.


[KLQ13] S. Kaliszewski, M. B. Landstad, and J. Quigg, Exotic group C∗-algebras in noncom-mutative duality, New York J. Math. 19 (2013), 689–711. MR3141810

[KMQW10] S. Kaliszewski, P. S. Muhly, J. Quigg, and D. P. Williams, Coactions and Fell bundles,New York J. Math. 16 (2010), 315–359. MR2740580 (2012d:46165)

[Kas88] G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math.91 (1988), no. 1, 147–201, DOI 10.1007/BF01404917. MR918241 (88j:58123)

[Kos65] J.-L. Koszul, Lectures on groups of transformations, Notes by R. R. Simha and R.

Sridharan. Tata Institute of Fundamental Research Lectures on Mathematics, No.32, Tata Institute of Fundamental Research, Bombay, 1965. MR0218485 (36 #1571)

[Mat14] M. Matsumura, A characterization of amenability of group actions on C∗-algebras,J. Operator Theory 72 (2014), no. 1, 41–47, DOI 10.7900/jot.2012sep07.1958.MR3246980

[Mur90] G. J. Murphy, C∗-algebras and operator theory, Academic Press, Inc., Boston, MA,1990. MR1074574 (91m:46084)

[Oka] R. Okayasu, Free group C∗-algebras associated with �p, Internat. J. Math. 25 (2014),no. 7, 1450065, 12, DOI 10.1142/S0129167X14500657. MR3238088

[Pal61] R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann.of Math. (2) 73 (1961), 295–323. MR0126506 (23 #A3802)

[Phi89] N. C. Phillips, Equivariant K-theory for proper actions, Pitman Research Notes inMathematics Series, vol. 178, Longman Scientific & Technical, Harlow; copublishedin the United States with John Wiley & Sons, Inc., New York, 1989. MR991566(90g:46105)

[QS92] J. C. Quigg and J. Spielberg, Regularity and hyporegularity in C∗-dynamical systems,Houston J. Math. 18 (1992), no. 1, 139–152. MR1159445 (93c:46122)

[Qui96] J. C. Quigg, Discrete C∗-coactions and C∗-algebraic bundles, J. Austral. Math. Soc.Ser. A 60 (1996), no. 2, 204–221. MR1375586 (97c:46086)

[Rie74] M. A. Rieffel, Induced representations of C∗-algebras, Advances in Math. 13 (1974),176–257. MR0353003 (50 #5489)

[Rie82] M. A. Rieffel, Applications of strong Morita equivalence to transformation group C∗-algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc.

Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 299–310.MR679709 (84k:46046)

[Rie90] Marc A. Rieffel, Proper actions of groups on C∗-algebras, Mappings of operatoralgebras (Philadelphia, PA, 1988) (Boston, MA), Birkhauser Boston, 1990.

[Rie04] M. A. Rieffel, Integrable and proper actions on C∗-algebras, and square-integrablerepresentations of groups, Expo. Math. 22 (2004), no. 1, 1–53, DOI 10.1016/S0723-0869(04)80002-1. MR2166968 (2006g:46108)

[Wil07] D. P. Williams, Crossed products of C∗-algebras, Mathematical Surveys and Mono-graphs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR2288954(2007m:46003)

School of Mathematical and Statistical Sciences, Arizona State University, Tempe,

Arizona 85287


Department of Mathematical Sciences, Norwegian University of Science and Tech-

nology, NO-7491 Trondheim, Norway


School of Mathematical and Statistical Sciences, Arizona State University, Tempe,

Arizona 85287



Reflexivity of Murray-von Neumann algebras

Zhe Liu

Dedicated to Professor Richard V. Kadison on the occasion of his 90th Birthday

Abstract. A Murray-von Neumann algebra Af(R) is the algebra of operatorsaffiliated with a finite von Neumann algebra R. Such an algebra contains bothbounded and unbounded operators on a Hilbert space. In this article, we study

reflexivity of Murray-von Neumann algebras. We discuss the stability of closedsubspaces of a Hilbert space H under closed, densely defined operators on H,

based on which we define LatS of a set S of closed, densely defined operators

on H, and AlgP of a set P of closed subspaces of H. We show that Murray-von

Neumann algebras Af(R) are reflexive, that is, Af(R) = AlgLatAf(R). We

also define RefaAf (R), and show that Murray-von Neumann algebras Af(R)

are algebraically reflexive, that is, Af(R) = RefaAf (R).

1. Introduction and Preliminaries

One of the productive avenues through the study of Hilbert spaces and thelinear transformations (“operators”) acting on them is indicated by the vague (buteasily understood) question: Which operators can put which vectors in which placesin the Hilbert space? A sample of such a question might be: What is the commonnull space of a certain given set of operators or the common eigenvectors for thatset? To amplify on the nature of such questions, we may be asking the questionabout a single operator or families (perhaps, algebras) of operators, about a singlevector or families (perhaps, subspaces) of vectors, and a variety of places. Anothersample of such a question might be: What are the stable (invariant) subspacesunder the operators in a given von Neumann algebra? (The answer would be:The ranges of projections in the commutant of that von Neumann algebra.) Theanswers, in general, will often be given in terms of conditions on the operators. Inthe illustration just given, the condition is that the set of operators form a vonNeumann algebra. The “variety of places” may be described in terms of conditionson the vectors. For example: Which non-zero vectors does a given operator T placein the one-dimensional space generated by that vector? The answer, of course, is“the eigenvectors for T .” For a given operator or a given family of operators, we maybe interested in finding the vectors they place at 0, that is, in finding the commonnull space. Of course, this is the main step in solving linear ordinary and partialdifferential equations. Such “action-location” questions are ubiquitous throughoutanalysis, and by contact and extension, throughout geometry and algebra.

Reflexivity is a topic that has become an important theme in the study ofoperator algebra in recent decades. (See [H][Ha][L][RR] for general references.)


175

176 ZHE LIU

It also illustrates some aspect of action-location questions where operators andplacement of vectors in subspaces are concerned. A main thrust in this article is thedevelopment of the basics of reflexivity in the framework of unbounded operators.As we shall see, the dominance of domain considerations adds a significant difficultyto the study of reflexivity when unbounded operators are involved.

Suppose H is a separable Hilbert space and B(H) denotes the algebra of allbounded linear operators on H. Let P be a set of (orthogonal) projections inB(H). Define

AlgP = {T ∈ B(H) : TE = ETE, for all E ∈ P}( = {T ∈ B(H) : T (Y ) ⊆ Y, where Y = E(H), for all E ∈ P}).

Then AlgP is a weak-operator closed subalgebra of B(H). Let S be a subset ofB(H). Define

LatS = {Projections E : TE = ETE, for all T ∈ S}( = {Projections on the closed subspaces Y of H : T (Y ) ⊆ Y, for all T ∈ S}).

Then LatS is a strong-operator closed lattice of projections. A subalgebra A ofB(H) is said to be reflexive if A = AlgLatA. We know that every von Neumannalgebra containing the identity is reflexive. Similarly, a lattice L of projectionsin B(H) is said to be reflexive if L = LatAlgL. For a subset S of B(H), letSx = {Ax : A ∈ S}. Define

RefaS = {T ∈ B(H) : Tx ∈ Sx for each x ∈ H}.So, T ∈ RefaS if and only if for each x ∈ H, there is an Ax ∈ S, depending on x,such that Tx = Axx. A subspace A of B(H) is said to be algebraically reflexive ifA = RefaA.

In this article, we study reflexivity and algebraic reflexivity of Murray-vonNeumann algebras. A Murray-von Neumann algebra is the algebra of operatorsaffiliated with a finite von Neumann algebra.

Definition 1. We say that a closed densely defined operator T on a Hilbertspace H is affiliated with a von Neumann algebra R when U ′T = TU ′ for eachunitary operator U ′ inR′, the commutant ofR. (We use D(T ) to denote the domainof T . Note that the equality U ′T = TU ′ means that D(U ′T )(= D(T )) = D(TU ′)and U ′Tx = TU ′x for each x ∈ D(T ) and U ′ maps D(T ) onto itself.)

Murray and von Neumann show, at the end of [M-v.N. 1], that the family ofoperators affiliated with a factor of type II1 (or, more generally, affiliated with afinite von Neumann algebra, those in which the identity operator is finite) admitssurprising operations of addition and multiplication that suit the formal algebraicmanipulations used by the founders of quantum mechanics in their mathematicalmodel. This is the case because of very special domain properties that are validfor finite families of operators affiliated with a finite von Neumann algebra. (Un-bounded operators, even those that are closed and densely defined, can often neitherbe added nor multiplied usefully. They may not have common dense domains.)

Let R be a finite von Neumann algebra acting on a Hilbert space H, andlet Af(R) denote the family of operators affiliated with R. Then Af(R) is a *algebra (with unit I, the identity operator) under the operations of addition +and multiplication · (in the “Murray-von Neumann” sense)([M-v.N. 1][Z1]). Moreprecisely, if A and B are in Af(R), then A+B and AB are preclosed (closable) and

REFLEXIVITY OF MURRAY-VON NEUMANN ALGEBRAS 177

their closures, denoted by A + B and A · B, respectively, are in Af(R). We referto Af(R) as the Murray-von Neumann algebra (associated with R). In [Z2], basicstructures of Af(R) (the center of Af(R), maximal abelian self-adjoint subalgebrasof Af(R)) are discussed.

We now define “Alg,” “Lat,” and “Refa,” in the unbounded-operator setting.

In this case (with unbounded operators present), we denote these sets by “Alg,”

“Lat,” and “Refa,” respectively. Due to subtleties involving domains of the un-bounded operators, we cannot adopt the standard definitions of the bounded casefor these sets. These domain difficulties have been dealt with by Richard V. Kadi-son and Simon A. Levin in an article [KL] in which they introduce a concept theycall “proper stability.” Using this concept and their “Operator-Projection Com-mutativity Principle” (appearing as our Theorem 19), we are able to press on andprove the basic facts about reflexivity in the unbounded-operator setting.

The author wishes to thank Professors Kadison and Levin for the advance viewof their work and for their permission to include their Operator-Projection Com-mutativity Principle. Our thanks are also due to our colleague Professor DeguangHan for valuable conversations while this research was in progress.

For a set P of projections in B(H), define

AlgP={Closed, densely defined T : T leaves E(H) properly stable for every E∈P}.And for a set S of closed, densely defined operators on H, define

LatS = {Projections E : E(H) is properly stable under every T ∈ S}.The definition of proper stability and related results will be discussed in Section 2.

For a Murray-von algebra Af(R), let Af(R)x = {Ax : A ∈ Af(R), x ∈ D(A)}.Define

RefaAf(R) = {Closed, densely defined T : Tx ∈ Af(R)x for every x ∈ D(T ),

and U ′(D(T )) ⊆ D(T ) for every unitary U ′ ∈ R′},

So, T ∈ RefaAf(R) if and only if for every x ∈ D(T ), there is an Ax ∈ Af(R)such that x ∈ D(Ax), Tx = Axx and U ′(D(T )) ⊆ D(T ) for every unitary U ′ ∈ R′.

Note that, for the definition of RefaAf(R), we cannot use “Tx ∈ Af(R)x for everyx ∈ H ” since T is, in general, not everywhere-defined and x may not be in thedomain of every element in Af(R).

We shall prove that the Murray-von Neumann algebra Af(R) is reflexive (The-

orem 23), that is, Af(R) = AlgLatAf(R), and that it is algebraically reflexive

(Theorem 24), that is, Af(R) = RefaAf(R).

2. Commutativity and Stability

Suppose that E is the projection from the Hilbert space H onto a closed sub-space Y . With T in B(H), clearly, T (Y ) ⊆ Y if and only if TEx = ETEx for eachx in H (we say that Y is invariant, or stable, under T in this case). Note, also, that

T ∗(I − E)− (I − E)T ∗(I − E) = ET ∗(I − E) = (TE − ETE)∗.

It follows that Y is invariant under T if and only if Y ⊥ (the orthogonal complementof Y ) is invariant under T ∗. Moreover, we have that T and E commute if and onlyif Y is invariant under both T and T ∗ (that is, Y and Y ⊥ are both invariant underT ). To see this, if Y is invariant under both T and T ∗, then ET = (T ∗E)∗ =

178 ZHE LIU

(ET ∗E)∗ = ETE = TE. Conversely, if TE = ET , then ET ∗ = T ∗E, so thatETE = ET = TE and ET ∗E = ET ∗ = T ∗E.

We ask the question of whether or not the above result (Operator-ProjectionCommutativity Principle for bounded operators) holds when replacing a boundedoperator T with a closed, densely defined operator C on the Hilbert space H. Sincesuch a operator C is, in general, not everywhere-defined, the meaning of “a closed,densely defined operator commutes with a projection (or, in general, a boundedoperator) on H” needs to be carefully defined.

Definition 2. If A and C are closed, densely defined operators on a Hilbertspace H, we say that C is an extension of A, written A ⊆ C, when D(A) ⊆ D(C)and Ax = Cx for each x ∈ D(A).

Commutativity for operators A and B in the purely algebraic sense is simplythe equality of AB and BA. In the case of operators on a Hilbert space (or, moregenerally, a normed space), if A and B are bounded (and everywhere-defined),commutativity remains just the equality of AB and BA. When A and B areunbounded, even closed, this simple equality no longer serves as the expression ofan adequate concept of commutativity. A hint of what can cause difficulties canbe seen by considering the case where B is the (everywhere-defined) operator 0.In this instance, AB is, again, 0. However, BA is the 0 operator defined only onD(A), the domain of A. Of course, we want to think of 0 as commuting witheach closed operator. We do have that 0A ⊆ A0. For a closed operator A anda bounded operator B, the relation BA ⊆ AB serves as an adequate concept of“commutativity” (of A and B).

Definition 3. If A is a closed, densely defined operator on a Hilbert spaceH and B is a bounded, everywhere-defined operator on H, we say that A and Bcommute when BA ⊆ AB.

We state some basic facts about unbounded operators (from a Hilbert space Hinto H) here without proofs. We use [KR] (in particular, Section 2.7) as our basicreference for results in the theory of unbounded operators.

Remark 4. If T is closed and B is bounded, then TB is closed (while BT , ingeneral, will not be closed nor even preclosed).

Remark 5. If A ⊆ B, then AC ⊆ BC and CA ⊆ CB.

Remark 6. If T0 is densely defined and T0 ⊆ T , then T ∗ ⊆ T ∗0 .

Remark 7. If T is densely defined, then T ∗ is closed.

Proposition 8. If T is densely defined and preclosed, then T ∗∗ = T . ([KRI];Theorem 2.7.8)

Proposition 9. Suppose that S and T are densely defined. Then T ∗ + S∗ ⊆(T + S)∗ if T + S is densely defined, and T ∗S∗ ⊆ (ST )∗ if ST is densely defined.([KRI]; Exercise 2.8.44)

Proposition 10. If C is closed and BC ⊆ CB for each B in a self-adjointsubset F of B(H), then TC ⊆ CT for each T in the von Neumann algebra generatedby F . ([KRI]; Lemma 5.6.13)


Proposition 11. If A and C are densely defined preclosed operators and Bis a bounded operator such that A = BC, then (BC)∗ = C∗B∗. ([KRII]; Lemma6.1.10)

Remark 12. Commutativity of a closed, densely defined operator C and aprojection E, now means that EC ⊆ CE. From this commutativity, CE mustbe densely defined since C is densely defined and D(C) = D(EC) ⊆ D(CE).Thus E(D(C)) ⊆ D(C), and, therefore, E(D(C)) ⊆ D(C) ∩ E(H). At the sametime, E(D(C)) is dense in E(H), as D(C) is dense in H and E is continuous onH. Also, if EC ⊆ CE, then C maps D(C) ∩ E(H) into E(H). To see this, ifx ∈ D(C) ∩ E(H), then x = Ex and Cx = CEx = ECx ∈ E(H). In Lemma 16,we shall prove that C(D(C) ∩E(H)) ⊆ E(H) if and only if CE = ECE (comparewith the “bounded” case: for T in B(H), T (E(H)) ⊆ E(H) if and only if TE =ETE and that TE = ET if and only if E(H) is invariant under both T andT ∗). In Theorem 19, we demonstrate that the “operator-projection commutativityprinciple” holds not only when the operator is bounded, but also when it is closedand densely defined, provided, in addition, that E(D(C)) ⊆ D(C) and E(D(C∗)) ⊆D(C∗). For the purposes of this theorem, we define “proper stability” of E(H)under C with the observations we have just made. In Proposition 20, we show thatE(H) is properly stable under C if and only if E(H)⊥, the orthogonal complementof E(H), is properly stable under C∗ (again, compare with the “bounded” case:E(H) is invariant under T ∈ B(H) if and only if E(H)⊥ is invariant under T ∗).Using these results in the “unbounded-operator” setting together with other resultsestablished in this paper and elsewhere, we prove our reflexivity theorems in Section3.

Definition 13. The range E(H) of a projection E on a Hilbert space H issaid to be properly stable under a closed, densely defined operator C on H when Cmaps D(C) ∩E(H) into E(H) and E(D(C)) ⊆ D(C).

Remark 14. Note that if E(D(C)) ⊆ D(C), then E(D(C)) = D(C) ∩ E(H).To see this, with E(D(C)) ⊆ D(C), clearly, E(D(C)) ⊆ D(C)∩E(H)). At the sametime, if y ∈ D(C)∩E(H), then y = Ey ∈ E(D(C)), thus D(C)∩E(H) ⊆ E(D(C)).

Proposition 15. If C is a closed, densely defined operator and B is a bounded,self-adjoint operator on a Hilbert space such that BC ⊆ CB, then BC∗ ⊆ C∗B.

Proof. Since BC ⊆ CB, from Remark 6, (CB)∗ ⊆ (BC)∗. From Proposition11, (BC)∗ = C∗B (note that BC is densely defined since D(BC) = D(C) andthat BC is preclosed since BC ⊆ CB and CB is closed). From Proposition 9, andthe fact that CB is densely defined (as BC ⊆ CB and BC is densely defined),BC∗ ⊆ (CB)∗. Thus BC∗ ⊆ (CB)∗ ⊆ (BC)∗ = C∗B. �

Lemma 16. If C is a closed, densely defined operator and E is a projection ona Hilbert space H, then C maps D(C)∩E(H) into E(H) if and only if CE = ECE.

Proof. Since D(CE) = D(ECE), it suffices to show that one of CE or ECEis an extension of the other to show that they are equal.

Suppose that C maps D(C) ∩ E(H) into E(H). If x ∈ D(CE), then Ex ∈D(C) ∩ E(H). By assumption, CEx ∈ E(H), hence ECEx = CEx. Since thisholds for each x in D(CE), CE ⊆ ECE. Therefore, CE = ECE.

180 ZHE LIU

Suppose, now, that CE = ECE. If x ∈ D(C) ∩ E(H), then Ex = x ∈ D(C)so that x ∈ D(CE) and x ∈ D(ECE). Moreover, Cx = CEx = ECEx ∈ E(H).Thus C maps D(C) ∩ E(H) into E(H) in this case. �

Lemma 17. If C is a closed, densely defined operator and E is a projection ona Hilbert space H such that CE and C∗E are densely defined, and C and C∗ map,respectively, D(C) ∩ E(H) and D(C∗) ∩ E(H) into E(H), then EC and EC∗ aredensely defined and preclosed, and CE ⊆ EC and C∗E ⊆ EC∗.

Proof. From Lemma 16, CE = ECE and C∗E = EC∗E. Since CE and C∗Eare closed, so are ECE and EC∗E. Hence, from Proposition 8, ECE = (ECE)∗∗

and EC∗E = (EC∗E)∗∗. Since CE is densely defined, EC∗ ⊆ (CE)∗ (Proposition9) . Hence EC∗ is preclosed (with dense domain D(C∗)), and similarly for EC.Thus, from Proposition 11 and Remark 5,

(EC)∗ = C∗E = EC∗E ⊆ (CE)∗E = (ECE)∗E.

It follows, from Remark 6 and Proposition 8, that

((ECE)∗E)∗ ⊆ (EC)∗∗ = EC.

Now, from Proposition 9,

E(ECE)∗∗ ⊆ ((ECE)∗E)∗ ⊆ EC.

ThusCE = ECE = E(ECE) = E(ECE)∗∗ ⊆ EC.

Arguing symmetrically, we obtain C∗E ⊆ EC∗. �

Lemma 18. If C is a closed, densely defined operator and E is a projection ona Hilbert space H such that C and E commute (that is, EC ⊆ CE), then CE = ECand C∗E = EC∗.

Proof. Since C and E commute, from Remark 12, C maps D(C)∩E(H) intoE(H) and CE is densely defined. From Proposition 15, C∗ and E commute; henceC∗ maps D(C∗) ∩ E(H) into E(H) and C∗E is densely defined. Thus Lemma 17applies and CE ⊆ EC. As CE is closed and EC ⊆ CE, it follows that EC ⊆CE ⊆ EC. Hence CE = EC. Applying this conclusion, with C∗ in place of C, weobtain C∗E = EC∗. �

Theorem 19. (Operator-Projection Commutativity Principle for unboundedoperators) A closed, densely defined operator C and a projection E commute (thatis, EC ⊆ CE) if and only if E(H) is properly stable under C and C∗.

Proof. Suppose that C and E commute, from Remark 12, C maps D(C) ∩E(H) into E(H) and E(D(C)) ⊆ D(C), that is, E(H) is properly stable underC. From Proposition 15, EC ⊆ CE implies that EC∗ ⊆ C∗E. Again, the properstability of E(H) under C∗ follows from the commutativity of C∗ and E.

Suppose, now, that E(H) is properly stable under C and C∗, that is,

C(D(C) ∩E(H)) ⊆ E(H), E(D(C)) ⊆ D(C),

C∗(D(C∗) ∩ E(H)) ⊆ E(H), E(D(C∗)) ⊆ D(C∗).

Since E(D(C)) ⊆ D(C), D(C) ⊆ D(CE); hence CE and, similarly, C∗E aredensely defined. From Lemma 17, EC and EC∗ are densely defined and preclosed.From Lemma 16, we also have CE = ECE and C∗E = EC∗E.


Our present goal is to show that EC ⊆ CE. If x ∈ D(EC), then x ∈ D(C) andEx ∈ D(C) (since E(D(C)) ⊆ D(C)). Thus x ∈ D(CE) and D(EC) ⊆ D(CE).We shall show that ECx = CEx for each x ∈ D(EC). Note that, with x ∈ D(EC),Ex ∈ D(EC)(= D(C)) and

(1) EC(Ex) = ECE(Ex) = CE(Ex) = CEx.

Note, too, that (I − E)x = x − Ex ∈ D(C) = D(EC) (since x ∈ D(C) andEx ∈ D(C)). Of course, (I − E)x ∈ D(CE) since E(I − E)x = 0 ∈ D(C). FromLemma 17, CE ⊆ EC. It follows that (I − E)x ∈ D(EC) and EC(I − E)x =CE(I − E)x = 0. Now, with EC ⊆ EC, it follows that

(2) EC(I − E)x = EC(I − E)x = CE(I − E)x = 0.

We conclude that, for each x in D(EC), x = Ex+ (I − E)x ∈ D(CE), and, from(1) and (2)

ECx = ECEx+ EC(I − E)x = CEx+ 0 = CEx.

Hence EC ⊆ CE. �

Proposition 20. The range E(H) of a projection E on a Hilbert space His properly stable under a closed, densely defined operator C on H if and only ifE(H)⊥, the orthogonal complement of E(H), is properly stable under C∗.

Proof. Suppose that E(H) is properly stable under C, that is, C maps D(C)∩E(H) into E(H) and E(D(C)) ⊆ D(C). From Remark 12, E(D(C)) is dense inE(H). From Remark 14, under the assumption that E(D(C)) ⊆ D(C), we haveE(D(C)) = D(C)∩E(H). Our goal is to show that C∗ maps D(C∗)∩E(H)⊥ intoE(H)⊥ and (I − E)(D(C∗)) ⊆ D(C∗).

For x in D(C∗)∩E(H)⊥ and y in D(C)∩E(H), since Cy ∈ C(D(C)∩E(H)) ⊆E(H), we have

〈C∗x, y〉 = 〈x,Cy〉 = 0.

The mapping y → 〈C∗x, y〉 is the 0-mapping on D(C) ∩ E(H) and it has a uniqueextension from D(C)∩E(H) to E(H) (since E(D(C)) = D(C) ∩E(H) is dense inE(H)). It follows that the mapping y → 〈C∗x, y〉 is 0 on E(H) and hence C∗x isin E(H)⊥.

It remains to show that (I−E)(D(C∗)) ⊆ D(C∗). For x in D(C∗), to show that(I−E)x is in D(C∗), we need to show that the linear functional w → 〈Cw, (I−E)x〉on D(C) is bounded (so that it has a unique bounded extension from D(C) toH andRiesz’s representation theorem provides a vector z ∈ H such that 〈Cw, (I−E)x〉 =〈w, z〉 and therefore (I − E)x ∈ D(C∗) and C∗((I − E)x) = z). We, first, considerthose w in D(C) ∩ E(H). Then, since Cw ∈ C(D(C) ∩ E(H)) ⊆ E(H),

〈Cw, (I − E)x〉 = 0.

So, the mapping w → 〈Cw, (I −E)x〉 is 0 on D(C)∩E(H). If w ∈ D(C)∩E(H)⊥,then

〈Cw, (I − E)x〉 = 〈Cw, x〉 − 〈Cw,Ex〉.The mapping w → 〈Cw, x〉 is bounded since x ∈ D(C∗). For 〈Cw,Ex〉, if Cw ∈E(H)⊥, 〈Cw,Ex〉 = 0. If Cw ∈ E(H), 〈Cw,Ex〉 = 〈ECw, x〉 = 〈Cw, x〉. So, themapping w → 〈Cw,Ex〉 is bounded and therefore the mapping w → 〈Cw, (I−E)x〉is bounded on w ∈ D(C) ∩ E(H)⊥.

182 ZHE LIU

If E(H)⊥ is properly stable under C∗, applying what we have just proved, weobtain that (E(H)⊥)⊥ is properly stable under C∗∗. Since (E(H)⊥)⊥ = E(H) andC∗∗ = C = C (C is closed), we have that E(H) is properly stable under C. �

Lemma 21. Let R be a von Neumann algebra acting on a Hilbert space H, andlet T be a closed, densely defined operator on H. If UT ⊆ TU for every unitaryoperator U in R, then UT = TU .

Proof. Since UT ⊆ TU , multiplying both sides on the right and on the leftby U∗, we obtain TU∗ ⊆ U∗T . Since this holds for the adjoint of every unitary inR, it still holds if U∗ is replaced by U , that is, TU ⊆ UT for every unitary U in R.It follows, from the assumption UT ⊆ TU , that UT = TU . �

Proposition 22. If R is a finite von Neumann algebra acting on a Hilbertspace H, H and K are self-adjoint (possibly unbounded) operators in Af(R), and{Eλ}λ∈R, {Fλ}λ∈R are the spectral resolutions of H, K, respectively, then H · K =K · H if and only if K · Eλ = Eλ · K for each λ in R, and if and only ifEλFλ′ = Fλ′Eλ for all λ and λ′ in R. ([Z2]; Proposition 26)

3. Main results

Theorem 23. Let R be a finite von Neumann algebra. The Murray-von algebra

Af(R) is reflexive. That is, Af(R) = AlgLatAf(R).

Proof. By definition,

LatAf(R) = {Projections E : E(H) is properly stable under every T in Af(R)}.

Since Af(R) is a self-adjoint algebra (that is, when it contains T , it contains T ∗),from Theorem 19,

LatAf(R) = {Projections E : ET ⊆ TE for every T in Af(R)}.

From Lemma 18, TE = ET if T and E commute. Recall that, by definition,E · T = ET in Murray-von Neumann algebras. It follows that E · T = ET =TE = TE = T · E (note that TE is closed) and

LatAf(R) = {Projections E : E · T = T · E for every T in Af(R)}= {Projections E : E ∈ R′} (Proposition 22)


Then

AlgLatAf(R)

= {Closed, densely defined T : T leaves E(H) properly stable

for every E ∈ LatAf(R)}= {Closed, densely defined T : T leaves E(H) properly stable for every E ∈ R′}= {Closed, densely defined T : T leaves E(H) and E(H)⊥ properly stable

for every E ∈ R′}= {Closed, densely defined T : both T and T ∗ leave E(H) properly stable

for every E ∈ R′} (Proposition 20)

= {Closed, densely defined T : ET ⊆TE for every projection E∈R′} (Theorem 19)

= {Closed, densely defined T : UT ⊆TU for every unitary U ∈R′} (Proposition 10)

= {Closed, densely defined T : UT = TU for every unitary U ∈ R′} (Lemma 21)

= Af(R). �Theorem 24. Let R be a finite von Neumann algebra. The Murray-von Neu-

mann algebra Af(R) is algebraically reflexive, that is, Af(R) = RefaAf(R).

Proof. Clearly, Af(R) ⊆ RefaAf(R). To show that RefaAf(R) ⊆ Af(R),

since Af(R) = AlgLatAf(R), we shall show that RefaAf(R) ⊆ AlgLatAf(R). Sup-

pose T is in RefaAf(R). From the proof of the preceding theorem, to show that T

is in AlgLatAf(R), it suffices to show that for every projection E in R′, E(H) isproperly stable under T , that is,

T (D(T ) ∩ E(H)) ⊆ E(H) and E(D(T )) ⊆ D(T ),

which is, from Lemma 16, equivalent to

TE = ETE and E(D(T )) ⊆ D(T ).

Since T ∈ RefaAf(R), U ′(D(T )) ⊆ D(T ) for every unitary U ′ ∈ R′ and henceE(D(T )) ⊆ D(T ) for every projection E ∈ R′. For every x ∈ D(TE)(= D(ETE)),Ex ∈ D(T ), and

TEx = AExEx = EAExEx = ETEx,

for some AEx ∈ Af(R). Therefore, T ∈ AlgLatAf(R). �

References

[H] P. R. Halmos, Reflexive lattices of subspaces, J. London Math. Soc. (2) 4 (1971),257–263. MR0288612 (44 #5808)

[Ha] D. Hadwin, A general view of reflexivity, Trans. Amer. Math. Soc. 344 (1994), no. 1,325–360, DOI 10.2307/2154719. MR1239639 (95f:47071)

[L] D. R. Larson, Reflexivity, algebraic reflexivity and linear interpolation, Amer. J. Math.110 (1988), no. 2, 283–299, DOI 10.2307/2374503. MR935008 (89d:47096)

[KR] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras.Vol. I. Elementary theory. Vol. II. Advanced theory. Graduate Studies in Mathematics.AMS, 1997. MR1468229, MR1468230.

[KL] R. V. Kadison and S. A. Levin, Commutativity and Null Spaces of Unbounded Oper-ators on Hilbert Space, preprint.

[M-v.N. 1] F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936),no. 1, 116–229, DOI 10.2307/1968693. MR1503275

184 ZHE LIU

[RR] H. Radjavi and P. Rosenthal, On invariant subspaces and reflexive algebras, Amer. J.Math. 91 (1969), 683–692. MR0251569 (40 #4796)

[Z1] Zhe Liu, On some mathematical aspects of the Heisenberg relation, Sci. ChinaMath. 54 (2011), no. 11, 2427–2452, DOI 10.1007/s11425-011-4266-x. MR2859703(2012m:46066)

[Z2] Zhe Liu, A double commutant theorem for Murray-von Neumann algebras, Proc. Nat.Acad. Sci. U.S.A. 109 (2012), 7676-7681.

Department of Mathematics, University of Central Florida, Orlando, Florida

32816



Hochschild cohomology for tensor products of factors

Florin Pop and Roger R. Smith

Dedicated to Richard V. Kadison on the occasion of his 90 th birthday.

Abstract. In this paper we review some of the basic results for Hochschildcohomology for von Neumann algebras, concentrating on the most recent the-orems concerning tensor products. We describe methods rather than givingproofs, but we include detailed references to the literature on this topic.

1. Introduction

The algebraic theory of cohomology was initiated by Hochschild [19–21], basedon multilinear maps into a bimodule and coboundary operators. SubsequentlyJohnson, Kadison and Ringrose [22, 23, 27, 28] developed an appropriate theoryfor Banach algebras and operator algebras by requiring the relevant cocycles andcoboundaries to be bounded. The cohomology group Hn(A, V ) for an operatoralgebra A and an A-bimodule V is the quotient of the space of bounded n-cocyclesby the subspace of bounded n-coboudaries. There are many choices for V , but wewill focus almost exclusively on the case where A is a von Neumann algebra M andV is M itself.

There are several accounts of the early theory: a survey paper by Ringrose [45],a later book by Sinclair and the second author [47] and a subsequent survey bythe same pair of authors [50]. For this reason we will give only a brief expositionof the main results up to the publication of [50] and concentrate on more recenttheorems on tensor products.

The starting point for cohomology in the functional analytic setting is thetheorem of Kadison and Sakai [26,46] which states that every derivation δ : M →M is inner. In cohomological terms this means that H1(M,M) = 0. A similarquestion for the bimodule B(H) is still open, and is known from [30] to be equivalentto Kadison’s similarity problem [25]. It should also be noted that vanishing of firstcohomology for all dual M -bimodules occurs precisely when M is hyperfinite [12].The Kadison-Sakai result leads to the question of whether the higher order groupsHn(M,M), n ≥ 2, also vanish. This is not fully settled but is known to be true in

2010 Mathematics Subject Classification. Primary 46L10; Secondary 46L07.Key words and phrases. von Neumann algebra, factor, C∗–algebra, cohomology group, com-

plete boundedness, Cartan, property Γ, tensor product.This paper is an expanded version of a talk given by the second author in the special session

“Operator Algebras and Their Applications: A Tribute to Richard V. Kadison” at the San AntonioA.M.S. meeting, January 2015. RRS was partially supported by NSF grant DMS-1101403.


185

186 FLORIN POP AND ROGER R. SMITH

all cases where these groups have been determined. We note, however, that beyondthe self-adjoint setting cohomology groups of arbitrarily specified dimension arepossible [17], although we will not pursue this direction.

In Section 2 we give the definition of cocycles and coboundaries and also ofHn(A, V ). We also give a brief review of the theory of completely bounded lin-ear and multilinear maps on operator algebras that has become indispensible inthe theory of cohomology. There are several good detailed sources for completeboundedness, see [15,34,38].

Since two cocycles that differ by a coboundary define the same element ofHn(M,M), when studying a particular cocycle we have the freedom to perturb itby a suitably chosen coboundary in order to improve its properties. This is the topicof Section 3 and Theorem 3.1 summarizes what can be achieved in this direction.The subsequent section is devoted to completely bounded cohomology. Theorem4.1 is the crucial one for studying this type of cohomology and is derived fromstructural results for completely bounded multilinear maps using the techniquesof [46]. It leads to vanishing of the completely bounded cohomology groups inTheorem 4.2 and the vanishing of Hn(M,M) for various classes of von Neumannalgebras in Theorem 4.3.

The final section is devoted to a discussion of the results of [41] on the vanishingof H2(M ⊗N,M ⊗N) for tensor products of type II1 factors, where once againcomplete boundedness will play a crucial role.

2. Preliminaries

Let A be a C∗-algebra and let V be a Banach space that is also an A-bimodule.The main examples for us are V = B(H) where A is faithfully represented on Hand also V = A. For n ≥ 1, we denote by Ln(A, V ) the space of n-linear boundedmaps φ : An → V where An is the n-fold Cartesian product A×A×· · ·×A of copiesof A and L0(A, V ) denotes V . The coboundary map ∂ : Ln(A, V ) → Ln+1(A, V )is defined as follows. For n = 0, ∂v is the derivation x �→ xv − vx, while for n ≥ 1and φ ∈ Ln(A, V ) we let

∂φ(x1, . . . , xn+1) = x1φ(x2, . . . , xn)

+n−1∑i=1

(−1)iφ(x1, . . . , xi−1, xixi+1, xi+2, . . . , xn)

+ (−1)nφ(x1, . . . , xn)xn+1,(2.1)

for xi ∈ A, 1 ≤ i ≤ n + 1. A routine calculation shows that ∂∂ = 0. The kernelof ∂ : Ln(A, V ) → Ln+1(A, V ) is the space of cocycles, and contains the space ofcoboundaries which is the image of ∂ : Ln−1(A, V ) → Ln(A, V ). The quotient ofthese spaces is the nth Hochschild cohomology group Hn(A, V ). For the case n = 1,the cocycles are the derivations and the coboundaries are the inner derivations, soall derivations are inner precisely when H1(A, V ) = 0. Having given the definitionfor a general module V , we will eventually concentrate on the situation where A isa von Neumann algebra M and V = M .

An important role in the theory is played by the completely bounded multi-linear maps. If φ : A → B(H) is a bounded map, then φ lifts to a bounded mapφn : Mn(A) → Mn(B(H)) by (aij) �→ (φ(aij)). Then φ is completely bounded ifsupn≥1 ‖φn‖ < ∞ and this supremum defines ‖φ‖cb. There is an extensive theory

COHOMOLOGY FOR TENSOR PRODUCTS OF FACTORS 187

of such maps (see [8, 9, 15, 34, 38]) and there is a very useful representation re-sult: if φ : A → B(H) is completely bounded, then there exists a representationπ : A → B(K) and maps V : K → H, W : H → K so that

(2.2) φ(a) = V π(a)W, a ∈ A.

Moreover, V and W can be chosen to satisfy the optimal estimate ‖V ‖ = ‖W‖ =

‖φ‖1/2cb .There is also a theory of completely bounded multilinear maps developed in

[8–11] for C∗-algebras and extended in [35] to general operator spaces. We givethe definition for bilinear maps and the extension to n-linear ones should be clearsince we are mimicing matrix multiplication.

If φ : A × B → B(H), then φ lifts to a bilinear map φn : Mn(A) ×Mn(B) →Mn(B(H)) by specifying the (i, j)-entry of φn((aij), (bij)) to be

∑nk=1 φ(aik, bkj),

and if supn≥1 ‖φn‖ < ∞ then φ is said to be completely bounded with this supre-mum defining ‖φ‖cb. The representation theorem in this case is as follows. Forsimplicity we state it for two variables but the extension to n variables only in-volves adding more representations, Hilbert spaces and connecting operators.

Theorem 2.1 ([8]). Let A1, A2 be C∗-algebras. A bounded bilinear map φ : A1×

A2 → B(H) is completely bounded if and only if it may be expressed by

(2.3) φ(x, y) = V1π1(x)V2π2(y)V3, x ∈ A1, y ∈ A2,

where πi : Ai → B(Ki), i = 1, 2, are ∗-representations and V3 : H → K2,V2 : K2 → K1, V1 : K1 → H are bounded operators satisfying

(2.4) ‖V1‖ = ‖V2‖ = ‖V3‖ = ‖φ‖1/3cb .

Since it is easy to check that any map with such a representation is completelybounded, it becomes a routine exercise to see that ∂ preserves complete bound-edness. Thus there is a parallel theory of completely bounded cohomology groupsHn

cb(M,M) in which the defining maps are required to be completely bounded.If A ⊆ B ⊆ B(H) are C∗-algebras, then φ : B → B(H) is said to be A-modular

if

(2.5) φ(ab) = aφ(b), φ(ba) = φ(b)a

for a ∈ A and b ∈ B. Bimodularity of φ : B × B → B(H) is defined similarly bythe requirements

aφ(b1, b2) = φ(ab1, b2),(2.6)

φ(b1a, b2) = φ(b1, ab2),(2.7)

φ(b1, b2a) = φ(b1, b2)a,(2.8)

for a ∈ A and b1, b2 ∈ B, with an obvious extension to A-multimodularity of mapsφ : Bn → B(H). As will be seen below, reduction to cocycles that are multimodularwith respect to a suitably chosen subalgebra is a crucial part of the theory.

3. Reduction of cocycles

If we have a bounded cocycle φ : Mn → M and a hyperfinite subalgebraR ⊆ M , then by averaging repeatedly over an amenable group of unitaries in Rthat generates a weakly dense C∗-algebra A, we find that φ is equivalent to a cocycle


ψ : Mn → M which vanishes whenever one of its arguments is in A. We illustratethis for derivations.

Let G be an amenable group of unitaries that generates a C∗-algebra A ⊆ M ,let μ be an invariant mean for G and let δ : M → M be a derivation. We abusenotation and write the action of μ as an integral. For u, v ∈ G,

(3.1)

∫G

u∗δ(uv) dμ(u) =

∫G

u∗uδ(v) dμ(u) +

∫G

u∗δ(u)v dμ(u)

so

δ(v) =

∫G

u∗δ(uv) dμ(u)−∫G

u∗δ(u)v dμ(u)

=

∫G

v(uv)∗δ(uv) dμ(u)−∫G

u∗δ(u)v dμ(u).(3.2)

Let t ∈ M be∫Gu∗δ(u) dμ(u). The invariance of μ gives

(3.3) δ(v) = vt− tv, v ∈ G.

If δ0(x) = δ(x)− xt+ tx for x ∈ M , then δ0 is equivalent to δ and δ∣∣A= 0. Then

(3.4) δ(am) = δ(a)m+ aδ(m) = aδ(m), a ∈ A, m ∈ M,

so δ is left A-modular, with a similar calculation for right A-modularity.More sophisticated averaging techniques [23] in the second dual allow us to

reduce cocycles to ones that are separately normal in each variable and are R-multimodular. One consequence is that Hn(R,R) = 0 for all hyperfinite von Neu-mann algebras.

While it is unknown whether cocycles φ : Mn → M are coboundaries into M ,there is a result on extended cobounding [28] which expresses a cocycle φ as ∂ψwhere ψ maps Mn−1 into a larger von Neumann algebra. We fix a masa B ⊆ M ′

and extend φ to a cocycle φ1 : C∗(M,B)n → B(H) on generators by

(3.5) φ1(m1b1, . . . ,mnbn) = φ(m1, . . . ,mn)b1 · · · bn, mi ∈ M bi ∈ B.

This extends to a cocycle on W ∗(M,B)n. Since W ∗(M,B)′ = B, we see thatW ∗(M,B) is hyperfinite so its cohomology is trivial. Consequently φ1 = ∂ψ1 forsome bounded map ψ1 : W ∗(M,B)n−1 → W ∗(M,B), and then take ψ to be ψ1

∣∣M.

A further refinement is that if N is a hyperfinite von Neumann algebra so thatM ⊆ N ⊆ B(H), and if EN : B(H) → N is a conditional expectation, then ψ1

can be replaced by EN ◦ ψ1, so that the coboundary has its range in N , and asubsequent averaging argument allows it to be separately normal. We summarizethese reductions, proved in [23,28] as

Theorem 3.1. Let φ : Mn → M be a bounded n-cocycle. Let R ⊆ M bea hyperfinite von Neumann subalgebra. Then φ is equivalent to an n-cocycle ψ :Mn → M that is separately normal in each variable, vanishes whenever any of thearguments lies in R, and is R-multimodular. If N is a hyperfinite von Neumannalgebra containing M , then there exists a separately normal R-multimodular mapψ : Mn−1 → N so that φ = ∂ψ.


4. Completely bounded cohomology

In this section, we restrict attention to the cohomology groups Hncb(M,M)

where the cocycles and coboundaries are assumed to be completely bounded. Thisis the part of the theory that is fully worked out as we will see below. To motivatethe next result, consider a completely bounded map φ : M → M of the specialform φ(x) = axb, x ∈ M , where a, b ∈ M are fixed. By the Dixmier approximationtheorem [29, Prop. 8.3.4] there is a central element z ∈ Z(M), sets of unitaries{ui,n : 1 ≤ i ≤ kn} and nonnegative constants {λi,n : 1 ≤ i ≤ kn} summing to 1 sothat

(4.1) ‖z −kn∑i=1

λi,nui,nbu∗i,n‖ <

1

n, n ≥ 1.

Then

(4.2) limn→∞

kn∑i=1

φ(x√λi,nui,n)

√λi,nu

∗i,n = axz

in norm and the resulting map on the right hand side is right M -modular since

(4.3) axyz = (axz)y, x, y ∈ M.

If we take sets of operators {mi} in M satisfying∑i

mim∗i = 1 and form the

completely bounded maps

(4.4) x �→∑i

φ(xmi)m∗i , x ∈ M,

then suitable choices of the mi’s ensure convergence strongly to a right M -modulemap ρφ. The compete boundedness of φ is essential for this to work. The conclusionis

Theorem 4.1 ([11]). There is a contractive projection ρ from the space CB(M,M)of completely bounded maps to the space CB(M,M)M of completely bounded rightM -module maps.

To see the relevance of this for cohomology, we look at the simplest situation,a derivation δ : M → M . By [26] there exists t ∈ B(H) so that t implements δ, soδ is completely bounded (we give a different proof of this at the end of this sectionthat may be of independent interest). Since right M -module maps have the simpleform x �→ ax for some a ∈ M , when we apply ρ to the equation

(4.5) δ(xy) = xδ(y) + δ(x)y, x, y ∈ M

thinking of x as fixed and y as the variable, we get

(4.6) axy = xay + δ(x)y, x, y ∈ M,

and putting y = 1 gives δ(x) = ax − xa, showing that δ is inner. This approachworks on higher order cohomology by working with the rightmost variable, andleads to the result that

Theorem 4.2 ([10,11]). Let M be a von Neumann algebra. Then

(4.7) Hncb(M,M) = 0, n ≥ 1.


Much of the recent work on cohomology relies on this fundamental result, andthe general approach is to show that a cocycle is equivalent to one that is completelybounded (at least in the rightmost variable) so that the techniques of Thereom 4.2can be applied. We summarize here the results that have been obtained in thismanner, with a brief discussion following.

Theorem 4.3. Let M be a von Neumann algebra. Then Hn(M,M) = 0 forn ≥ 1 in the following situations:

(i) M is type I, II∞ or III.(ii) M is a type II1 factor and is McDuff (M ∼= M ⊗R where R is the hyper-

finite II1 factor [31,32]).(iii) M is a II1 factor with a Cartan subalgebra A (A is a masa and its nor-

malizing unitaries in M generate M).(iv) M is a II1 factor with property Γ and separable predual.

Some comments on the parts of this theorem:

(i) Type I von Neumann algebras M are hyperfinite so Theorem 3.1 showsthat each cocycle φ is equivalent to ψ which vanishes whenever one of thearguments is from M , i.e. ψ = 0. This result is in [27]. When M istype II∞ or III then M ∼= M ⊗B(H) and so a cocycle φ which is B(H)-modular has the form ψ ⊗ I and is easily seen to be completely bounded.These two cases were proved in [4,10].

(ii) When M ∼= M ⊗R then, as in (i), we reduce to a cocycle of the formψ⊗ I. Since R contains arbitrarily large matrix subfactors, the cocycle iscompletely bounded. This case was proved in [10].

(iii) The key observation is that if A is a Cartan subalgebra of a II1 factor withseparable predual then A∨ JAJ is a masa in B(H) [43], allowing certaincocycles to be shown to be completely bounded. This was used in [40]to prove the case n = 2, and further refinements settled the cases n = 3[5] and n ≥ 4 [48,49]. Subsequently Cameron [1] was able to remove theseparable predual hypothesis to obtain the general case for factors withCartan subalgebras.

(iv) Recall that a II1 factor M has property Γ if, given x1, . . . , xn ∈ M andε > 0, there exists a unitary u ∈ M with τ (u) = 0 and

(4.8) ‖[u, xi]‖2 < ε, 1 ≤ i ≤ n.

An equivalent formulation due to Dixmier [13] is the following: given aninteger n, elements x1, . . . , xm ∈ M and ε > 0, there exist orthogonalprojections p1, . . . pn ∈ M summing to 1 with τ (pi) =

1n so that

(4.9) ‖[xi, pj ]‖2 < ε, 1 ≤ i ≤ n, 1 ≤ j ≤ m.

In [6,7] these projections were used to show that cocycles are equivalentto ones that are completely bounded, establishing that Hn(M,M) = 0for n ≥ 1. Subsequently property Γ was extended to the non-factor casein [44] and vanishing of the cohomology was proved in a similar manner.

As we mentioned after Theorem 4.1, we would like to give an alternative proofof the complete boundedness of derivations δ : M → M . This will be deduced fromthe following proposition concerning column boundedness. A similar inequality wasproved by Christensen [3], but with constant 14 instead of 4.


Proposition 4.4. If M ⊆ B(H) is a von Neumann algebra and δ : M → B(H)is a derivation, then

(4.10) ||n∑

i=1

δ(ai)∗δ(ai)|| ≤ 4 ||δ||2 · ||

n∑i=1

a∗i ai||, ai ∈ M, 1 ≤ i ≤ n.

Proof. Fix an element a ∈ M with polar decomposition a = vh, where vis a partial isometry and h = (a∗a)1/2 ≥ 0. For every unitary u ∈ M we haveδ(a) = δ(vu∗uh) = vu∗δ(uh) + δ(vu∗)uh, and so

δ(a)∗δ(a) = [δ(uh)∗uv∗ + hu∗δ(vu∗)∗] · [vu∗δ(uh) + δ(vu∗)uh]

= δ(uh)∗uv∗vu∗δ(uh) + δ(uh)∗uv∗δ(vu∗)uh

+ hu∗δ(vu∗)∗vu∗δ(uh) + hu∗δ(vu∗)∗δ(vu∗)uh.(4.11)

For arbitrary x, y ∈ M we have (x−y)∗(x−y) ≥ 0, hence x∗x+y∗y ≥ x∗y+y∗x.It follows that

δ(a)∗δ(a) ≤ 2 [δ(uh)∗uv∗vu∗δ(uh) + hu∗δ(vu∗)∗δ(vu∗)uh]

≤ 2 δ(uh)∗δ(uh) + 2 ||δ||2h2.(4.12)

Fix ξ ∈ H, ||ξ|| ≤ 1. Then [37, Theorem 7.3] ensures the existence of an ultra-filter ω on N, a sequence of unitary operators un ∈ M, and a state f on M suchthat lim

ω||δ(una)ξ||2 ≤ ||δ||2f(a2) for all self-adjoint elements a ∈ M. We obtain

(4.13) 〈δ(a)∗δ(a)ξ, ξ〉 ≤ 2 ||δ(unh)ξ||2 + 2 ||δ||2⟨h2ξ, ξ

⟩,

and by taking the limit along ω we get

(4.14) 〈δ(a)∗δ(a)ξ, ξ〉 ≤ 2 ||δ||2f(h2) + 2 ||δ||2⟨h2ξ, ξ

⟩.

For a1, . . . , an ∈ M we have⟨n∑

i=1

δ(ai)∗δ(ai)ξ, ξ

⟩≤ 2 ||δ||2f(

n∑i=1

a∗i ai) + 2 ||δ||2⟨

n∑i=1

a∗i aiξ, ξ

⟩

≤ 4 ||δ||2||n∑

i=1

a∗i ai||(4.15)

and the conclusion follows. �

The next result proves complete boundedness of derivations (see [26]).

Theorem 4.5. If M ⊆ B(H) is a von Neumann algebra and δ : M → M is aderivation, then δ is completely bounded and ||δ||cb ≤ 4 ||δ||.

Proof. Let X be an n × n matrix over M and fix ε > 0. Since M column-norms itself [39], choose a contractive n× 1 column C with entries in M such that||δn(X)|| ≤ ||δn(X)C||+ ε. Then, by using Proposition 4.4, we have

||δn(X)|| ≤ ||δn(X)C||+ ε = ||δn(XC)−Xδn(C)||+ ε

≤ 4 ||δ|| · ||X|| · ||C||+ ε ≤ 4 ||δ|| · ||X||+ ε.(4.16)

Since ε > 0 was arbitrary, the conclusion follows. �


5. Tensor products

In this section we describe some recent work [41] on the second cohomologyof tensor products M ⊗N . We restrict to the case where both algebras are II1factors with separable preduals for simplicity, but the results are true for tensorproducts of general II1 von Neumann algebras. From [42], this allows us to specifysubalgebras A ⊆ R ⊆ M and B ⊆ S ⊆ N such that A and B are masas in Mand N respectively while R and S are hyperfinite subfactors with trivial relativecommutants in their containing factors. These subalgebras play an important rolein proving the main result of this section:

Theorem 5.1. ([41]) Let M and N be II1 factors with separable preduals. Then

(5.1) H2(M ⊗N,M ⊗N) = 0.

We will sketch the proof of this result, setting out the required lemmas. Asalways in cohomology, complete boundedness is an important component.

Lemma 5.2 ([10]). Let M ⊆ B(H) and S ⊆ B(K) be II1 factors with Shyperfinite, and let φ : M ⊗S → B(H ⊗ K) be bounded, normal and(I ⊗ S)-modular. Then φ is completely bounded.

Sketch. If ψ is the restriction of φ to M ⊗ I then the (I ⊗ S)-modularityimplies that ψ maps into S′ so φ maps M ⊗min S into C∗(S′, S). From [14] there isan isomorphism ρ : S′ ⊗min S → C∗(S′, S) given by ρ(s′s) = s′⊗s. Then φ

∣∣M ⊗min S

can be realized as ρ◦(ψ⊗ idS), and complete boundedness follows from normality ofφ, the w∗-density of M ⊗min S in M ⊗S, and the Kaplansky density theorem. �

We also need a result on complete boundedness of certain multimodular bilinearmaps.

Lemma 5.3. Let φ : (M ⊗S) × (M ⊗S) → M ⊗S be a bounded separatelynormal (R⊗S)-multimodular bilinear map. Then φ is completely bounded.

This result depends on an inequality [47, 5.4.5 (ii)] stemming from Grothen-dieck’s inequality [18,36],

(5.2)

∥∥∥∥∥n∑

i=1

φ(xi, yi)

∥∥∥∥∥ ≤ 2‖φ‖∥∥∥∥∥

n∑i=1

xix∗i

∥∥∥∥∥1/2 ∥∥∥∥∥

n∑i=1

y∗i yi

∥∥∥∥∥1/2

,

and also the fact that I ⊗ S norms M ⊗S in the sense of [39].It is not known in general whether every derivation δ : M → B(H) is inner,

but there are many circumstances where positive results in this direction have beenobtained. We quote two of these:

Lemma 5.4 (Theorem 3.1 in [3]). Each completely bounded derivation δ : M →B(H) is inner and is implemented by an operator in B(H).

Lemma 5.5 (special case of Theorem 5.1 in [2]). If M ⊆ N is an inclusionof finite von Neumann algebras, then each derivation δ : M → N is inner and isimplemented by an element of N .

In considering a 2-cocycle φ : (M ⊗N)× (M ⊗N) → M ⊗N , adding cobound-aries allows us to reduce to the situation where the following four conditions aresatisfied:


(C1) φ is separately normal in each variable;(C2) φ(x, y) = 0 whenever x or y lies in R⊗S;(C3) φ is R⊗S-multimodular;(C4) φ(m1 ⊗ I,m2 ⊗ I) = φ(I ⊗ n1, I ⊗ n2) = 0 for m1,m2 ∈ M , n1, n2 ∈ N .

The first three are standard reductions, as in Theorem 3.1. To see that (C4)can be achieved, first observe that (I ⊗ S)-modularity ensures that φ maps (M ⊗I)×(M⊗I) into (I⊗S)′∩(M ⊗N) = M⊗I, from which φmaps (M ⊗S)×(M ⊗S)into M ⊗S. Then φ

∣∣M ⊗S

is completely bounded by Lemma 5.2. Then there is

a normal (R⊗S)-modular map α : M ⊗S → M ⊗S so that φ∣∣M ⊗S

= ∂α, and

similarly a normal (R⊗S)-modular map β : R⊗N → R⊗N so that φ∣∣R⊗N

= ∂β.

These maps extend to α, β : M ⊗N → M ⊗N by

(5.3) α = α ◦ EM ⊗S , β = β ◦ ER⊗N

and one checks that ψ = φ− ∂(α+ β) satisfies (C1)-(C4).We now digress briefly for a discussion of the basic construction. If P ⊆ Q is a

containment of finite von Neumann algebras where Q has a faithful normal trace τ ,then we view Q as being in its standard representation on L2(Q, τ ), or just L2(Q).There is a projection eP of L2(Q) onto L2(P ) and the basic construction 〈Q, eP 〉 isthe von Neumann algebra generated by Q and eP . The map x → x∗ on Q extendsto a conjugate linear isometry J : L2(Q) → L2(Q) and 〈Q, eP 〉′ = JPJ . (See [24]or [51, Chapter 4] for details of the basic construction). If P is hyperfinite then sois 〈Q, eP 〉, while if P is a masa in Q then JPJ is a masa in JQJ = Q′.

If we apply this to A⊗B⊆R⊗S⊆M ⊗N , then 〈M ⊗N, eA⊗B〉′=J(A⊗B)Jis a masa in (M ⊗N)′, so from Theorem 3.1 a cocycle φ : (M ⊗N) × (M ⊗N) →M ⊗N can be expressed as ∂λ where λ : M ⊗N → 〈M ⊗N, eA⊗B〉. Since〈M ⊗N, eR⊗S〉 is hyperfinite, we can use a conditional expectation E〈M ⊗N,eR ⊗ S〉to define γ : M ⊗N → 〈M ⊗N, eR⊗S〉 by γ = E〈M ⊗N,eR ⊗S〉 ◦λ, and we also have

φ = ∂γ. Moreover, by Theorem 3.1 we can take γ to be normal.We introduce three auxiliary maps. They are not obviously bounded so at the

outset we only define them on the algebraic tensor product M ⊗ N by

f(m⊗ n) = φ(m⊗ I, I ⊗ n) + γ(m⊗ n),(5.4)

g(m⊗ n) = φ(I ⊗ n,m⊗ I) + γ(m⊗ n),(5.5)

h(m⊗ n) = g(m⊗ n)− f(m⊗ n)

= φ(I ⊗ n,m⊗ I)− φ(m⊗ I, I ⊗ n).(5.6)

The objective is to show that these maps are bounded. This is perhaps sur-prising for the map m⊗ n �→ φ(m⊗ I, I ⊗ n), but our assumption that φ satisfies(C1)-(C4) puts severe restrictions on the form that φ can take. It may be helpfulto look at a particular example of such a φ.

Let α : M → M be a completely bounded normal R-modular map satisfyingα(I) = 0, say α(m) = ER(m)−m. Similarly define β : N → N by β(n) = ES(n)−n.If we put ξ = α ⊗ β and φ = ∂ξ, then it is easy to see that φ satisfies (C1)-(C4).Moreover,

φ(m⊗ I, I ⊗ n) = (m⊗ I)(α(I)⊗ β(n))− α(m)⊗ β(n)

+ (α(m)⊗ β(I))(I ⊗ n) = −ξ(m⊗ n),(5.7)

and we see boundedness of m⊗ n �→ φ(m⊗ I, I ⊗ n) from the boundedness of ξ.


The next lemma lists some properties of these functions f , g, and h. The mapγ below is defined in the paragraph preceding (5.4).

Lemma 5.6. The following properties hold:

(i) The restrictions γ|M⊗S and γ|R⊗N are completely bounded derivations,spatially implemented by elements of 〈M⊗N, eR⊗S〉.

(ii) The restrictions f |M⊗I , f |I⊗N , g|M⊗I , and g|I⊗N are equal to the respec-tive restrictions of γ to these subalgebras, and are all bounded derivationsspatially implemented by elements of 〈M⊗N, eR⊗S〉.

(iii) The restrictions h|M⊗I and h|I⊗N are both 0.

Sketch. Since φ∣∣M⊗I

= 0 by (C4) and φ = ∂γ, we see that γ∣∣M⊗I

is a deriva-

tion and the same is true for γ∣∣M ⊗S

by (R⊗S)-modularity. Then complete bound-

edness of γ∣∣M ⊗S

is a consequence of Lemma 5.2. Thus γ∣∣M ⊗S

is implemented by

an element t ∈ B(L2(M ⊗N)) and further by an element of 〈M ⊗N, eR⊗S〉 byapplying the conditional expectation onto this algebra. The same argument worksfor γ

∣∣R⊗N

.

The other parts follow routinely from (i). �

The next result is a rather long algebraic calculation based on conditions (C1)-(C4) and the cocycle condition ∂φ = 0. We refer to [41, Prop. 3.3] for the details.

Proposition 5.7. The map f of (5.4) is a derivation on M ⊗N .

The next result is the key observation, namely that the mapping m ⊗ n �→φ(m⊗ I, I⊗n) is not only bounded but also has a normal extension to M ⊗N . Weinclude the proof from [41].

Proposition 5.8. There exists a bounded normal map ξ : M⊗N → M⊗Nsuch that

(5.8) ξ(m⊗ n) = φ(m⊗ I, I ⊗ n), m ∈ M, n ∈ N.

Proof. From Proposition 5.7, f is a derivation on M ⊗ N with values in〈M⊗N, eR⊗S〉 = 〈M, eR〉⊗〈N, eS〉. By Lemma 5.6 (ii), f |M⊗I is a completelybounded derivation implemented by an element t ∈ 〈M, eR〉⊗〈N, eS〉. Define aderivation δ : M ⊗N → 〈M, eR〉⊗〈N, eS〉 by

(5.9) δ(m⊗ n) = f(m⊗ n)− [t(m⊗ n)− (m⊗ n)t], m ∈ M, n ∈ N.

Then δ|M⊗I = 0 from (5.9), so δ is (M ⊗ I)-modular. From Lemma 5.6 (ii), f |1⊗N

is a derivation implemented by an element of 〈M, eR〉⊗〈N, eS〉, so from (5.9) thereis an element b in this algebra such that

(5.10) δ(I ⊗ n) = b(I ⊗ n)− (I ⊗ n)b, n ∈ N.

The (M ⊗ I)-modularity of δ shows that, for m ∈ M and n ∈ N ,

(5.11) (m⊗ I)δ(I ⊗ n) = δ(m⊗ n) = δ(I ⊗ n)(m⊗ I),

and we conclude that the range of δ|I⊗N lies in (M ⊗ I)′ ∩ 〈M, eR〉⊗〈N, eS〉. Thisalgebra is (M ′ ∩ 〈M, eR〉)⊗〈N, eS〉, equal to (JMJ ∩ (JRJ)′)⊗〈N, eS〉, and in turnequal to I⊗〈N, eS〉. The latter algebra is hyperfinite, so if we take a conditional


expectation onto it and apply this to (5.10), then we conclude that the element bof (5.10) may be assumed to lie in I⊗〈N, eS〉. Then b commutes with M ⊗ I, so

δ(m⊗ n) = (m⊗ I)δ(I ⊗ n) = (m⊗ I)[b(I ⊗ n)− (I ⊗ n)b]

= b(m⊗ n)− (m⊗ n)b, m ∈ M, n ∈ N.(5.12)

Thus δ has a unique bounded normal extension to M⊗N , and (5.9) shows thatthe same is then true for f . Since ξ = f − γ on M ⊗ N from (5.4), and γ isalready bounded and normal on M⊗N , this gives a bounded normal extension ofξ to M⊗N . �

The proof of this proposition and equation (5.4) show that f has a boundednormal extension to M ⊗N and so is a derivation. Since φ = ∂γ = ∂f −∂ξ = −∂ξ,and ξ has its range in M ⊗N , we see that φ is a coboundary, establishing Theorem5.1.

If Fn is the free group on n generators, n ≥ 2, then the II1 factors L(Fn) donot have a Cartan subalgebra [52], do not have property Γ [33], and are primeso are not tensor products of II1 factors [16]. Thus none of the theorems in thispaper apply to L(Fn) and so nothing is known of the cohomology beyond the firstcohomology group [26,46] in this case. Any progress on the higher cohomology offree group factors would be very interesting.

References

[1] J. M. Cameron, Hochschild cohomology of II1 factors with Cartan maximal abelian subalge-bras, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 2, 287–295, DOI 10.1017/S0013091507000053.MR2506393 (2010h:46090)

[2] E. Christensen, Extension of derivations, J. Funct. Anal. 27 (1978), no. 2, 234–247, DOI10.1016/0022-1236(78)90029-0. MR481217 (80d:46114)

[3] E. Christensen, Extensions of derivations. II, Math. Scand. 50 (1982), no. 1, 111–122.MR664512 (83m:46092)

[4] E. Christensen, E. G. Effros, and A. Sinclair, Completely bounded multilinear maps and C∗-algebraic cohomology, Invent. Math. 90 (1987), no. 2, 279–296, DOI 10.1007/BF01388706.MR910202 (89k:46084)

[5] E. Christensen, F. Pop, A. M. Sinclair, and R. R. Smith, On the cohomology groupsof certain finite von Neumann algebras, Math. Ann. 307 (1997), no. 1, 71–92, DOI10.1007/s002080050023. MR1427676 (98c:46130)

[6] E. Christensen, F. Pop, A. M. Sinclair, and R. R. Smith, Hochschild cohomology of fac-tors with property Γ, Ann. of Math. (2) 158 (2003), no. 2, 635–659, DOI 10.4007/an-nals.2003.158.635. MR2018931 (2004h:46072)

[7] E. Christensen, F. Pop, A. M. Sinclair, and R. R. Smith, Property Γ factors and theHochschild cohomology problem, Proc. Natl. Acad. Sci. USA 100 (2003), no. 7, 3865–3869,DOI 10.1073/pnas.0737489100. MR1963813 (2004f:46070)

[8] E. Christensen and A. M. Sinclair, Representations of completely bounded multilinear op-erators, J. Funct. Anal. 72 (1987), no. 1, 151–181, DOI 10.1016/0022-1236(87)90084-X.MR883506 (89f:46113)

[9] E. Christensen and A. M. Sinclair, A survey of completely bounded operators, Bull. LondonMath. Soc. 21 (1989), no. 5, 417–448, DOI 10.1112/blms/21.5.417. MR1005819 (91b:46051)

[10] E. Christensen and A.M. Sinclair, On the Hochschild cohomology for von Neumann algebras,unpublished manuscript.

[11] E. Christensen and A. M. Sinclair, Module mappings into von Neumann algebras and injec-tivity, Proc. London Math. Soc. (3) 71 (1995), no. 3, 618–640, DOI 10.1112/plms/s3-71.3.618.MR1347407 (96m:46107)

[12] A. Connes, On the cohomology of operator algebras, J. Functional Analysis 28 (1978), no. 2,248–253. MR0493383 (58 #12407)


[13] J. Dixmier, Quelques proprietes des suites centrales dans les facteurs de type II1 (French),Invent. Math. 7 (1969), 215–225. MR0248534 (40 #1786)

[14] E. G. Effros and E. C. Lance, Tensor products of operator algebras, Adv. Math. 25 (1977),no. 1, 1–34. MR0448092 (56 #6402)

[15] E. G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs.New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.MR1793753 (2002a:46082)

[16] L. Ge, Applications of free entropy to finite von Neumann algebras. II, Ann. of Math. (2)147 (1998), no. 1, 143–157, DOI 10.2307/120985. MR1609522 (99c:46068)

[17] F. L. Gilfeather and R. R. Smith, Cohomology for operator algebras: cones and suspen-sions, Proc. London Math. Soc. (3) 65 (1992), no. 1, 175–198, DOI 10.1112/plms/s3-65.1.175.MR1162492 (93i:46137)

[18] U. Haagerup, The Grothendieck inequality for bilinear forms on C∗-algebras, Adv. in Math.56 (1985), no. 2, 93–116, DOI 10.1016/0001-8708(85)90026-X. MR788936 (86j:46061)

[19] G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46(1945), 58–67. MR0011076 (6,114f)

[20] G. Hochschild, On the cohomology theory for associative algebras, Ann. of Math. (2) 47(1946), 568–579. MR0016762 (8,64c)

[21] G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14(1947), 921–948. MR0022842 (9,267b)

[22] B. E. Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence,R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR0374934 (51 #11130)

[23] B. E. Johnson, R. V. Kadison, and J. R. Ringrose, Cohomology of operator algebras. III.Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972), 73–96. MR0318908(47 #7454)

[24] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25, DOI10.1007/BF01389127. MR696688 (84d:46097)

[25] R. V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 77(1955), 600–620. MR0072442 (17,285c)

[26] R. V. Kadison, Derivations of operator algebras, Ann. of Math. (2) 83 (1966), 280–293.

MR0193527 (33 #1747)[27] R. V. Kadison and J. R. Ringrose, Cohomology of operator algebras. I. Type I von Neumann

algebras, Acta Math. 126 (1971), 227–243. MR0283578 (44 #809)[28] R. V. Kadison and J. R. Ringrose, Cohomology of operator algebras. II. Extended cobounding

and the hyperfinite case, Ark. Mat. 9 (1971), 55–63. MR0318907 (47 #7453)[29] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I,

Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich,Publishers], New York, 1983. Elementary theory. MR719020 (85j:46099)

[30] E. Kirchberg, The derivation problem and the similarity problem are equivalent, J. OperatorTheory 36 (1996), no. 1, 59–62. MR1417186 (97f:46108)

[31] D. McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21(1970), 443–461. MR0281018 (43 #6737)

[32] D. McDuff, On residual sequences in a II1 factor, J. London Math. Soc. (2) 3 (1971), 273–280.MR0279597 (43 #5318)

[33] F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943),716–808. MR0009096 (5,101a)

[34] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathemat-ics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., NewYork, 1986. MR868472 (88h:46111)

[35] V. I. Paulsen and R. R. Smith, Multilinear maps and tensor norms on operator systems,J. Funct. Anal. 73 (1987), no. 2, 258–276, DOI 10.1016/0022-1236(87)90068-1. MR899651(89m:46099)

[36] G. Pisier, Grothendieck’s theorem for noncommutative C∗-algebras, with an appendix onGrothendieck’s constants, J. Funct. Anal. 29 (1978), no. 3, 397–415, DOI 10.1016/0022-1236(78)90038-1. MR512252 (80j:47027)

[37] G. Pisier, Similarity problems and completely bounded maps, Lecture Notes in Mathematics,vol. 1618, Springer-Verlag, Berlin, 1996. MR1441076 (98d:47002)


[38] G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture NoteSeries, vol. 294, Cambridge University Press, Cambridge, 2003. MR2006539 (2004k:46097)

[39] F. Pop, A. M. Sinclair, and R. R. Smith, Norming C∗-algebras by C∗-subalgebras, J. Funct.Anal. 175 (2000), no. 1, 168–196, DOI 10.1006/jfan.2000.3601. MR1774855 (2001h:46105)

[40] F. Pop and R. R. Smith, Cohomology for certain finite factors, Bull. London Math. Soc. 26(1994), no. 3, 303–308, DOI 10.1112/blms/26.3.303. MR1289052 (95g:46133)

[41] F. Pop and R. R. Smith, Vanishing of second cohomology for tensor products of

type II1 von Neumann algebras, J. Funct. Anal. 258 (2010), no. 8, 2695–2707, DOI10.1016/j.jfa.2010.01.013. MR2593339 (2012b:46125)

[42] S. Popa, On a problem of R. V. Kadison on maximal abelian ∗-subalgebras in factors, Invent.Math. 65 (1981/82), no. 2, 269–281, DOI 10.1007/BF01389015. MR641131 (83g:46056)

[43] S. Popa, Notes on Cartan subalgebras in type II1 factors, Math. Scand. 57 (1985), no. 1,171–188. MR815434 (87f:46114)

[44] W. Qian and J. Shen, Hochschild cohomology of type II1 von Neumann algebras with PropertyΓ. arXiv:1407.0664.

[45] J. R. Ringrose, Cohomology of operator algebras, Lectures on operator algebras (Tulane Univ.Ring and Operator Theory Year, 1970–1971, Vol. II; dedicated to the memory of David M.Topping), Springer, Berlin, 1972, pp. 355–434. Lecture Notes in Math., Vol. 247. MR0383102(52 #3983)

[46] S. Sakai, Derivations of W ∗-algebras, Ann. of Math. (2) 83 (1966), 273–279. MR0193528(33 #1748)

[47] A. M. Sinclair and R. R. Smith, Hochschild cohomology of von Neumann algebras, LondonMathematical Society Lecture Note Series, vol. 203, Cambridge University Press, Cambridge,1995. MR1336825 (96d:46094)

[48] A. M. Sinclair and R. R. Smith, Hochschild cohomology for von Neumann algebras withCartan subalgebras, Amer. J. Math. 120 (1998), no. 5, 1043–1057. MR1646053 (99j:46087)

[49] A. M. Sinclair and R. R. Smith, The Hochschild cohomology problem for von Neu-mann algebras, Proc. Natl. Acad. Sci. USA 95 (1998), no. 7, 3376–3379 (electronic), DOI10.1073/pnas.95.7.3376. MR1622277 (99b:46110)

[50] A. M. Sinclair and R. R. Smith, A survey of Hochschild cohomology for von Neu-

mann algebras, Operator algebras, quantization, and noncommutative geometry, Con-temp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 383–400, DOI10.1090/conm/365/06712. MR2106829 (2005h:46089)

[51] A. M. Sinclair and R. R. Smith, Finite von Neumann algebras and masas, London Mathe-matical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008.MR2433341 (2009g:46116)

[52] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free prob-ability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), no. 1,172–199, DOI 10.1007/BF02246772. MR1371236 (96m:46119)

Department of Mathematics and Computer Science, Wagner College, Staten Is-

land, New York 10301


Department of Mathematics, Texas A&M University, College Station, Texas 77843



On the optimal paving over MASAsin von Neumann algebras

Sorin Popa and Stefaan Vaes

Dedicated to Dick Kadison for his 90th birthday

Abstract. We prove that if A is a singular MASA in a II1 factor M and ω is

a free ultrafilter, then for any x ∈ M A, with ‖x‖ ≤ 1, and any n ≥ 2, thereexists a partition of 1 with projections p1, p2, . . . , pn ∈ Aω (i.e. a paving) suchthat ‖Σn

i=1pixpi‖ ≤ 2√n− 1/n, and give examples where this is sharp. Some

open problems on optimal pavings are discussed.

1. Introduction

A famous problem formulated by R.V. Kadison and I.M. Singer in 1959 askedif the diagonal MASA (maximal abelian ∗-subalgebra) D of the algebra B(�2N), ofall linear bounded operators on the Hilbert space �2N, satisfies the paving property,requiring that for any contraction x = x∗ ∈ B(�2N) with 0 on the diagonal, andany ε > 0, there exists a partition of 1 with projections p1, . . . , pn ∈ D, such that‖∑

i pixpi‖ ≤ ε. This problem has been settled in the affirmative by A. Marcus,D. Spielman and N. Srivastava in [MSS13], with an actual estimate n ≤ 124ε−4

for the paving size, i.e., for the minimal number n = n(x, ε) of such projections.In a recent paper [PV14], we considered a notion of paving for an arbitrary

MASA in a von Neumann algebra A ⊂ M , that we called so-paving, which requiresthat for any x = x∗ ∈ M and any ε > 0, there exist n ≥ 1, a net of partitions of1 with n projections p1,i, . . . , pn,i ∈ A, a net of elements ai ∈ A with ‖ai‖ ≤ ‖x‖for all i and projections qi ∈ M such that ‖qi(Σn

k=1pk,ixpk,i − ai)qi‖ ≤ ε, ∀i, andqi → 1 in the so-topology.

This property is in general weaker than the classic Kadison-Singer norm paving,but it coincides with it for the diagonal MASA D ⊂ B(�2N). We conjectured in[PV14] that any MASAA ⊂ M satisfies so-paving. We used the results in [MSS13]to check this conjecture for all MASAs in type I von Neumann algebras, and allCartan MASAs in amenable von Neumann algebras and in group measure spacefactors arising from profinite actions, with the estimate 124ε−4 for the so-pavingsize derived from [MSS13] as well.

We also showed in [PV14] that if A is the range of a normal conditional expec-tation, E : M → A, and ω is a free ultrafilter on N, then so-paving for A ⊂ M is

The first author was supported in part by NSF Grant DMS-1401718.The second author was supported by ERC Consolidator Grant 614195 from the European

Research Council under the European Union’s Seventh Framework Programme.


199

200 SORIN POPA AND STEFAAN VAES

equivalent to the usual Kadison-Singer paving for the ultrapower MASA Aω ⊂ Mω,with the norm paving size for Aω ⊂ Mω coinciding with the so-paving size forA ⊂ M . In the case A is a singular MASA in a II1 factor M , norm-paving forthe ultrapower inclusion Aω ⊂ Mω has been established in [P13], with paving size1250ε−3. This estimate was improved to < 16ε−2 + 1 in [PV14], while also shownto be ≥ ε−2 for arbitrary MASAs in II1 factors.

In this paper we prove that the paving size for singular MASAs in II1 factorsis in fact < 4ε−2 + 1, and that for certain singular MASAs this is sharp. Moreprecisely, we prove that for any contraction x ∈ Mω with 0 expectation onto Aω,and for any n ≥ 2, there exists a partition of 1 with n projections pi ∈ Aω suchthat ‖Σn

i=1pixpi‖ ≤ 2√n− 1/n. In fact, given any finite set of contractions F ⊂

Mω % Aω, we can find a partition p1, . . . , pn ∈ Aω that satisfies this estimate forall x ∈ F , so even the multipaving size for singular MASAs is < 4ε−2 + 1.

To construct pavings satisfying this estimate, we first use Theorem 4.1(a) in[P13] to get a unitary u ∈ Aω with un = 1, τ (uk) = 0, 1 ≤ k ≤ n − 1, such thatany word with alternating letters from {uk | 1 ≤ k ≤ n− 1} and F ∪ F ∗ has trace0. This implies that for each x ∈ F the set X = {ui−1xu−i+1 | i = 1, 2, . . . , n}satisfies the conditions τ (Πm

k=1(x2k−1x∗2k)) = 0 = τ (Πm

k=1(x∗2k−1x2k)), for all m and

all xk ∈ X with xk �= xk+1 for all k. We call L-freeness this property of a subsetof a II1 factor. We then prove the general result, of independent interest, thatany L-free set of contractions {x1, . . . , xn} satisfies the norm estimate ‖Σn

i=1xi‖ ≤2√n− 1. We do this by first “dilating” {x1, . . . , xn} to an L-free set of unitaries

{U1, . . . , Un} in a larger II1 factor, for which we deduce the Kesten-type estimate‖Σn

i=1Ui‖ = 2√n− 1 from results in [AO74]. This implies the inequality for the

L-free contractions as well. By applying this to {ui−1xu1−i | i = 1, . . . , n} andtaking into account that 1

nΣni=1u

i−1xu1−i = Σni=1pixpi, where p1, . . . , pn are the

minimal spectral projections of u, we get ‖Σni=1pixpi‖ ≤ 2

√n− 1/n, ∀x ∈ F .

We also notice that if M is a II1 factor, A ⊂ M is a MASA and v ∈ M aself-adjoint unitary of trace 0 which is free with respect to A, then ‖Σn

i=1pivpi‖ ≥2√n− 1/n for any partition of 1 with projections in Aω, with equality if and only

if τ (pi) = 1/n, ∀i. A concrete example is when M = L(Z ∗ (Z/2Z)), A = L(Z)(which is a singular MASA in M by [P81]) and v = v∗ ∈ L(Z/2Z) ⊂ M denotesthe canonical generator. This shows that the estimate 4ε−2 + 1 for the paving sizeis in this case optimal.

The constant 2√n− 1 is known to coincide with the spectral radius of the

n-regular tree, and with the first eigenvalue less than n of n-regular Ramanujangraphs. Its occurence in this context leads us to a more refined version of a conjec-ture formulated in [PV14], predicting that for any MASA A ⊂ M which is rangeof a normal conditional expectation, any n ≥ 2 and any contraction x = x∗ ∈ Mwith 0 expectation onto A, the infimum ε(A ⊂ M ;n, x) over all norms of pavingsof x, ‖Σn

i=1pixpi‖, with n projections p1, . . . , pn in Aω, Σipi = 1, is bounded aboveby 2

√n− 1/n, and that in fact sup{ε(A ⊂ M ;n, x) | x = x∗ ∈ M % A, ‖x‖ ≤ 1} =

2√n− 1/n. Such an optimal estimate would be particularly interesting to establish

for the diagonal MASA D ⊂ B(�2Z).

2. Preliminaries

A well known result of H. Kesten in [K58] shows that if Fk denotes the freegroup with k generators h1, . . . , hk, and λ is the left regular representation of Fk

OPTIMAL PAVING OVER MASAS IN VON NEUMANN ALGEBRAS 201

on �2Fk, then the norm of the Laplacian operator L = Σki=1(λ(hi) + λ(h−1

i )) is

equal to 2√2k − 1. It was also shown in [K58] that, conversely, if k elements

h1, . . . , hk in a group Γ satisfy ‖Σki=1λ(hi) + λ(h−1

i )‖ = 2√2k − 1, then h1, . . . , hk

are freely independent, generating a copy of Fk inside Γ. The calculation of thenorm of L in [K58] uses the formalism of random walks on groups, but it reallyamounts to calculating the higher moments τ (L2n) and using the formula ‖L‖ =limm(τ (L2m))1/2m, where τ denotes the canonical (normal faithful) tracial state onthe group von Neumann algebra L(Fk).

Kesten’s result implies that whenever u1, . . . , uk are freely independent Haarunitaries in a type II1 factor M (i.e., u1, . . . , uk generate a copy of L(Fk) inside M),then one has ‖Σk

i=1ui+u∗i ‖ = 2

√2k − 1. In particular, if M is the free group factor

L(Fk) and ui = λ(hi), where h1, . . . , hk ∈ Fk as above, then ‖Σki=1αiui + αiu

∗i ‖ =

2√2k − 1, for any scalars αi ∈ C with |αi| = 1.Estimates of norms of linear combinations of elements satisfying more general

free independence relations in group II1 factors L(Γ) have later been obtainedin [L73], [B74], [AO74]1. These estimates involve elements in L(Γ) (viewed asconvolvers on �2Γ) that are supported on a subset {g1, . . . , gn} ⊂ Γ satisfying thefollowing weaker freeness condition, introduced in [L73]: whenever k ≥ 1 andis �= js, js �= is+1 for all s, we have that

gi1g−1j1

· · · gikg−1jk

�= e .

In [B74] and [AO74], this is called the Leinert property and it is proved to be equiv-alent with {g−1

1 g2, . . . , g−11 gn} freely generating a copy of Fn−1. The most general

calculation of norms of elements x = Σiciλ(gi) ∈ L(Γ), supported on a Leinert set{gi}i, with arbitrary coefficients ci ∈ C, was obtained by Akemann and Ostrandin [AO74]. The calculation shows in particular that if {g1, . . . , gn} satisfies Lein-ert’s freeness condition then ‖Σn

i=1λ(gi)‖ = 2√n− 1. Since h1, . . . , hk ∈ Γ freely

independent implies {hi, h−1i | 1 ≤ i ≤ k} is a Leinert set, the result in [AO74]

does recover Kesten’s theorem as well. Like in [K58], the norm of an element ofthe form L = Σn

i=1ciλ(gi) in [AO74] is calculated by evaluating limn τ ((L∗L)n)1/2n

(by computing the generating function of the moments of L∗L).An argument similar to [K58] was used in [Le96] to prove that, conversely,

if some elements g1, . . . , gn in a group Γ satisfy ‖Σni=1λ(gi)‖ = 2

√n− 1, then

g1, . . . , gn is a Leinert set. On the other hand, note that if g1, . . . , gn are n arbitraryelements in an arbitrary group Γ and we denote L = Σn

i=1λ(gi) the correspondingLaplacian, then the n’th moment τ ((L∗L)n) is bounded from below by the n’thmoment of the Laplacian obtained by taking gi to be the generators of Fn. Thus,we always have ‖Σn

i=1λ(gi)‖ ≥ 2√n− 1. More generally, if v1, . . . , vn are unitaries

in a von Neumann algebra M with normal faithful trace state τ , such that any wordvi1v

∗j1vi2v

∗j2. . . .vimv∗jm , ∀m ≥ 1, ∀1 ≤ ik, jk ≤ n, has trace with non-negative real

part, then ‖Σni=1vi‖ ≥ 2

√n− 1. In particular, for any unitaries u1, . . . , un ∈ M

one has ‖Σni=1ui ⊗ ui‖ ≥ 2

√n− 1.

For the reader’s convenience, we state below some norm calculations from[AO74], formulated in the form that will be used in the sequel:

1See also the more “rough” norm estimates for elements in L(Fn) obtained by R. Powers in1967 in relation to another problem of Kadison, but published several years later in [Po75], andwhich motivated in part the work in [AO74].


Proposition 2.1 ([AO74]). If v1, v2, . . . , vn−1 ∈ M are freely independentHaar unitaries, then

(2.1) ‖1 + Σn−1i=1 vi‖ = 2

√n− 1.

Also, if α0, . . . , αn−1 ∈ C, Σi|αi|2 = 1, then

(2.2) ‖α01 + Σn−1i=1 αivi‖ ≤ 2

√1− 1/n.

Note that (2.1) above shows in particular that if p, q ∈ M are projectionswith τ (p) = 1/2 and τ (q) = 1/n, for some n ≥ 3, and they are freely inde-pendent, then ‖qpq‖ = 1/2 +

√n− 1/n. Indeed, any two such projections can

be thought of as embedded into L(F2) with p and q lying in the MASAs of thetwo generators, p ∈ A1, respectively q ∈ A2. Denote v = 2p − 1. Let q1 =q, q2, . . . , qn ∈ A2 be mutually orthogonal projections of trace 1/n and denoteu = Σn

j=1λj−1qj , where λ = 2 exp(2πi/n). It is then easy to see that the ele-

ments vk = vukvu−k, k = 1, 2, . . . , n − 1 are freely independent Haar unitaries.By (2.1) we thus have ‖Σn−1

k=0ukvu−k‖ = ‖1 + Σn−1

k=1vukvu−k‖ = 2

√n− 1. But

Σn−1k=0u

kvu−k = n(Σnj=1qjvqj), implying that

‖qvq‖ = ‖q(2p− 1)q‖ = 2√n− 1/n = 2

√τ (q)(1− τ (q))

or equivalently

‖qpq‖ = 1/2 +√n− 1/n = τ (p) +

√τ (q)(1− τ (q)).

The computation of the norm of the product of freely independent projectionsq, p of arbitrary trace in M (in fact, of the whole spectral distribution of qpq) wasobtained by Voiculescu in [Vo86], as one of the first applications of his multiplica-tive free convolution (which later became a powerful tool in free probability). Werecall here these norm estimates, which in particular show that the first of the abovenorm calculations holds true for projections q of arbitrary trace (see also [ABH87]for the case τ (q) = 1/n, τ (p) = 1/m, for integers n ≥ m ≥ 2):

Proposition 2.2 ([Vo86]). If p, q ∈ M are freely independent projections withτ (q) ≤ τ (p) ≤ 1/2, then

(2.3) ‖qpq‖ = τ (p) + τ (q)− 2τ (p)τ (q) + 2√

τ (p) τ (1− p) τ (q) τ (1− q).

If in addition τ (p) = 1/2 and we denote v = 2p− 1, then

(2.4) ‖qvq‖ = 2√τ (q) τ (1− q).

3. L-free sets of contractions and their dilation

Recall from [P13] that two selfadjoint sets X,Y ⊂ M % C1 of a tracial vonNeumann algebra M are called freely independent sets2 if the trace of any wordwith letters alternating from X and Y is equal to 0. Also, a subalgebra B ⊂ M iscalled freely independent of a set X, if X and B%C1 are freely independent as sets.Several results were obtained in [P13] about constructing a “large subalgebra” Binside a given subalgebra Q ⊂ M that is freely independent of a given countable setX. Motivated by a condition appearing in one such result, namely [P13, Theorem4.1], and by a terminology used in [AO74], we consider in this paper the followingfree independence condition for arbitrary elements in tracial algebras:

2We specifically consider this condition for subsets X,Y ⊂ M C1, not to be confused withthe freeness of the von Neumann algebras generated by X and Y .


Definition 3.1. Let (M, τ ) be a von Neumann algebra with a normal faithfultracial state. A subset X ⊂ M is called L-free3 if

τ (x1x∗2 · · ·x2k−1x

∗2k) = 0 and τ (x∗

1x2 · · ·x∗2k−1x2k) = 0 ,

whenever k ≥ 1, x1, . . . , x2k ∈ X and xi �= xi+1 for all i = 1, . . . , 2k − 1.

Note that if the subset X in the above definition is taken to be contained in theset of canonical unitaries {ug | g ∈ Γ} of a group von Neumann algebra M = L(Γ),i.e. X = {ug | g ∈ F} for some subset F ⊂ Γ, then L-freeness of X amounts toF being a Leinert set. But the key example of an L-free set that is importantfor us here occurs from a diffuse algebra B that is free independent from a setY = Y ∗ ⊂ M % C1: given any y1, . . . , yn ∈ Y and any unitary element u ∈ U(B)with τ (uk) = 0, 1 ≤ k ≤ n− 1, the set {uk−1yku

−k+1 | 1 ≤ k ≤ n} is L-free.Note that we do need to impose both conditions on the traces being zero in

Definition 3.1, because we cannot deduce τ (x∗1x2x

∗3x1) = 0 from τ (y1y

∗2y3y

∗4) = 0

for all yi ∈ X with y1 �= y2, y2 �= y3, y3 �= y4. However, if X ⊂ U(M) consists ofunitaries, then only one set of conditions is sufficient. We in fact have:

Lemma 3.2. Let X = {u1, . . . , un} ⊂ U(M). Then the following conditions areequivalent

(a) X is an L-free set.(b) τ (ui1u

∗j1· · ·uiku

∗jk) = 0 whenever k ≥ 1 and is �= js, js �= is+1 for all s.

(c) u∗1u2, . . . , u

∗1un are free generators of a copy of L(Fn−1).

Proof. This is a trivial verification. �

Corollary 3.3. If {u1, . . . , un} is an L-free set of unitaries in U(M), then‖Σn

i=1ui‖ = 2√n− 1. Moreover, if α1, . . . , αn ∈ C with Σn

i=1|αi|2 ≤ 1, then∥∥∥ n∑i=1

αiui

∥∥∥ ≤ 2√1− 1/n.

Proof. Since ‖Σni=1αiui‖ = ‖α11 + Σn

i=2αiu∗1ui‖, the statement follows by

applying (2.2) to the freely independent Haar unitaries vj = u∗1uj , 2 ≤ j ≤ n. �

Proposition 3.4. Let M be a finite von Neumann algebra with a faithful tracialstate τ . If {x1, . . . , xn} ⊂ M is an L-free set with ‖xi‖ ≤ 1 for all i, then there existsa tracial von Neumann algebra (M, τ ), a trace preserving unital embedding M ⊂ Mand an L-free set of unitaries {U1, . . . , Un} ⊂ U(M) with M = Mn+1(C)⊗M sothat, denoting by (eij)i,j=0,...,n the matrix units of Mn+1(C), we have e00Uie00 = xi

for all i.

Proof. Define M = M ∗L(Fn(n−1)) and denote by ui,j , i �= j, free generatorsof L(Fn(n−1)). For every i ∈ {1, . . . , n}, define

ci =√1− xix∗

i and di = −√1− x∗

i xi .

Put M = Mn+1(C)⊗M and define the unitary elements Ui ∈ U(M) given by

Ui = (e00 ⊗ xi) + (eii ⊗ x∗i ) + (e0i ⊗ ci) + (ei0 ⊗ di) +

∑j �=i

(ejj ⊗ ui,j) .

3Note that this notion is not the same as (and should not be confused with) the notion ofL-sets used in [Pi92].


Note that Ui is the direct sum of the unitary(xi cidi x∗

i

)in positions 0 and i, and the unitary ⊕

j �=i

ui,j

in the positions j �= i. By construction, we have that e00Uie00 = xie00. So, itremains to prove that {U1, . . . , Un} is L-free.

Take k ≥ 1 and indices is, js such that is �= js, js �= is+1 for all s. We mustprove that

(3.1) τ (Ui1U∗j1 · · ·UikU

∗jk) = 0 .

Consider V := Ui1U∗j1· · ·UikU

∗jk

as a matrix with entries in M. Every entry of thismatrix is a sum of “words” with letters

{xi, x∗i , ci, di | i = 1, . . . , n} ∪ {ui,j , u

∗i,j | i �= j} .

We prove that every word that appears in a diagonal entry Vii of V has zero trace.The following types of words appear.

1◦ Words without any of the letters ua,b or u∗a,b. These words only appear as follows:

• in the entry V00 as xi1x∗j1· · ·xikx

∗jk, which has zero trace;

• if i1 = jk = i, in the entry Vii as w = dix∗j1xi2x

∗j2· · ·xik−1

x∗jk−1

xikd∗i . Then

we have

τ (w) = τ (x∗j1xi2 · · ·x∗

jk−1xik d

∗i di)

= τ (x∗j1xi2 · · ·x∗

jk−1xik)− τ (x∗

j1xi2 · · ·x∗jk−1

xik x∗i xi)

= 0− τ (xi1x∗j1 · · ·xikx

∗jk) = 0 ,

because i = i1 and i = jk.

2◦ Words with exactly one letter of the type ua,b or u∗a,b. These words have zero

trace because τ (Mua,bM) = {0}.

3◦ Words w with two or more letters of the type ua,b or u∗a,b. Consider two consec-

utive such letters in w, i.e. a subword of w of the form

uεi,j w0 uε′

i′,j′

with ε, ε′ = ±1 and where w0 is a word with letters from {xi, x∗i , ci, di | i =

1, . . . , n}. We distinguish three cases.

• (ε′, i′, j′) �= (−ε, i, j).

• ui,j w0 u∗i,j .

• u∗i,j w0 ui,j .

To prove that τ (w) = 0, it suffices to prove that in the last two cases, we havethat τ (w0) = 0.


A subword of the form ui,j w0 u∗i,j can only arise from the jj-entry of

UisU∗js · · ·UitU

∗jt with is = jt = i , js = it = j

(and thus, t ≥ s+ 2). In that case,

w0 = c∗j xis+1x∗js+1

· · ·xit−1x∗jt−1

cj .

Thus,

τ (w0) = τ (xis+1x∗js+1

· · ·xit−1x∗jt−1

cjc∗j )

= τ (xis+1x∗js+1

· · ·xit−1x∗jt−1

)− τ (xis+1x∗js+1

· · ·xit−1x∗jt−1

xjx∗j )

= 0− τ (xjsx∗is+1

· · ·x∗jt−1

xit) = 0 ,

because j = js and j = it.

Finally, a subword of the form u∗i,j w0 ui,j can only arise from the jj-entry

of

U∗js−1

Uis · · ·U∗jt−1

Uit with js−1 = it = i , is = jt−1 = j

(and thus, t ≥ s+ 2). In that case,

w0 = dj x∗jsxis+1

· · ·x∗jt−2

xit−1d∗j .

As above, it follows that τ (w0) = 0.

So, we have proved that every word that appears in a diagonal entry Vii of Vhas trace zero. Then also τ (V ) = 0 and it follows that {U1, . . . , Un} is an L-free setof unitaries. �

Corollary 3.5. Let (M, τ ) be a finite von Neumann algebra with a faithfulnormal tracial state. If {x1, . . . , xn} ⊂ M is L-free with ‖xi‖ ≤ 1 for all i, then∥∥∥ n∑

i=1

xi

∥∥∥ ≤ 2√n− 1 .

More generally, given any complex scalars α1, . . . , αn with Σni=1|αi|2 ≤ 1, we have∥∥∥ n∑

i=1

αixi

∥∥∥ ≤ 2√1− 1/n .

Proof. Assuming n ≥ 2, with the notations from Proposition 3.4 and by using

Corollary 3.3, we have∥∥∥∑n

i=1 αiUi

∥∥∥ ≤ 2√1− 1/n. Reducing with the projection

e00, it follows that ∥∥∥ n∑i=1

αixi

∥∥∥ ≤ 2√1− 1/n .

�

4. Applications to paving problems

Like in [P13], [PV14], if A ⊂ M is a MASA in a von Neumann algebra andx ∈ M, then we denote by n(A ⊂ M;x, ε) the smallest n for which there existprojections p1, . . . , pn ∈ A and a ∈ A such that ‖a‖ ≤ ‖x‖,

∑ni=1 pi = 1 and∥∥∥∑n

i=1 pixpi−a∥∥∥ ≤ ε‖x‖ (with the convention that n(A ⊂ M;x, ε) = ∞ if no such

finite partition exists), and call it the paving size of x.


Recall also from [D54] that a MASA A in a von Neumann algebra M is calledsingular, if the only unitary elements in M that normalize A are the unitaries inA.

Theorem 4.1. Let An ⊂ Mn be a sequence of singular MASAs in finite vonNeumann algebras and ω a free ultrafilter on N. Denote M =

∏ω Mn and A =∏

ω An. Given any countable set of contractions X ⊂ M%A and any integer n ≥ 2,there exists a partition of 1 with projections p1, . . . , pn ∈ A such that∥∥∥ n∑

j=1

pjxpj

∥∥∥ ≤ 2√n− 1/n, for all x ∈ X .

In particular, the paving size of A ⊂ M,

n(A ⊂ M; ε)def= sup{n(A ⊂ M;x, ε) | x = x∗ ∈ M%A} ,

is less than 4ε−2 + 1, for any ε > 0.

Proof. By Theorem 4.1(a) in [P13], there exists a diffuse abelian von Neu-mann subalgebra A0 ⊂ A such that for any k ≥ 1, any word with alternating lettersx = x0Π

ki=1(vixi) with xi ∈ X, 1 ≤ i ≤ k − 1, x0, xk ∈ X ∪ {1}, vi ∈ A0 % C1, has

trace equal to 0.This implies that if p1, . . . , pn ∈ A are projections of trace 1/n summing up to

1 and we denote u = Σnj=1λ

j−1pj , where λ = exp(2πi/n), then for any x ∈ X the

set {ui−1xu−i+1 | i = 1, 2, . . . , n} is L-free. Since 1nΣ

ni=1u

i−1xu1−i = Σni=1pixpi,

where p1, . . . , pn are the minimal spectral projections of u, by Proposition 3.4 itfollows that for all x ∈ X we have

‖Σni=1pixpi‖ =

1

n‖Σn

i=1ui−1xu−i+1‖ ≤ 2

√n− 1/n.

To derive the last part, let ε > 0 and denote by n the integer with the propertythat 2n−1/2 ≤ ε < 2(n − 1)−1/2. If x ∈ M % A, ‖x‖ ≤ 1, and p1, . . . , pn ∈ Aare mutually orthogonal projections of trace 1/n that satisfy the free independencerelation with X = {x} as above, then n < 4ε−2 + 1 and we have

‖Σni=1pixpi‖ ≤ 2

√n− 1/n ≤ ε,

showing that n(A ⊂ M;x, ε) < 4ε−2 + 1. �

Remark 4.2. The above result suggests that an alternative way of measuringthe so-paving size over a MASA in a von Neumann algebra A ⊂ M admitting anormal conditional expectation, is by considering the quantity

ε(A ⊂ M ;n)def= sup

x∈(Mωh �Aω)1

(inf{‖Σni=1pixpi‖ | pi ∈ P(Aω),Σipi = 1}).

With this notation, the above theorem shows that for a singular MASA in aII1 factor A ⊂ M , one has ε(A ⊂ M ;n) ≤ 2

√n− 1/n, ∀n ≥ 2, a formulation that’s

slightly more precise than the estimate ns(A ⊂ M ; ε) = n(Aω ⊂ Mω; ε) < 4ε−2+1.Also, the conjecture (2.8.2◦ in [PV14]) about the so-paving size can this waybe made more precise, by asking whether ε(A ⊂ M ;n) ≤ 2

√n− 1/n, ∀n, for

any MASA with a normal conditional expectation A ⊂ M . It seems particularlyinteresting to study this question in the classical Kadison-Singer case of the diagonalMASA D ⊂ B = B(�2N), and more generally for Cartan MASAs A ⊂ M . So


far, the solution to the Kadison-Singer paving problem in [MSS13] shows thatε(D ⊂ B;n) ≤ 12n−1/4.

Also, while by [CEKP07] one has n(D ⊂ B; ε) ≥ ε−2 and by [PV14] one hasns(A ⊂ M ; ε) = n(Aω ⊂ Mω; ε) ≥ ε−2, for any MASA in a II1 factor A ⊂ M , itwould be interesting to decide whether ε(D ⊂ B;n) and ε(A ⊂ M ;n) are in factbounded from below by 2

√n− 1/n, ∀n.

For a singular MASA in a II1 factor, A ⊂ M , combining 4.1 with such a lowerbound would show that ε(A ⊂ M ;n) = 2

√n− 1/n, ∀n. While we could not prove

this general fact, let us note here that for certain singular MASAs this equalityholds indeed.

Proposition 4.3. 1◦ Let M be a II1 factor and A ⊂ M a MASA. Assumev ∈ M is a unitary element with τ (v) = 0 such that A is freely independent of the set{v, v∗} (i.e., any alternating word in A%C1 and {v, v∗} has trace 0). Then for anypartition of 1 with projections p1, . . . , pn ∈ Aω we have ‖Σn

i=1pivpi‖ ≥ 2√n− 1/n,

with equality iff all pi have trace 1/n. Also, ε(A ⊂ M ;n) ≥ 2√n− 1/n, ∀n.

2◦ If M = L(Z∗(Z/2Z)), A = L(Z) and v = v∗ denotes the canonical generatorof L(Z/2Z), then ε(A ⊂ M ; v, n) = ε(A ⊂ M ;n) = 2

√n− 1, ∀n.

Proof. The free independence assumption in 1◦ implies that Aω % C and{v, v∗} are freely independent sets as well. This in turn implies that for each i, theprojections pi and vpiv

∗ are freely independent, and so by Proposition 2.2 one has‖pivpi‖ = ‖pivpiv∗‖ = 2

√τ (pi)(1− τ (pi)). Thus, if one of the projections pi has

trace τ (pi) > 1/n, then ‖Σjpjvpj‖ ≥ ‖pivpi‖ > 2√n− 1/n, while if τ (pi) = 1/n,

∀i, then ‖Σjpjvpj‖ = 2√n− 1/n.

By applying 1◦ to part 2◦, then using 4.1 and the fact that A = L(Z) is singularin M = L(Z ∗ (Z/2Z)) (cf. [P81]), proves the last part of the statement. �

References

[AO74] C. A. Akemann and P. A. Ostrand, Computing norms in group C∗-algebras, Amer. J.Math. 98 (1976), no. 4, 1015–1047. MR0442698 (56 #1079)

[ABH87] J. Anderson, B. Blackadar, and U. Haagerup, Minimal projections in the reduced groupC∗-algebra of Zn ∗ Zm, J. Operator Theory 26 (1991), no. 1, 3–23. MR1214917(94c:46110)

[B74] M. Bozejko, On Λ(p) sets with minimal constant in discrete noncommutative groups,Proc. Amer. Math. Soc. 51 (1975), 407–412. MR0390658 (52 #11481)

[CEKP07] P. Casazza, D. Edidin, D. Kalra, and V. I. Paulsen, Projections and the Kadison-Singer problem, Oper. Matrices 1 (2007), no. 3, 391–408, DOI 10.7153/oam-01-23.

MR2344683 (2009a:46105)[D54] J. Dixmier, Sous-anneaux abeliens maximaux dans les facteurs de type fini (French),

Ann. of Math. (2) 59 (1954), 279–286. MR0059486 (15,539b)[KS59] R. V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959),

383–400. MR0123922 (23 #A1243)[K58] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959),

336–354. MR0109367 (22 #253)[Le96] F. Lehner, A characterization of the Leinert property, Proc. Amer. Math. Soc.

125 (1997), no. 11, 3423–3431, DOI 10.1090/S0002-9939-97-03966-X. MR1402870(97m:22001)

[L73] M. Leinert, Faltungsoperatoren auf gewissen diskreten Gruppen (German), StudiaMath. 52 (1974), 149–158. MR0355480 (50 #7954)

[MSS13] A. W. Marcus, D. A. Spielman, and N. Srivastava, Interlacing families II: Mixed char-acteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015),no. 1, 327–350, DOI 10.4007/annals.2015.182.1.8. MR3374963


[Pi92] G. Pisier, Multipliers and lacunary sets in non-amenable groups, Amer. J. Math. 117(1995), no. 2, 337–376, DOI 10.2307/2374918. MR1323679 (96e:46078)

[P81] S. Popa, Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras, J. OperatorTheory 9 (1983), no. 2, 253–268. MR703810 (84h:46077)

[P13] S. Popa, A II1 factor approach to the Kadison-Singer problem, Comm. Math. Phys.332 (2014), no. 1, 379–414, DOI 10.1007/s00220-014-2055-4. MR3253706

[PV14] S. Popa and S. Vaes, Paving over arbitrary MASAs in von Neumann algebras, Anal.

PDE 8 (2015), no. 4, 1001–1023, DOI 10.2140/apde.2015.8.1001. MR3366008[Po75] R. T. Powers, Simplicity of the C∗-algebra associated with the free group on two gen-

erators, Duke Math. J. 42 (1975), 151–156. MR0374334 (51 #10534)[Vo86] D. Voiculescu, Multiplication of certain noncommuting random variables, J. Operator

Theory 18 (1987), no. 2, 223–235. MR915507 (89b:46076)

Mathematics Department, UCLA, Los Angeles, California 90095-1555


Department of Mathematics, KU Leuven, Leuven (Belgium)



Matricial bridges for“Matrix algebras converge to the sphere”

Marc A. Rieffel

In celebration of the successful completionby Richard V. Kadison of 90 circumnavigations of the sun

Abstract. In the high-energy quantum-physics literature one finds state-ments such as “matrix algebras converge to the sphere”. Earlier I provided ageneral setting for understanding such statements, in which the matrix alge-bras are viewed as quantum metric spaces, and convergence is with respect toa quantum Gromov-Hausdorff-type distance. In the present paper, as prepara-tion of discussing similar statements for convergence of “vector bundles” overmatrix algebras to vector bundles over spaces, we introduce and study suit-able matrix-norms for matrix algebras and spaces. Very recently Latremoliereintroduced an improved quantum Gromov-Hausdorff-type distance betweenquantum metric spaces. We use it throughout this paper. To facilitate thecalculations we introduce and develop a general notion of “bridges with con-ditional expectations”.

1. Introduction

In several earlier papers [11, 12, 14] I showed how to give a precise meaningto statements in the literature of high-energy physics and string theory of the kind“matrix algebras converge to the sphere”. (See the references to the quantumphysics literature given in [1, 2, 4, 5, 11, 13, 15].) I did this by introducing anddeveloping a concept of “compact quantum metric spaces”, and a correspondingquantum Gromov-Hausdorff-type distance between them. The compact quantumspaces are unital C*-algebras, and the metric data is given by putting on the alge-bras seminorms that play the role of the usual Lipschitz seminorms on the algebrasof continuous functions on ordinary compact metric spaces. The natural settingfor “matrix algebras converge to the sphere” is that of coadjoint orbits of compactsemi-simple Lie groups.

But physicists need much more than just the algebras. They need vector bun-dles, gauge fields, Dirac operators, etc. So I now seek to give precise meaning tostatements in the physics literature of the kind “here are the vector bundles over

2010 Mathematics Subject Classification. Primary 46L87; Secondary 53C23, 58B34, 81R15,81R30.

The research reported here was supported in part by National Science Foundation grantDMS-1066368.


209

210 MARC A. RIEFFEL

the matrix algebras that correspond to the monopole bundles on the sphere”. (See[13] for many references.) In [13] I studied convergence of ordinary vector bundleson ordinary compact metric spaces for ordinary Gromov-Hausdorff distance. Fromthat study it became clear that one needed Lipschitz-type seminorms on all thematrix algebras over the underlying algebras, with these seminorms coherent in thesense that they form a “matrix seminorm” (defined below). The purpose of thispaper is to define and develop the properties of such matrix seminorms for the set-ting of coadjoint orbits, and especially to study how these matrix seminorms meshwith the non-commutative analogs of Gromov-Hausdorff distance.

Very recently Latremoliere introduced an improved version of quantum Gromov-Hausdorff distance [8] that he calls “the Gromov-Hausdorff propinquity”. We showthat his propinquity works very well for our setting of coadjoint orbits, and sopropinquity is the form of quantum Gromov-Hausdorff distance that we use in thispaper. Latremoliere defines his propinquity in terms of an improved version of the“bridges” that I had used in my earlier papers. For our matrix seminorms we needcorresponding “matricial bridges”, and we show how to construct natural ones forthe setting of coadjoint orbits.

It is crucial to obtain good upper bounds for the lengths of the bridges that weconstruct. In the matricial setting the calculations become somewhat complicated.In order to ease the calculations we introduce a notion of “bridges with conditionalexpectations”, and develop their general theory, including the matricial case, andincluding bounds for their lengths in the matricial case.

The main theorem of this paper, Theorem 6.10, states in a quantitative waythat for the case of coadjoint orbits the lengths of the matricial bridges goes to 0as the size of the matrix algebras goes to infinity.

We also discuss a closely related class of examples coming from [12], for whichwe construct bridges between different matrix algebras associated to a given coad-joint orbit. This provides further motivation for our definitions and theory ofbridges with conditional expectation.

Contents

1. Introduction2. The first basic class of examples3. The second basic class of examples4. Bridges with conditional expectations5. The corresponding matricial bridges6. The application to the first class of basic examples7. The application to the second class of basic examples8. TreksReferences

2. The first basic class of examples

In this section we describe the first of the two basic classes of examples under-lying this paper. It consists of the main class of examples studied in the papers[11,14]. We begin by describing the common setting for the two basic classes ofexamples.

MATRICIAL BRIDGES 211

Let G be a compact group (perhaps even finite, at first). Let U be an irreducibleunitary representation of G on a (finite-dimensional) Hilbert space H. Let B =L(H) denote the C∗-algebra of all linear operators on H (a “full matrix algebra”,with its operator norm). There is a natural action, α, of G on B by conjugation byU , that is, αx(T ) = UxTU

∗x for x ∈ G and T ∈ B. Because U is irreducible, the

action α is “ergodic”, in the sense that the only α-invariant elements of B are thescalar multiples of the identity operator.

Let P be a rank-one projection in B(H) (traditionally specified by giving anon-zero vector in its range). For any T ∈ B we define its Berezin covariant symbol[11], σT , with respect to P , by

σT (x) = tr(Tαx(P )),

where tr denotes the usual (un-normalized) trace on B. (When the αx(P )’s areviewed as giving states on B via tr, they form a family of “coherent states” [11]if a few additional conditions are satisfied.) Let H denote the stability subgroupof P for α. Then it is evident that σT can be viewed as a (continuous) functionon G/H. We let λ denote the action of G on G/H, and so on A = C(G/H), byleft-translation. If we note that tr is α-invariant, then it is easily seen that σ is aunital, positive, norm-nonincreasing, α-λ-equivariant map from B into A.

Fix a continuous length function, �, on G (so G must be metrizable). Thus �is non-negative, �(x) = 0 iff x = eG (the identity element of G), �(x−1) = �(x), and�(xy) ≤ �(x) + �(y). We also require that �(xyx−1) = �(y) for all x, y ∈ G. Thenin terms of α and � we can define a seminorm, LB, on B by the formula

(2.1) LB(T ) = sup{‖αx(T )− T‖/�(x) : x ∈ G and x �= eG}.Then (B, LB) is an example of a compact C*-metric-space, as defined in definition4.1 of [14], and in particular LB satisfied the conditions given there for being a“Lip-norm”.

Of course, from λ and � we also obtain a seminorm, LA, on A by the evidentanalog of formula 2.1, except that we must permit LA to take the value ∞. It isshown in proposition 2.2 of [10] that the set of functions for which LA is finite (theLipschitz functions) is a dense ∗-subalgebra of A. Also, LA is the restriction to A ofthe seminorm on C(G) that we get from � when we view C(G/H) as a subalgebraof C(G), as we will often do when convenient. From LA we can use equation 2.2below to recover the usual quotient metric [16] on G/H coming from the metric onG determined by �. One can check easily that LA in turn comes from this quotientmetric. Thus (A, LA) is the compact C*-metric-space associated to this ordinarycompact metric space. Then for any bridge from A to B we can use LA and LB

to define the length of the bridge in the way given by Latremoliere, which we willdescribe soon below.

For any two unital C*-algebras A and B a bridge from A to B in the sense ofLatremoliere [8] is a quadruple (D, πA, πB, ω) for which D is a unital C*-algebra, πAand πB are unital injective homomorphisms ofA and B intoD, and ω is a self-adjointelement of D such that 1 is an element of the spectrum of ω and ‖ω‖ = 1. Actually,Latremoliere only requires a looser but more complicated condition on ω, but theabove condition will be appropriate for our examples. Following Latremoliere wewill call ω the “pivot” for the bridge. We will often omit mentioning the injectionsπA and πB when it is clear what they are from the context, and accordingly we willoften write as though A and B are unital subalgebras of D.

212 MARC A. RIEFFEL

For our first class of examples, in which A and B are as described in theparagraphs above, we take D to be the C*-algebra

D = A⊗ B = C(G/H,B).We take πA to be the injection of A into D defined by

πA(a) = a⊗ 1B

for all a ∈ A, where 1B is the identity element of B. The injection πB is definedsimilarly. From the many calculations done in [11,14] it is not surprising that wedefine the pivot ω to be the function in C(G/H,B) defined by

ω(x) = αx(P )

for all x ∈ G/H. We notice that ω is actually a non-zero projection in D, and so itsatisfies the requirements for being a pivot. We will denote the bridge (D, ω) by Π.

For any bridge between two unital C*-algebras A and B and any choice of semi-norms LA and LB on A and B, Latremoliere [8] defines the “length” of the bridgein terms of these seminorms. For this he initially puts relatively weak requirementson the seminorms, but for the purposes of the matricial bridges that we will definelater, we need somewhat different weak requirements. To begin with, Latremoliereonly requires his seminorms, say LA on a unital C*-algebra A, to be defined onthe subspace of self-adjoint elements of the algebra. However, we need LA to bedefined on all of A. To somewhat compensate for this we require that LA be a∗-seminorm, that is, that LA(a∗) = LA(a) for all a ∈ A. As with Latremoliere,our LA is permitted to take value +∞. Latremoliere also requires the subspace onwhich LA takes finite values to be dense in the algebra. We do not really need thishere, but for us there would be no harm in assuming it, and all interesting examplesprobably will satisfy this. Finally, Latremoliere requires that the null space of LA

(i.e where it takes value 0) be exactly R1A. We must loosen this to simply requiringthat LA(1A) = 0, but permitting LA to also take value 0 on elements not in C1A.We think of such seminorms as “semi-Lipschitz seminorms”. To summarize all ofthis we make:

Definition 2.1. By a slip-norm on a unital C*-algebra A we mean a ∗-seminorm, L, on A that is permitted to take the value +∞, and is such thatL(1A) = 0.

Because of these weak requirements on LA, various quantities in this papermay be +∞, but most interesting examples will satisfy stronger requirements thatwill result in various quantities being finite.

Latremoliere defines the length of a bridge, for given LA and LB, by firstdefining its “reach” and its “height”. We apply his definitions to slip-norms.

Definition 2.2. Let A and B be unital C*-algebras and let Π = (D, ω) be abridge from A to B . Let LA and LB be slip-norms on A and B. Set

L1A = {a ∈ A : a = a∗ and LA(a) ≤ 1},

and similarly for L1B. (We can view these as subsets of D.) Then the reach of Π is

given by:reach(Π) = HausD(L1

Aω , ωL1B),

where HausD denotes the Hausdorff distance with respect to the norm of D, andwhere the product defining L1

Aω and ωL1B is that of D.


Latremoliere shows just before definition 3.14 of [8] that, under conditions thatinclude the case in which (A, LA) and (B, LB) are C*-metric spaces, the reach of Πis finite.

To define the height of Π we need to consider the state space, S(A), of A, andsimilarly for B and D. Even more, we set

S1(ω) = {φ ∈ S(D) : φ(ω) = 1},the “level-1 set of ω”. The elements of S1(ω) are “definite” on ω in the sense [7]that for any φ ∈ S1(ω) and d ∈ D we have

φ(dω) = φ(d) = φ(ωd).

Let ρA denote the metric on S(A) determined by LA using the formula

(2.2) ρA(μ, ν) = sup{|μ(a)− ν(a)| : LA(a) ≤ 1}.(Without further conditions on LA we must permit ρA to take the value +∞. Also,it is not hard to see that the supremum can be taken equally well just over L1

A.)Define ρB on S(B) similarly.

Notation 2.3. We denote by SA1 (ω) the restriction of the elements of S1(ω)

to A. We define SB1 (ω) similarly.

Definition 2.4. Let A and B be unital C*-algebras and let Π = (D, ω) be abridge from A to B . Let LA and LB be slip-norms on A and B. The height of thebridge Π is given by

height(Π) = max{HausρA(SA1 (ω), S(A)), HausρB(S

B1 (ω), S(B))},

where the Hausdorff distances are with respect to the indicated metrics (and value+∞ is allowed). The length of Π is then defined by

length(Π) = max{reach(Π), height(Π)}.In Section 6 we will show how to obtain a useful upper bound on the length of

Π for our first class of examples.

3. The second basic class of examples

Our second basic class of examples has the same starting point as the first class,consisting of G, H and U as before. We also need a second irreducible representationof G, and for each of these two representations we pick a rank-one projection. Thekey feature that we require is that the stability subgroups of these two projectionsfor the action of G coincide. The more concrete class of examples motivating thissituation, but for which we will not need the details, is that in [11,12,14] in whichG is a compact semi-simple Lie group, λ is a positive integral weight, and our tworepresentations of G are the representations with highest weights mλ and nλ forpositive integers m and n, m �= n. Furthermore, the projections P are required tobe those along highest weight vectors. The key feature of this situation that we doneed to remember here is that the stability subgroups H for the two projectionscoincide.

Accordingly, for our slightly more general situation, we will denote our tworepresentations by (Hm, Um) and (Hn, Un), where now m and n are just labels.Our two C*-algebras will be Bm = L(Hm) and Bn = L(Hn). We will denote theaction of G on these two algebras just by α, since the context should always makeclear which algebra is being acted on. The corresponding projections will be Pm

214 MARC A. RIEFFEL

and Pn. The crucial assumption that we make is that the stability subgroups ofthese two projections coincide. We will denote this common stability subgroup byH as before.

We construct a bridge from Bm to Bn as follows. We let A = C(G/H) as inour first class of examples, and we define D by

D = Bm ⊗A⊗ Bn = C(G/H,Bm ⊗ Bn).

We view Bm as a subalgebra of D by sending b ∈ Bm to b⊗ 1A⊗ 1Bn , and similarlyfor Bn. From the many calculations done in [12] it is not surprising that we definethe pivot, ω, to be the function in C(G/H,Bm ⊗ Bn) defined by

ω(x) = αx(Pm)⊗ αx(P

n).

We let Lm be the Lip-norm defined on Bm determined by the action α and thelength function � as in formula (2.1), and similarly for Ln on Bn. In terms of theseLip-norms the length of any bridge from Bm to Bn is defined. Thus the length ofthe bridge described above is defined. In Section 7 we will see how to obtain usefulupper bounds on the length of this bridge.

4. Bridges with conditional expectations

We will now seek a somewhat general framework for obtaining useful estimatesfor the lengths of bridges such as those of our two basic classes of examples. Todiscover this framework we will explore some properties of our two basic classes ofexamples. We will summarize what we find at the end of this section.

On G/H there is a unique probability measure that is invariant under lefttranslation by elements of G. We denote the corresponding linear functional onA = C(G/H) by τA, and sometimes refer to it as the canonical tracial state onA. On B = L(H) there is a unique tracial state, which we denote by τB. Thesecombine to form a tracial state, τD = τA ⊗ τB on D = A ⊗ B. Similarly, we havethe unique tracial states τm and τn on Bm and Bn, which combine with τA to givea tracial state on D = Bm ⊗A⊗ Bn.

For D = A⊗B, the tracial state τB determines a conditional expectation, EA,from D onto its subalgebra A, defined on elementary tensors by

EA(a⊗ b) = aτB(b)

for any a ∈ A and b ∈ B. (This is an example of a “slice map” as discussed in [3],where conditional expectations are also discussed.) This conditional expectationhas the property that for any d ∈ D we have

τA(EA(d)) = τD(d),

and it is the unique conditional expectation with this property. (See corollaryII.6.10.8 of [3].) In the same way the tracial state τA determines a canonicalconditional expectation, EB from D onto its subalgebra B.

For the case in which D = Bm ⊗A ⊗ Bn, the tracial state τA ⊗ τn on A⊗ Bn

determines a canonical conditional expectation, Em, from D onto Bm in the sameway as above, and the tracial state τm ⊗ τA determines a canonical conditionalexpectation, En, from D onto Bn.


These conditional expectations relate well to the pivots of the bridges. For thecase in which D = A⊗ B we find that for any F ∈ D = C(G/H,B) we have

EA(Fω)(x) = τB(F (x)αx(P )).

In particular, for any T ∈ B we have

EA(Tω)(x) = τB(Tαx(P ))

for all x ∈ G/H. Aside from the fact that we are here using the normalized traceinstead of the standard trace on the matrix algebra B, the right-hand side is exactlythe definition of the Berezin covariant symbol of T that plays such an importantrole in [11,14] (beginning in section 1 of [11]), and that is denoted there by σT .This indicates that for general A and B a map b �→ EA(bω) might be of importanceto us. For our specific first basic class of examples we note the following favorableproperties:

(1) Self-adjointness i.e. EA(F ∗ω) = (EA(ωF ))∗ for all F ∈ D.(2) EA(Fω) = EA(ωF ) for all F ∈ D.(3) Positivity, i.e. if F ≥ 0 then EA(Fω) ≥ 0.(4) EA(1Dω) = r−11A where B is an r × r matrix algebra.

However, if we consider EB instead EA, then for any F ∈ D we have

EB(Fω) =

∫G/H

F (x)αx(P ) dx,

and we see that in general properties 1-3 above fail, although property 4 still holds,with the same constant r. But if we restrict F to be any f ∈ A, we see thatproperties 1-3 again hold. Even more, the expression∫

G/H

f(x)αx(P ) dx

is, except for normalization of the trace, the formula involved in the Berezin con-travariant symbol that in [11] is denoted by σ.

For our second class of examples, in which D = Bm ⊗A⊗Bn, we find that forF ∈ D = C(G/H,Bm ⊗ Bn) we have

Em(Fω) =

∫G/H

(ιA ⊗ τn)(F (x)(αx(Pm)⊗ αx(P

n))) dx.

Again we see that properties 1-3 above are not in general satisfied. But if we restrictF to be any T ∈ Bn then the above formula becomes∫

G/H

αx(Pm)τn(Tαx(P

n)) dx,

which up to normalization of the trace is exactly the second displayed formula insection 3 of [12]. It is not difficult to see that properties 1-3 above are again satisfiedunder this restriction.

We remark that it is easily seen that the maps T �→ Em(Tω) from Bn to Bm

and S �→ En(Sω) from Bm to Bn are each other’s adjoints when they are viewed asbeing between the Hilbert spaces L2(Bm, τm) and L2(Bn, τn). A similar statementholds for our first basic class of examples.

With these observations in mind, we begin to formulate a somewhat generalframework. As before, we assume that we have two unital C*-algebras A and B,and a bridge Π = (D, ω) from A to B. We now require that we are given conditional

216 MARC A. RIEFFEL

expectations EA and EB from D onto its subalgebras A and B. (We do not requirethat they be associated to any tracial states.) We require that they relate well to ω.To begin with, we will just require that ω ≥ 0 so that ω1/2 exists. Then the map

d �→ EA(ω1/2dω1/2)

from D to A is positive.Once we have slip-norms LA and LB on A and B, we need to require that

the conditional expectations are compatible with these slip-norms. To begin with,we require that if LB(b) = 0 for some b ∈ B then LA(EA(ω1/2bω1/2)) = 0. Butone of the conditions on a Lip-norm is that it takes value 0 exactly on the scalarmultiples of the identity element, and the case of Lip-norms is important to us. ForLip-norms we see that the above requirement implies that EA(ω) ∈ C1A, and soEA(ω) = rω1A for some positive real number rω. We require the same of EB withthe same real number, so that we require that

EA(ω) = rω1A = EB(ω).

We require further that rω �= 0. We then define a map, ΦA, from D to A by

ΦA(d) = r−1ω EA(ω1/2dω1/2).

In a similar way we define ΦB from D to B. We see that ΦA and ΦB are unitalpositive maps, and so are of norm 1 (as seen by composing them with states). Thenthe main compatibility requirement that we need is that for all b ∈ B we have

LA(ΦA(b)) ≤ LB(b),

and similarly for A and B reversed. Notice that this implies that if b ∈ L1B then

ΦA(b) ∈ L1A.

We now show how to obtain an upper bound for the reach of the bridge Π whenthe above requirements are satisfied. Let b ∈ L1

B be given. As an approximation toωb by an element of the form aω for some a ∈ L1

A we take a = ΦA(b). It is indeedin L1

A by the requirements made just above. This prompts us to set

(4.1) γB = sup{‖ΦA(b)ω − ωb‖D : b ∈ L1B},

and we see that ωb is then in the γB-neighborhood of L1Aω. Note that without

further assumptions on LB we could have γB = +∞. Interchanging the roles of Aand B, we define γA similarly. We then see that

reach(Π) ≤ max{γA, γB}.We will explain in Sections 6 and 7 why this upper bound is useful in the contextof [11,12,14].

We now consider the height of Π. For this we need to consider S1(ω) as definedin Section 2. Let μ ∈ S(A). Because ΦA is positive and unital, its compositionwith μ is in S(D). When we evaluate this composition at ω to see if it is in S1(ω),we obtain μ(r−1

ω EA(ω2)), and we need this to equal 1. Because μ(r−1ω EA(ω)) = 1,

it follows that we need μ(r−1ω EA(ω − ω2)) = 0. If this is to hold for all μ ∈ S(A),

we must have EA(ω − ω2) = 0. If EA is a faithful conditional expectation, as istrue for our basic examples, then because ω ≥ ω2 it follows that ω2 = ω so that ω isa projection, as is also true for our basic examples. These arguments are reversible,and so it is easy to see that if ω is a projection, then for every μ ∈ S(A) we obtainan element, φμ, of S1(ω), defined by

φμ(d) = μ(r−1ω EA(ωdω)) = μ(ΦA(d)).


This provides us with a substantial collection of elements of S1(ω).Consequently, since to estimate the height of Π we need to estimate the distance

from each μ ∈ S(A) to SA1 (ω), we can hope that φμ restricted to A is relatively

close to μ. Accordingly, for any a ∈ A we compute

|μ(a)− φμ(a)| = |μ(a)− μ(ΦA(a))| ≤ ‖a− ΦA(a)‖.Set

δA = sup{‖a− ΦA(a)‖ : a ∈ L1A}.

Then we see thatρLA(μ, φμ|A) ≤ δA.

We define δB in the same way, and obtain the corresponding estimate for the dis-tances from elements of S(B) to the restriction of S1(ω) to B. In this way we seethat

height(Π) ≤ max{δA, δB}.(Notice that δA involves what ΦA does on A, whereas γA involves what ΦB doeson A.)

While this bound is natural within this context, it turns out not to be so usefulfor our two basic classes of example. In Proposition 4.6 below we will give a differentbound that does turn out to be useful for our basic examples. But perhaps otherexamples will arise for which the above bound is useful.

We now summarize the main points discussed in this section.

Definition 4.1. Let A and B be unital C*-algebras and let Π = (D, ω) bea bridge from A to B. We say that Π is a bridge with conditional expectations ifconditional expectations EA and EB from D onto A and B are specified, satisfyingthe following properties:

(1) The conditional expectations are faithful.(2) The pivot ω is a projection.(3) There is a constant, rω, such that

EA(ω) = rω1D = EB(ω).

For such a bridge with conditional expectations we define ΦA on D by

ΦA(d) = r−1ω EA(ωdω).

We define ΦB similarly, with the roles of A and B reversed. We will often writeΠ = (D, ω, EA, EB) for a bridge with conditional expectations.

I should mention here that at present I do not see how the class of examplesconsidered by Latremoliere that involves non-commutative tori [9] fits into thesetting of bridges with conditional expectations, though I have not studied thismatter carefully. It would certainly be interesting to understand this better. I alsodo not see how the general case of ordinary compact metric spaces, as discussed intheorem 6.6 of [8], fits into the setting of bridges with conditional expectations

Definition 4.2. With notation as above, let LA and LB be slip-norms on Aand B. We say that a bridge with conditional expectations Π = (D, ω, EA, EB) isadmissible for LA and LB if

LA(ΦA(b)) ≤ LB(b)

for all b ∈ B, andLB(ΦB(a)) ≤ LA(a)

for all a ∈ A .

218 MARC A. RIEFFEL

We define the reach, height and length of a bridge with conditional expectations(D, ω, EA, EB) to be those of the bridge (D, ω).

From the earlier discussion we obtain:

Theorem 4.3. Let LA and LB be slip-norms on unital C*-algebras A and B,and let Π = (D, ω, EA, EB) be a bridge with conditional expectations from A to Bthat is admissible for LA and LB. Then

reach(Π) ≤ max{γA, γB},where

γA = sup{‖aω − ωΦB(a)‖D : a ∈ L1A},

and similarly for γB, while

height(Π) ≤ max{δA, δB},where

δA = sup{‖a− ΦA(a)‖ : a ∈ L1A},

and similarly for δB. Consequently

length(Π) ≤ max{γA, γB, δA, δB}.(Consequently the propinquity between (A, LA) and (B, LB), as defined in [8], is nogreater than the right-hand side above.)

We could axiomitize the above situation in terms of just ΦA and ΦB, withoutrequiring that they come from conditional expectations, but at present I do notknow of examples for which this would be useful. It would not suffice to requirethat ΦA and ΦB just be positive (and unital) because for the matricial case discussedin the next section they would need to be completely positive.

The following result is very pertinent to our first class of basic examples.

Proposition 4.4. With notation as above, suppose that our bridge Π has thequite special property that ω commutes with every element of A, or at least thatEA(ωaω) = EA(aω) for all a ∈ A. Then ΦA(a) = a for all a ∈ A. ConsequentlyδA = 0, and the restriction of S1(ω) to A is all of S(A).

Proof. This depends on the conditional expectation property of EA. Fora ∈ A we have

ΦA(a) = r−1ω EA(aω) = ar−1

ω EA(ω) = a.

�

The following steps might not initially seem useful, but in Sections 6 and 7we will see in connection with our basic examples that they are quite useful. Ournotation is as above. Let ν ∈ S(B). Then as seen above, ν ◦ ΦB ∈ S1(ω), and soits restriction to A is in S(A). But then ν ◦ ΦB ◦ ΦA ∈ S1(ω). Let us denote it byψν . Then the restriction of ψν to B can be used as an approximation to ν by anelement of S1(ω). Now for any b ∈ B we have

|ν(b)− ψν(b)| = |ν(b)− (ν ◦ ΦB ◦ ΦA)(b)| ≤ ‖b− ΦB(ΦA(b))‖.

Notation 4.5. In terms of the above notation we set

δB = sup{‖b− ΦB(ΦA(b))‖ : b ∈ L1B}.


We note that LB(ΦB(ΦA(b))) ≤ LB(b) because of the admissibility requirementsof Definition 4.2. It follows that

ρLB(ν, ψν) ≤ δB.

We define δA in the same way, and obtain the corresponding estimate for thedistances from elements of S(A) to the restriction of S1(ω) to A. In this way weobtain:

Proposition 4.6. For notation as above,

height(Π) ≤ max{min{δA, δA},min{δB, δB}}.We will see in Section 6 that for our first class of basic examples, ΦB ◦ ΦA

is exactly a term that plays an important role in [11, 14]. It is essentially an“anti-Berezin-transform”.

5. The corresponding matricial bridges

Fix a positive integer q. We let Mq denote the C*-algebra of q × q matriceswith complex entries. For any C*-algebra A we let Mq(A) denote the C*-algebraof q × q matrices with entries in A. We often identify it in the evident way withthe C*-algebra Mq ⊗A.

Let A and B be unital C*-algebras, and let Π = (D, ω) be a bridge from Ato B. Then Mq(A) can be viewed as a subalgebra of Mq(D), as can Mq(B). Letωq = 1q ⊗ ω, where 1q is the identity element of Mq, so ωq can be viewed as thediagonal matrix in Mq(D) with ω in each diagonal entry. Then it is easily seen thatΠq = (Mq(D), ωq) is a bridge from Mq(A) to Mq(B).

Definition 5.1. The sequence {Πq} is called the matricial bridge correspondingto the bridge Π.

In order to measure the length of Πq we need slip-norms LAq and LB

q on Mq(A)and Mq(B). It is reasonable to want these slip-norms to be coherent in some senseas q varies. The discussion that we will give just after Theorem 6.8 suggests thatthe coherence requirement be that the sequences {LA

q } and {LBq } form “matrix

slipnorms”. To explain what this means, for any positive integers m and n we letMmn denote the linear space of m × n matrices with complex entries, equippedwith the norm obtained by viewing such matrices as operators from the Hilbertspace Cn to the Hilbert space Cm. We then note that for any A ∈ Mn(A), anyα ∈ Mmn, and any β ∈ Mnm the usual matrix product αAβ is in Mm(A). Thefollowing definition, for the case of Lip-norms, is given in definition 5.1 of [18] (andsee also [6,14,17,19]).

Definition 5.2. A sequence {LAn } is a matrix slip-norm for A if LA

n is a ∗-seminorm (with value +∞ permitted) on Mn(A) for each integer n ≥ 1, and thisfamily of seminorms has the following properties:

(1) For any A ∈ Mn(A), any α ∈ Mmn, and any β ∈ Mnm, we have

LAm(αAβ) ≤ ‖α‖LA

n (A)‖β‖.(2) For any A ∈ Mm(A) and any C ∈ Mn(A) we have

LAm+n

([A 00 C

])= max(LA

m(A), LAn (C)).

(3) LA1 is a slip-norm.

220 MARC A. RIEFFEL

We remark that the properties above imply that for n ≥ 2 the null-space of LAn

contains all of Mn, not just the scalar multiples of the identity. This is why ourdefinition of slip-norms does not require that the null-space is exactly the scalarmultiples of the identity.

Now let Π = (D, ω, EA, EB) be a bridge with conditional expectations. Forany integer q ≥ 1 set EA

q = ιq ⊗ EA, where ιq is the identity map from Mq onto

itself. Define EBq similarly. Then it is easily seen that EA

q and EBq are faithful

conditional expectations from Mq(D) onto its subalgebras Mq(A) and Mq(B) re-spectively. Furthermore, EA

q (ωq) is the diagonal matrix each diagonal entry of

which is EA(ω) = rω1D, and from this we see that EAq (ωq) = rω1Mq(A). Thus

rωq= rω. It is also clear that ωq is a projection. Putting this all together, we

obtain:

Proposition 5.3. Let Π = (D, ω, EA, EB) be a bridge with conditional expec-tations from A to B. Then

Πq = (Mq(D), ωq, EAq , EB

q )

is a bridge with conditional expectations from Mq(A) to Mq(B). It has the sameconstant rω as does Π.

We can then set ΦAq = ιq ⊗ΦA, and similarly for ΦB

q . Because Πq has the same

constant rω as does Π, we see that for any D ∈ Mq(D) we have

ΦAq (D) = r−1

ω EAq (ωqDωq).

Suppose now that A and B have matrix slip-norms {LAn } and {LB

n}. We remarkthat a matrix slip-norm {LA

n } is in general not at all determined by LA1 . Thus a

bridge that is admissible for LA1 as in Definition 4.2 need not relate well to the

seminorms LAn for higher n.

Definition 5.4. With notation as above, let {LAn } and {LB

n} be matrix slip-norms on A and B. We say that a bridge with conditional expectations Π =(D, ω, EA, EB) is admissible for {LA

n } and {LBn} if for all integers n ≥ 1 the bridge

Πn is admissible for LAn and LB

n ; that is, for all integers n ≥ 1 we have

LAn (Φ

An (B)) ≤ LB

n(B)

for all B ∈ Mn(B), whereΦA

n (B) = r−1ω EA

n (ωnBωn),

and similarly with the roles of A and B reversed.

We assume now that Π = (D, ω, EA, EB) is admissible for {LAn } and {LB

n}.Since for a fixed integer q the bridge Πq is admissible for LA

q and LBq , the length of

Πq is defined. We now show how to obtain an upper bound for the length of Πq interms of the data used in the previous section to get an upper bound on the lengthof Π.

We consider first the reach of Πq. Set, much as earlier,

L1qA = {A ∈ Mq(A) : A = A∗ and LA

q (A) ≤ 1},

and similarly for L1qB . Then the reach of Πq is defined to be

HausMq(D){L1qA ωq , ωqL1q

B }.


Suppose that B ∈ L1qB . Then ΦA

q (B) ∈ L1qA by the admissibility requirement. So

we want to bound

‖ΦAq (B)ωq − ωqB‖Mq(D).

I don’t see any better way to bound this in terms of the data used in Theorem 4.3for Π than by using an entry-wise estimate as done in the third paragraph beforelemma 14.2 of [14]). We use the fact that for a q × q matrix C = [cjk] with entriesin a C*-algebra we have ‖C‖ ≤ qmaxjk{‖cjk‖} (as is seen by expressing C as thesum of the q matrices whose only non-zero entries are the entries cjk for which j−kis a given constant mod q). In this way, for B ∈ Mq(B) with B = [bjk] we find thatthe last displayed term above is

≤ qmaxjk

{‖ΦA(bjk)ω − ωbjk‖D}.

The small difficulty is that the bjk’s need not be self-adjoint. But for any b ∈ B, if wedenote its real and imaginary parts by br and bi, then because LB is a ∗-seminormit follows that LB(br) ≤ LB(b) and similarly for bi. Consequently

‖ΦA(b)ω − ωb‖D ≤ ‖ΦA(br)ω − ωbr‖D + ‖ΦA(bi)ω − ωbi‖D≤ γBLB(br) + γBLB(bi) ≤ 2γBLB(b).

Thus the term displayed just before is

≤ 2qγBLB(bjk).

But {LBn} is a matrix slip-norm, and by the first property of such seminorms given

in Definition 5.2, we have

max{LB(bjk)} ≤ LBq (B).

Thus for B ∈ L1qB we see that

‖ΦAq (B)ωq − ωqB‖Mq(D) ≤ 2qγB,

so that ωqB is in the 2qγB-neighborhood of L1qA ωq. In the same way Aωq is in the

2qγA-neighborhood of ωqL1qB for every A ∈ L1q

A . We find in this way that

reach(Πq) ≤ 2qmax{γA, γB}.We now consider the height of Πq. We argue much as in the discussion of

height before Definition 4.1. For any μ ∈ S(Mq(A)) its composition with ΦAq is an

element, φμ, of S1(ωq), specifically defined by

φμ(D) = μ(ΦAq (D)) = μ(r−1

ω EAq (ωqDωq)).

We take φμ|Mq(A) as an approximation to μ, and estimate the distance between

these elements of S(Mq(A)). For A ∈ L1qA we calculate

|μ(A)− φμ(A)| = |μ(A)− μ(ΦAq (A))| ≤ ‖A− ΦA

q (A)‖.Again I don’t see any better way to bound this in terms of the data used in Theorem4.3 for Π than by using an entry-wise estimates. For A ∈ Mq(A) with A = [ajk] wefind (by using arguments as above to deal with the fact that the ajk’s need not beself-adjoint) that the last displayed term above is

≤ qmaxjk

{‖ajk − ΦA(ajk)‖} ≤ 2qδA max{LA(ajk)}.

222 MARC A. RIEFFEL

But again {LAn } is a matrix slip-norm, and so by the first property of such seminorms

given in Definition 5.2 we have

max{LA(ajk)} ≤ LAq (A).

Since our assumption is that A ∈ L1qA , we see in this way that

ρLAq(μ, φμ|A) ≤ 2qδA.

Thus S(Mq(A)) is in the 2qδA-neighborhood of the restriction to Mq(A) of S1(ωq).We find in the same way that S(Mq(B)) is in the 2qδB-neighborhood of the

restriction to Mq(B) of S1(ωq). Consequently,

height(Πq) ≤ 2qmax{δA, δB}.

We can instead use δA in the way done in Proposition 4.6. Using reasoningmuch like that used above, we find that for any B ∈ Mq(B) we have:

‖B − ΦBq (Φ

Aq (B))‖ ≤ qmax

jk{‖bjk − ΦB(ΦA(bjk))‖

≤ 2qδB max{LB(bjk)} ≤ 2qδBLBq (B)

Consequently we see that

height(Πq) ≤ 2qmax{δA, δB}.

We summarize what we have found by:

Theorem 5.5. Let {LAn } and {LB

n} be matrix slip-norms on unital C*-algebrasA and B, and let Π = (D, ω, EA, EB) be a bridge with conditional expectations fromA to B that is admissible for {LA

n } and {LBn}. For any fixed positive integer q let

Πq be the corresponding bridge with conditional expectations from Mq(A) to Mq(B).Then

reach(Πq) ≤ 2qmax{γA, γB},

where as before

γA = sup{‖aω − ωΦB(a)‖D : a ∈ L1A},

and similarly for γB; while

height(Πq) ≤ 2qmax{min{δA, δA},min{δB δB}},

where as before

δA = sup{‖ΦA(a)− a‖ : a ∈ L1A}

and

δA = sup{‖a− ΦA(ΦB(a))‖ : a ∈ L1A},

and similarly for δB and δB. Consequently

length(Πq) ≤ 2qmax{γA, γB,min{δA, δA},min{δB δB}}.


6. The application to the first class of basic examples

We now apply the above general considerations to our first class of basic ex-amples, described in Section 2, and we will use the same notation as is used inthat section. We proceed to obtain an upper bound for the length of the bridgeΠ = (D, ω), where D = C(G/H,B) and ω(x) = αx(P ). We begin by consideringits reach.

As seen in Section 4 , for any F ∈ D = C(G/H,B) we have

EA(ωFω)(x) = τB(F (x)αx(P )).

From this it is easily seen that r−1ω is the dimension of H, and so r−1

ω τB is the usualunnormalized trace on B, which we now denote by trB. In particular, for any T ∈ Bwe have

(6.1) ΦA(T )(x) = r−1ω EA(ωTω)(x) = trB(Tαx(P ))

for all x ∈ G/H. But this is exactly the covariant Berezin symbol of T (for thisgeneral context) as defined early in section 1 of [11] and denoted there by σT . Itis natural to put on D the action λ ⊗ α of G. One then easily checks that ΦA isequivariant for λ⊗ α and λ. From this it is easy to verify that

LA(ΦA(T )) = LA(σT ) ≤ LB(T )

for all T ∈ B, which is exactly the content of proposition 1.1 of [11]. Thus thatpart of admissibility is satisfied.

Now

(ΦA(T )ω − ωT )(x) = αx(P )(σT (x)1B − T ).

Consequently

‖ΦA(T )ω − ωT‖D = sup{‖αx(P )(σT (x)1B − T )‖B : x ∈ G/H}.As discussed in the text before proposition 8.2 of [14], by equivariance this is

= sup{‖P (σαx(T )1B − αx(T ))‖B : x ∈ G}.Then because αx is isometric on B for LB (as well as for the norm), we find for ourpresent example that γB, as defined in equation (4.1), is given by(6.2)γB = sup{‖ΦA(T )ω − ωT‖D : T ∈ L1

B} = sup{‖P (tr(PT )1B − T )‖B : T ∈ L1B}.

This last term is exactly the definition of γB given in proposition 8.2 of [14].We next consider γA. For any f ∈ A we have

(6.3) ΦB(f) = r−1ω EB(f) = dH

∫G/H

f(x)αx(P ) dx,

where dH = dim(H). But this is exactly the formula used for the Berezin con-travariant symbol, as indicated in Section 4. Early in section 2 of [11] this ΦB isdenoted by σf , that is,

(6.4) ΦB(f) = σf

for the present class of examples. One easily checks that ΦB is equivariant for λ⊗αand α. From this it is easy to verify that

LB(ΦB(f)) = LB(σf ) ≤ LA(f)

for all f ∈ A, as shown in section 2 of [11]. Thus we obtain:

224 MARC A. RIEFFEL

Proposition 6.1. The bridge with conditional expectationsΠ = (D, ω, EA, EB) is admissible for LA and LB.

Much as in the statement of proposition 8.1 of [14] set

(6.5) γA = dH

∫ρG/H(e, y)‖Pαy(P )‖dy.

In the proof of proposition 8.1 of [14] (given just before the statement of the propo-sition, and where γA is denoted just by γA) it is shown, with different notation,that

(6.6) ‖fω − ωσf‖D ≤ γALA(f).

Thus if f ∈ L1A then

‖fω − ωΦB(f)‖D ≤ γA.

It follows that γA ≤ γA. We have thus obtained:

Proposition 6.2. For the present class of examples, with notation as above,we have

reach(Π) ≤ max{γA, γB} ≤ max{γA, γB}where γA is defined in equation 6.5 and γB is defined in equation 6.2 (and 4.1)above.

Even more, for the case mentioned at the beginning of Section 3 (and centralto [11,12,14]) in which G is a compact semisimple Lie group and λ is a positiveintegral weight, for each positive integer m let (Hm, Um) be the irreducible repre-sentation of G with highest weight mλ. Then let Bm = L(Hm) with action α ofG, and let Pm be the projection on the highest weight vector in Hm. All the Pm’swill have the same α-stability group, H. As before, we let A = C(G/H). Then foreach m we can construct as in Section 4 the bridge with conditional expectations,Πm = (Dm, ωm, EA

m, EBm). From a fixed length function � on G we will obtain Lip-

norms {LBm} which together with {LA} give meaning to the lengths of the bridgesΠm. In turn the constants γA

m, γAm, γBm

, δAm, δBm

will be defined.Now it follows from the discussion of γA above that γA

m ≤ γAm for each m. But

section 10 of [14] gives a proof that the sequence γAm converges to 0 as m goes to

∞. It follows that γAm converges to 0 as m goes to ∞. Then section 12 of [14] gives

a proof that the sequence γBm

converges to 0 as m goes to ∞. Putting togetherthese results for γA

m and γBm

, we obtain:

Proposition 6.3. The reach of the bridge Πm goes to 0 as m goes to ∞.

We now consider the height of Π. For δA something quite special happens. Itis easily seen that A = C(G/H) is the center of D = A⊗B, and so all elements ofA commute with ω. Thus we can apply Proposition 4.4 to conclude that δA = 0.

In order to deal with B we use δB of Notation 4.5 and the discussion surroundingit. For any T ∈ B we have

ΦB(ΦA(T )) = r−1ω EB(ω(r−1

ω EA(ωTω))ω)

= dH

∫αx(P )(dHτB(αx(P )Tαx(P ))αx(P )dx

= dH

∫αx(P )(trB(αx(P )T )dx = σ(σT ).


where for the last term we use notation from [11,14]. The term σ(σT ) plays an

important role there. See theorem 6.1 of [11] and theorem 11.5 of [14]. The δB ofour Notation 4.5 is for the present class of examples exactly the δB of notation 8.4of [14]. For use in the next section we here denote it by δB, that is:

Notation 6.4. For G, A = C(G/H), B = L(H), σ, σ, etc as above, we set

δB = sup{‖T − σ(σT )‖ : T ∈ L1B}.

When we combine this with Propositions 4.6 and 6.2 we obtain:

Theorem 6.5. For the present class of examples, with notation as above, wehave

height(Π) ≤ δB.

Consequently

length(Π) ≤ max{γA, γB,min{δB, δB} ≤ max{γA, γB,min{δB, δB}}.

We will indicate in Section 8 Latremoliere’s definition of his propinquity be-tween compact quantum metric spaces, but it is always no larger than the lengthof any bridge between the two spaces. He denotes his propinquity simply by Λ,but we will denote it here by “Prpq”. Consequently, from the above theorem weobtain:

Corollary 6.6. With notation as above,

Prpq((A, LA), (B, LB)) ≤ max{γA, γB,min{δB, δB}}.

For the case of highest weight representations discussed just above, theorem11.5 of [14] gives a proof that the sequence δBm (in our notation) converges to 0 asm goes to ∞. It follows from the above proposition that:

Proposition 6.7. The height of the bridge Πm goes to 0 as m goes to ∞.

Combining this with Proposition 6.3, we obtain:

Theorem 6.8. The length of the bridge Πm goes to 0 as m goes to ∞. Conse-quently Prpq((A, LA), (Bm, LBm

)) goes to 0 as m goes to ∞.

We now treat the matricial case, beginning with the general situation in whichG is some compact group. We must first specify our matrix slip-norms. This isessentially done in example 3.2 of [18] and section 14 of [14]. As discussed in Section2, we have the actions λ and α on A = C(G/H) and B = B(H) respectively. For anyn let λn and αn be the corresponding actions ιn⊗λ and ιn⊗α on Mn⊗A = Mn(A)and Mn⊗B = Mn(B). We then use the length function � and formula 2.1 to defineseminorms LA

n and LBn on Mn(A) and Mn(B). It is easily verified that {LA

n } and{LB

n} are matrix slip-norms. Notice that here LA1 = LA and LB

1 = LB are actuallyLip-norms, and so, by property 1 of Definition 5.2, for each n the null-spaces of LA

n

and LBn are exactly Mn.

Now fix q and take n = q. From our bridge with conditional expectationsΠ = (D, ω, EA, EB) we define the bridge with conditional expectations Πq =(Mq(D), ωq, E

Aq , EB

q ) from Mq(A) to Mq(B) in the way done in Proposition 5.3.

We then set ΦAq = ιq ⊗ ΦA, and similarly for ΦB

q , as done right after Proposition5.3.

226 MARC A. RIEFFEL

Because λ and α and ΦA and ΦB act entry-wise on Mq(D), and because ΦA

and ΦB are equivariant for λ⊗α and λ and for λ⊗α and α respectively, it is easilyseen that Πq is admissible for {LA

q } and {LBq }.

We are thus in position to apply Theorem 5.5. From it and Theorem 6.5 weconclude that:

Theorem 6.9. With notation as above, we have

reach(Πq) ≤ 2qmax{γA, γB} ≤ 2qmax{γA, γB},

where γA is defined by formula 6.5. Furthermore

height(Πq) ≤ 2qmin{δB, δB},

where δB is defined in Notation 6.4. Thus

length(Πq) ≤ 2qmax{γA, γB,min{δB, δB}}.

We remark that we could improve slightly on the above theorem by using acalculation given in section 14 of [14] in the middle of the discussion there of Wu’sresults. Let F ∈ Mq(A) be given, with F = {fjk}, and set

tjk = σfjk = dH

∫fjk(y)αy(P )dy,

and let T = {tjk}. Then

(Fωq − ωqT )(x) = {αx(P )(fjk(x)− dH

∫fjk(y)αy(P )dy)}

= {dH∫(fjk(x)− fjk(y))αx(P )αy(P )dy}.

To obtain a bound on γAq we need to take the supremum of the norm of this

expression over all x and over all F with LAq (F ) ≤ 1. By translation by x, in the

way done shortly before proposition 8.1 of [14], it suffices to consider

sup{‖{dH∫(fjk(e)− fjk(y))Pαy(P )dy}‖}

≤ dH

∫‖F (e)− F (y)‖‖Pαy(P )‖dy

≤ LAq (F )dH

∫ρG/H(e, y)‖Pαy(P )‖dy = LA

q (F )γA.

In this way we see that γAq ≤ γA, with no factor of 2q needed.

We can apply Theorem 6.9 to the situation considered before Proposition 6.3in which G is a compact semisimple Lie group, λ is a positive integral weight, and(Hm, Um) is the irreducible representation of G with highest weight mλ for eachpositive m, with Bm = L(Hm). We can then form the bridge with conditionalexpectations, Πm = (Dm, ωm, EA

m, EBm) that is discussed there. For any positive

integer q we then have the matricial version involving Mq(A), Mq(Bm), and thecorresponding bridge Πq

m. On applying Theorem 6.9 together with the results

mentioned above about the convergence of the quantities γAm, γB

m, and δBm to 0, weobtain one of the two main theorems of this paper:



length(Πqm) ≤ 2qmax{γA

m, γBm,min{δBm, δBm}},

where γAm is defined as in formula 6.5, and where δBm is defined as in Notation 6.4.

Consequently length(Πqm) converges to 0 as m goes to ∞, for each fixed q.

We remark that because of the factor q in the right-hand side of the abovebound for length(Πq

m), we do not obtain convergence to 0 that is uniform in q. Ido not have a counter-example to the convergence being uniform in q, but it seemsto me very possible that the convergence will not be uniform.

7. The application to the second class of basic examples

We now apply our general considerations to our second basic class of examples,described in Section 3. We use the notation of that section. We also use muchof the notation of Section 6, but now we have two representations, (Hm, Um) and(Hn, Un) (where for the moment m and n are just labels). We have correspondingC*-algebras Bm and Bn, and projections Pm and Pn.

We let Lm be the Lip-norm defined on Bm determined by the action α and thelength function � as in equation 2.1, and similarly for Ln on Bn. In terms of theseLip-norms the length of any bridge from Bm to Bn is defined.

As in Section 3 we consider the bridge Π = (D, ω) for which

D = Bm ⊗A⊗ Bn = C(G/H, Bm ⊗ Bn),

and the pivot, ω, in C(G/H, Bm ⊗ Bn), is defined by

ω(x) = αx(Pm)⊗ αx(P

n).

We view Bm as a subalgebra of D by sending T ∈ Bm to T ⊗1A⊗1Bn , and similarlyfor Bn.

As seen in Section 4, the tracial state τA⊗τn on A⊗Bn determines a canonicalconditional expectation, Em, from D onto Bm, and the tracial state τm⊗ τA deter-mines a canonical conditional expectation, En, from D onto Bn . We find that forany F ∈ D we have

Em(F ) =

∫G/H

(ιm ⊗ τn)(F (x)) dx,

and similarly for En, where here ιm is the identity map from Bm to itself. Fromthis it is easily seen that

r−1ω = dmdn

where dm is the dimension of Hm and similarly for dn. Thus Π = (D, ω, Em, En)is a bridge with conditional expectations.

Then

Φm(F )(x) = r−1ω Em(ωFω)(x) = dmdnE

m(ωFω)(x).

But, if we set αx(Pm ⊗ Pn) = αx(P

m)⊗ αx(Pn), we have

Em(ωFω)(x) =

∫(ιm ⊗ τn)(αx(P

m ⊗ Pn)F (x)αx(Pm ⊗ Pn))dx.

In particular, for any T ∈ Bn we have

Em(ωTω)(x) =

∫αx(P

m)τn(Tαx(Pn))dx,

228 MARC A. RIEFFEL

and so, since dnτBn is the usual unnormalized trace trn on Bn, we have

Φm(T ) = dm

∫αx(P

m)trn(Tαx(Pn))dx.

This is essentially the formula obtained in Section 4, and is exactly the seconddisplayed formula in section 3 of [12]. Even more, with notation as in Section 6,especially the ΦA of equation (6.1), except for our different Bm and Bn etc, we seethat we can write

(7.1) Φm(T ) = ΦBm

(ΦA(T )) = σm(σnT ).

We have a similar equation for Φn(T ), and we see that we depend on the context tomake clear on which of the two algebras Bm and Bn we consider ΦA to be defined.

As in the proof of Proposition 6.1, we can use the fact that Em and En areequivariant, where the action of G on D is given by α⊗ λ⊗ α, to obtain:

Proposition 7.1. The bridge with conditional expectations Π is admissible forLm and Ln.

The formula (7.1) suggests the following steps for obtaining a bound on thereach of Π in terms of the data of the previous section. Let S ∈ Bm, f ∈ C(G/H),and T ∈ Bn. Then, for the norm of Bm ⊗ Bm and for any x ∈ G/H, we have

‖(Sω − ωT )(x)‖ = ‖(Sαx(Pm))⊗ αx(P

n)− αx(Pm)⊗ (αx(P

n)T )‖≤ ‖(Sαx(P

m))⊗ αx(Pn)− f(x)(αx(P

m)⊗ (αx(Pn))‖

+ ‖f(x)(αx(Pm)⊗ (αx(P

n))− αx(Pm)⊗ (αx(P

n)T )‖= ‖Sαx(P

m)− f(x)αx(Pm)‖+ ‖f(x)αx(P

n)− αx(Pn)T‖.(7.2)

Notice that the last two norms are in Bm and Bn respectively.We will also use the ΦB of equation (6.3), but now, to distinguish it from the

Φm above, we indicate that it is defined on A (and maps to Bm) by writing ΦBm

A .For fixed T ∈ Bn let us set f(x) = ΦA(T ) = trn(αx(P

n)T ), and then let usset S = ΦBm

A (f) = dm∫f(x)αx(P

m). Thus S = σmf by equation 6.4. When we

substitute these into the inequality (7.2), we obtain

‖(ΦBm

A (f)ω − ωT )(x)‖ ≤ ‖(ΦBm

A (f)(x)− f(x))αx(Pm)‖+ ‖αx(P

n)(f(x)− T )‖.In view of the definition of f , we recognize that the supremum over x ∈ G/H of thesecond term on the right of the inequality sign is the kind of term involved in thesupremum in the right-hand side of equality (6.2). Consequently that second termabove is no greater than γBn

Ln(T ). To indicate that this comes from equality (6.2)we write γBn

A instead of just γBn

. Because ΦBm

A (f) = σmf , we also recognize that the

supremum over x ∈ G/H of the first term above on the right of the inequality signis exactly (after taking adjoints to get Pm on the correct side) the left hand side ofinequality (6.6), where the ω there is that of Section 6. Consequently that term isno greater than γA

mLA(f), where the subscript m on γAm indicates that Pm should

be used in equation (6.5). But from the admissibility in Proposition 6.1 involvingΦBm

A = ΦBm

and ΦA we have

Lm(S) = Lm(ΦBm

A (f)) ≤ LA(f) ≤ Ln(T ).

Notice that it follows that if T ∈ L1Bn then S ∈ L1

Bm . Anyway, on taking thesupremum over x ∈ G/H, we obtain

‖ΦBm

A (f)ω − ωT‖D ≤ (γAm + γBn

A )Ln(T ).


We see in this way that the distance from ωT to L1Bmω is no bigger than γA

m+ γBn

A .The role ofA in Theorem 4.3 is here being played by Bm. So to reduce confusion

we will here write γmn for the γA of Theorem 4.3, showing also the dependence on

n. Thus by definition

γmn = sup{‖Tω − ωΦn(T )‖D : T ∈ L1

Bm}.We define γn

m similarly. Then in terms of this notation, what we have found aboveis that

γnm ≤ γA

m + γBn

AWe now indicate the dependence of Π on m and n by writing Πm,n. The situationjust above is essentially symmetric in m and n, and so, on combining this with thefirst inequality of Theorem 4.3, we obtain:

Proposition 7.2. With notation as above, we have

reach(Πm,n) ≤ max{γnm, γm

n } ≤ max{γAm + γBn

A , γAn + γBm

A },We apply this to the case in which G is a compact semisimple Lie group and λ

is a positive integral weight, and our two representations of G are of highest weightsmλ and nλ. As mentioned in the previous section, section 10 of [14] gives a proofthat the sequence γA

m converges to 0 as m goes to ∞, while section 12 of [14] givesa proof that the sequence γBm

A converges to 0 as m goes to ∞. We thus see thatwe obtain:

Proposition 7.3. The reach of the bridge Πm,n goes to 0 as m and n go to ∞simultaneously.

We now obtain an upper bound for the height of Πm,n. For this we will againuse Proposition 4.6. We calculate as follows, using equation (7.1). For T ∈ Bn wehave

Φn(Φm(T )) = Φn(σm(σnT )) = σn(σm(σm(σn

T )))

Thus

‖T − Φn(Φm(T ))‖(7.3)

≤ ‖T − σn(σnT )‖+ ‖σn(σn

T )− σn((σm ◦ σm)(σnT )‖

≤ δBn

A LBn

(T ) + ‖σnT − σm(σm(σn

T ))‖,

where the first term of the last line comes from Notation 6.4 and we write δBn

A for

the δBn

there. But σnT is just an element of A, and in inequality 11.2 of [14] it is

shown that for any f ∈ A we have

‖f − σm(σm(f)‖ ≤ δAmLA(f),

where δAm is defined in equation 11.1 of [14] by

(7.4) δAm =

∫G/H

ρG/H(e, x)dmtr(Pmαx(Pm)) dx.

(In equation 11.1 of [14] δAm is denoted just by δAm. Also, σm ◦ σm is, within oursetting, the usual Berezin transform.) Thus we see that the second term of the last

line of inequality (7.3) is no bigger than δAmLA(σnT ). But L

A(σnT ) ≤ LBn

(T ). Fromall of this we see that if T ∈ L1

Bn then

‖T − Φn(Φm(T ))‖ ≤ δBn

A + δAm.

230 MARC A. RIEFFEL

Again, the role of B in Notation 4.5 is being played here by Bn, and so to reduce

confusion we will here write δnm for the δB of Notation 4.5. We then see that forour present class of examples, which depend on m and n, we have

δnm ≤ δBn

A + δAm.

The situation is essentially symmetric in m and n, and so, combining this withPropositions 4.6 and 7.2, we obtain:


height(Πm,n) ≤ max{δnm, δmn } ≤ max{δBn

A + δAm, δBm

A + δAn }.

Consequently

length(Πm,n) ≤ max{γAm + γBn

A , γAn + γBm

A , δBn

A + δAm, δBm

A + δAn }.

We apply this to the case in which G is a compact semisimple Lie group andλ is a positive integral weight, and our two representations of G are of highestweights mλ and nλ. As mentioned in the previous section, theorem 11.5 of [14]

gives a proof that the sequence δBm

A (in our notation) converges to 0 as m goes to∞, while theorem 3.4 of [11] shows that the sequence δAm (where it was denoted byγm) converges to 0 as m goes to ∞. Thus when we combine this with Proposition7.3 we obtain:

Theorem 7.5. The height of the bridge Πm,n goes to 0 as m and n go to ∞simultaneously. Consequently the length of the bridge Πm,n goes to 0 as m and ngo to ∞ simultaneously, and thus Prpq((Bm, Lm), (Bn, Ln)) goes to 0 as m and ngo to ∞ simultaneously.

We now consider the matricial case. For any natural number q we apply theconstructions of Section 5 to obtain the bridge with conditional expectations

Πqm,n = (Mq(D), ωq, E

mq , En

q )

from Mq(Bm) to Mq(Bn). From this we then obtain the corresponding maps Φmq

and Φnq .

We have the actions αq of G on Mq(Bm) and Mq(Bn), much as discussed afterTheorem 6.8. From these actions and the length function � we obtain the slip-normsLmq and Ln

q . As q varies, these result in matrix slip-norms. One shows that Πqm,n is

admissible for Lmq and Ln

q by arguing in much the same way as done after formula(7.1).

We are thus in a position to apply Theorem 5.5, and, for the case of highest-weight representations of a semisimple Lie group, the convergence to 0 indicatedabove for the various constants. We obtain the second main theorem of this paper:


length(Πqm,n) ≤ 2qmax{γA

m + γBn

A , γAn + γBm

A , δBn

A + δAm, δBm

A + δAn }

where γAm is defined as in formula 6.5 while γBm

A is the γBm

of equation (6.2), and

where δBm

A = δBm

is defined in Notation 6.4 while δAm is defined by equation (7.4),and similarly for n. Consequently, for the case of highest-weight representations,length(Πq

m,n) converges to 0 as m and n go to ∞ simultaneously, for each fixed q.


8. Treks

Latremoliere defines his propinquity in terms of “treks”. We will not give herethe precise definition (for which see definition 3.20 of [8]), but the notion is quiteintuitive. A trek is a finite “path” of bridges, so that the “range” of the first bridgeshould be the “domain” of the second, etc. The length of a trek is the sum of thelengths of the bridges in it. The propinquity between two quantum compact metricspaces is the infimum of the lengths of all the treks between them. Latremoliereshows in [8] that propinquity is a metric on the collection of isometric isomorphismclasses of quantum compact metric spaces. Notably, he proves the striking fact thatif the propinquity between two quantum compact metric spaces is 0 then they areisometrically isomorphic.

There is an evident trek associated with our second class of examples. In thissection we will briefly examine this trek. Let the notation be as in the early partsof the previous section. Thus we have A = C(G/H), and the operator algebras Bm

and Bn. In Section 6 we have the bridge Πm = (A⊗ Bm, ωm) from A to Bm, andthe corresponding bridge Πn from A to Bn. But by reversing the roles of A andBm we obtain a bridge from Bm to A. We do this by still viewing A and Bm assubalgebras of Dm = C(G/H,Bm), but we now let A act on the right of Dm andwe let Bm act on the left. We will denote this bridge by D−1

m , which is consistentwith the notation of Latremoliere at the beginning of the proof of proposition 4.7 of[8]. Of course D−1

m has the “same” conditional expectations EA and EBm

as thoseof Πm. We will write EA as EA

m to distinguish it from the EA from Dn, which wewill denote by EA

n . Then D−1m is a bridge with conditional expectations, which is

easily seen to be admissible for LBm

and LA. It is then easily seen that

length(D−1m ) = length(Dm).

The pair Γm,n = (D−1m , Dn) then forms a trek from Bm to Bn, and

length(Γm,n) = length(D−1m ) + length(Dn).

From Theorem 6.5 it follows that

length(Γm,n) ≤ max{γAm, γBm

,min{δBm

, δBm}}

+ max{γAn , γBn

,min{δBn

, δBn}}.

Note that δAm and δAn do not appear in the above expression, in contrast to theirappearance in the estimate in Theorem 7.4 for length(Πm,n). This opens the pos-sibility that in some cases length(Γm,n) gives a smaller bound for Prpq(Bm,Bn)than does length(Πm,n), and, even more, that this might give examples for whichthe lengths of certain multi-bridge treks are strictly smaller that the lengths ofany single-bridge treks. But I have not tried to determine if this happens for theexamples in this paper.

We can view the situation slightly differently as follows. Although Latremolieredoes not mention it, it is natural to define the reach of a trek as the sum of thereaches of the bridges it contains, and similarly for the height of a trek. One couldthen give a new definition of the length of a trek as simply the max of its reachand height. This definition is no bigger that the original definition, and mightbe smaller. I have not examined how this might affect the arguments in [8], butI imagine that the effect would not be very significant. Anyway, for the above

232 MARC A. RIEFFEL

examples we see from Proposition 6.2 that we would have

reach(Γm,n) ≤ max{γAm, γBm

A }+max{γAn , γBn

A },

so that the bound for reach(Πm,n) given in Proposition 7.2 is no bigger than thatabove for reach(Γm,n). But from Theorem 6.5 we see that

height(Γm,n) ≤ δBm

A + δBn

A

(where δBm

A = δBm

), and this can clearly be less than the right-most bound forheight(Πm,n) given in Theorem 7.4.

References

[1] Nirmalendu Acharyya and Veronica Errasti Diez,Monopoles, Dirac operator and index theoryfor fuzzy SU(3)/(U(1)× U(1)), Phys. Rev. D 90 (2014) 125034 arXiv:1411.3538.

[2] Hajime Aoki, Yoshiko Hirayama, and Satoshi Iso, Construction of a topological charge onfuzzy S2 × S2 via a Ginsparg-Wilson relation, Phys. Rev. D 80 (2009), no. 12, 125006, 14,DOI 10.1103/PhysRevD.80.125006. MR2669812 (2011g:81301)

[3] B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-

Verlag, Berlin, 2006. Theory of C∗-algebras and von Neumann algebras; Operator Algebrasand Non-commutative Geometry, III. MR2188261 (2006k:46082)

[4] Athanasios Chatzistavrakidis and George Zoupanos, Higher-dimensional unified theories withfuzzy extra dimensions, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010),Paper 063, 47, DOI 10.3842/SIGMA.2010.063. MR2725020 (2011j:81170)

[5] S. Digal and T. R. Govindarajan, Topological stability of broken symmetry on fuzzy spheres,Modern Phys. Lett. A 27 (2012), no. 14, 1250082, 9, DOI 10.1142/S0217732312500824.MR2922516

[6] Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Mono-graphs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.MR1793753 (2002a:46082)

[7] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras.Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Provi-dence, RI, 1997. Elementary theory; Reprint of the 1983 original. MR1468229 (98f:46001a)

[8] Frederic Latremoliere, The quantum Gromov-Hausdorff propinquity, Trans. Amer. Math. Soc.368 (2016), no. 1, 365–411, DOI 10.1090/tran/6334. MR3413867

[9] Frederic Latremoliere, Convergence of fuzzy tori and quantum tori for the quantum Gromov-Hausdorf propinquity: an explicit approach, Munster J. Math., to appear, arXiv:1312.0069.

[10] Marc A. Rieffel, Metrics on states from actions of compact groups, Doc. Math. 3 (1998),215–229 (electronic). MR1647515 (99k:46126)

[11] Marc A. Rieffel, Matrix algebras converge to the sphere for quantum Gromov-Hausdorffdistance, Mem. Amer. Math. Soc. 168 (2004), no. 796, 67–91, DOI 10.1090/memo/0796.Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the

sphere for quantum Gromov-Hausdorff distance. MR2055928[12] Marc A. Rieffel, Distances between matrix algebras that converge to coadjoint orbits, Super-

strings, geometry, topology, and C∗-algebras, Proc. Sympos. Pure Math., vol. 81, Amer. Math.Soc., Providence, RI, 2010, pp. 173–180, DOI 10.1090/pspum/081/2681764. MR2681764(2011j:46126)

[13] Marc A. Rieffel, Vector bundles and Gromov-Hausdorff distance, J. K-Theory 5 (2010), no. 1,39–103, DOI 10.1017/is008008014jkt080. MR2600284 (2011c:53085)

[14] Marc A. Rieffel, Leibniz seminorms for “Matrix algebras converge to the sphere”, Quanta ofMaths (Providence, R.I.), Clay Mathematics Proceedings, vol. 11, Amer. Math. Soc., 2011,arXiv:0707.3229, pp. 543–578. MR2732064 (2011j:46125)

[15] Harold Steinacker, Emergent geometry and gravity from matrix models: an introduction, Clas-sical Quantum Gravity 27 (2010), no. 13, 133001, 46, DOI 10.1088/0264-9381/27/13/133001.MR2654039 (2011i:83056)

[16] Nik Weaver, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999.MR1832645 (2002g:46002)


[17] Wei Wu, Non-commutative metrics on matrix state spaces, J. Ramanujan Math. Soc. 20(2005), no. 3, 215–254. MR2181130 (2008c:46108)

[18] Wei Wu, Non-commutative metric topology on matrix state space, Proc. Amer. Math. Soc.134 (2006), no. 2, 443–453 (electronic), DOI 10.1090/S0002-9939-05-08036-6. MR2176013(2006f:46072)

[19] Wei Wu, Quantized Gromov-Hausdorff distance, J. Funct. Anal. 238 (2006), no. 1, 58–98,DOI 10.1016/j.jfa.2005.02.017. MR2234123 (2007h:46088)

Department of Mathematics, University of California, Berkeley, California 94720-

3840



Structure and applicationsof real C∗-algebras

Jonathan Rosenberg

Dedicated to Dick Kadison, with admiration and appreciation

Abstract. For a long time, practitioners of the art of operator algebras alwaysworked over the complex numbers, and nobody paid much attention to realC∗-algebras. Over the last thirty years, that situation has changed, and it’sbecome apparent that real C∗-algebras have a lot of extra structure not evidentfrom their complexifications. At the same time, interest in real C∗-algebrashas been driven by a number of compelling applications, for example in theclassification ofmanifolds of positive scalar curvature, in representation theory,and in the study of orientifold string theories. We will discuss a number ofinteresting examples of these, and how the real Baum-Connes conjecture playsan important role.

1. Real C∗-algebras

Definition 1.1. A real C∗-algebra is a Banach ∗-algebra A over R isometrically∗-isomorphic to a norm-closed ∗-algebra of bounded operators on a real Hilbertspace.

Remark 1.2. There is an equivalent abstract definition: a real C∗-algebra isa real Banach ∗-algebra A satisfying the C∗-identity ‖a∗a‖ = ‖a‖2 (for all a ∈ A)and also having the property that for all a ∈ A, a∗a has spectrum contained in[0,∞), or equivalently, having the property that ‖a∗a‖ ≤ ‖a∗a+b∗b‖ for all a, b ∈ A[32,48].

Books dealing with real C∗-algebras include [25,39,63], though they all havea slightly different emphasis from the one presented here.

Theorem 1.3 (“Schur’s Lemma”). Let π be an irreducible representation ofa real C∗-algebra A on a real Hilbert space H. Then the commutant π(A)′ of therepresentation must be R, C, or H.

2010 Mathematics Subject Classification. Primary 46L35; Secondary 19K35, 19L64, 22E46,81T30, 46L85, 19L50.

Key words and phrases. Real C∗-algebra, orientifold, KR-theory, twisting, T-duality, realBaum-Connes conjecture, assembly map, positive scalar curvature, Frobenius-Schur indicator.

This work was partially supported by NSF grant DMS-1206159. The author would like tothank Jeffrey Adams and Ran Cui for helpful discussions about Section 3.2 and Patrick Brosnanfor helpful discussions about Example 2.9.


235

236 JONATHAN ROSENBERG

Proof. Since π(A)′ is itself a real C∗-algebra (in fact a real von Neumannalgebra), it is enough to show it is a division algebra over R, since by Mazur’sTheorem [42] (variants of the proof are given in [32, Theorem 3.6] and [9]), R,C, and H are the only normed division algebras over R.1 Let x be a self-adjointelement of π(A)′. If p ∈ π(A)′ is a spectral projection of x, then pH and (1− p)Hare both invariant subspaces of π(A). By irreducibility, either p = 1 or 1 − p = 1.So this shows x must be of the form λ · 1 with λ ∈ R. Now if y ∈ π(A)′, y∗y = λ · 1with λ ∈ R, and similarly, yy∗ is a real multiple of 1. Since the spectra of y∗y andyy∗ must coincide except perhaps for 0, y∗y = yy∗ = λ = ‖y‖2 and either y = 0 orelse y is invertible (with inverse ‖y‖−2y∗). So π(A)′ is a division algebra. �

Corollary 1.4. The irreducible ∗-representations of a real C∗-algebra can beclassified into three types: real, complex, and quaternionic. (All of these can occur,as one can see from the examples of R, C, and H acting on themselves by lefttranslation.)

Given a real C∗-algebra A, its complexification AC = A + iA is a complexC∗-algebra, and comes with a real-linear ∗-automorphism σ with σ2 = 1, namelycomplex conjugation (with A as fixed points). Alternatively, we can consider θ(a) =σ(a∗) = (σ(a))∗. Then θ is a (complex linear) ∗-antiautomorphism of AC withθ2 = 1. Thus we can classify real C∗-algebras by classifying their complexificationsand then classifying all possibilities for σ or θ. This raises a number of questions:

Problem 1.5. Given a complex C∗-algebra A, is it the complexification of areal C∗-algebra? Equivalently, does it admit a ∗-antiautomorphism θ with θ2 = 1?

The answer to this in general is no. For example, Connes [11,12] showed thatthere are factors not anti-isomorphic to themselves, hence admitting no real form.Around the same time (ca. 1975), Philip Green (unpublished) observed that a sta-ble continuous-trace algebra over X with Dixmier-Douady invariant δ ∈ H3(X,Z)cannot be anti-isomorphic to itself unless there is a self-homeomorphism of X send-ing δ to −δ. Since it is easy to arrange for this not to be the case, there arecontinuous-trace algebras not admitting a real form.

By the way, just because a factor is anti-isomorphic to itself, that does notmean it has an self-antiautomorphism of period 2, and so it may not admit a realform. Jones constructed an example in [33].

Secondly we have:

Problem 1.6. Given a complex C∗-algebra A that admits a real form, howmany distinct such forms are there? Equivalently, how many conjugacy classes arethere of ∗-antiautomorphisms θ with θ2 = 1?

In general there can be more than one class of real forms. For example, M2(C)is the complexification of two distinct real C∗-algebras, M2(R) and H. From thisone can easily see that K(H) and B(H), the compact and bounded operators on aseparable infinite-dimensional Hilbert space, each have two distinct real forms. Forexample, K(H) is the complexification of both K(HR) and K(HH). This makes the

1Historical note: According to [37], Mazur presented this theorem in Lwow in 1938. Becauseof space limitations in Comptes Rendus, he never published the proof, but his original proof isreproduced in [73] as well as in [41], which also includes a copy of Mazur’s original hand-typedmanuscript, with the proof included.

REAL C∗-ALGEBRAS 237

following theorem due independently to Størmer and to Giordano and Jones all themore surprising and remarkable.

Theorem 1.7 ([20, 22, 67]). The hyperfinite II1 factor R has a unique realform.

This has a rather surprising consequence: if RR is the (unique) real hyperfiniteII1 factor, then RR ⊗R H ∼= RR (since this is a real form of R ⊗ M2(C) ∼= R). Infact we also have

Theorem 1.8 ([20]). The injective II∞ factor has a unique real form.

In the commutative case, there is no difference between antiautomorphisms andautomorphisms. Thus we get the following classification theorem.

Theorem 1.9 ([4, Theorem 9.1]). Commutative real C∗-algebras are classifiedby pairs consisting of a locally compact Hausdorff space X and a self-homeomorphismτ of X satisfying τ2 = 1. The algebra associated to (X, τ ), denoted C0(X, τ ), is

{f ∈ C0(X) | f(τ (x)) = f(x) ∀x ∈ X}.

Proof. If A is a commutative real C∗-algebra, then AC∼= C0(X) for some

locally compact Hausdorff space. The ∗-antiautomorphism θ discussed above be-comes a ∗-automorphism of AC (since the order of multiplication is immaterial) andthus comes from a self-homeomorphism τ of X satisfying τ2 = 1. We recover A as

{f ∈ AC | σ(f) = f} = {f ∈ AC | θ(f) = f∗}= {f ∈ C0(X) | f(τ (x)) = f(x) ∀x ∈ X}.

In the other direction, given X and τ , the indicated formula certainly gives acommutative real C∗-algebra. �

One could also ask about the classification of real AF algebras. This amountsto answering Problems 1.5 and 1.6 for complex AF algebras A (inductive limits offinite dimensional C∗-algebras). Since complex AF algebras are completely classi-fied by K-theory (K0(A) as an ordered group, plus the order unit if A is unital)[17], one would expect a purely K-theoretic solution. This was provided by Gior-dano [21], but the answer is considerably more complicated than in the complexcase. Of course this is hardly surprising, since we already know that even the sim-plest noncommutative finite dimensional complex C∗-algebra, M2(C), has two twodistinct real structures. Giordano also showed that his invariant is equivalent toone introduced by Goodearl and Handelman [26]. We will not attempt to give theprecise statement except to say that it involves all three of KO0, KO2, and KO4.(For example, one can distinguish the algebras M2(R) and H, both real forms ofM2(C), by looking at KO2.) Also, unlike the complex case, one usually has to dealwith torsion in the K-groups.

For the rest of this paper, we will focus on the case of separable type I C∗-algebras, especially those that arise in representation theory. Recall that if A is aseparable type I (complex) C∗-algebra, with primitive ideal space PrimA (equipped

with the Jacobson topology), then the natural map A → PrimA, sending theequivalence class of an irreducible representation π to its kernel kerπ, is a bijection,

and enables us to put a T0 locally quasi-compact topology on A.Suppose A is a complex C∗-algebra with a real form. As we have seen, that

means A is equipped with a conjugate-linear ∗-automorphism σ with σ2 = 1, or


alternatively, with a linear ∗-antiautomorphism θ with θ2 = 1. We can think of θas an isomorphism θ : A → At (where At is the opposite algebra, the same complexvector space with the same involution ∗, but with multiplication reversed) such thatthe composite θt ◦ θ is the identity (from A to (At)t = A).

Now let π be an irreducible ∗-representation of A. We can think of π as amap A → B(H), and obviously, π induces a related map (which as a map of sets isexactly the same as π) πt : At → B(H)t. Composing with θ and with the standard∗-antiautomorphism τ : B(H)t → B(H) (the “transpose map”) coming from theidentification of B(H) as the complexification of B(HR), we get a composite map

Aθ ��

θ∗(π)

��At πt�� B(H)t

τ �� B(H) .

One can also see that doing this twice brings us back where we started, so wehave seen:

Proposition 1.10. If A is a complex C∗-algebra (for our purposes, separa-ble and type I, though this is irrelevant here) with a real structure (given by a

∗-antiautomorphism θ of period 2), then θ induces an involution on A.

Proof. The involution sends [π] �→ [θ∗(π)]. To show that this is an involution,let’s compute θ∗(θ∗(π)). By definition, this is the composite

Aθ ��

��Atθ∗(π)

t

�� B(H)tτ �� B(H)

or

Aθ �� At θ ��

θ∗(π)t

Aπ �� B(H)

τ �� B(H)tτ �� B(H) ,

but θ ◦ θ and τ ◦ τ are each the identity, so this is just π again. �

Note that in the commutative case A = C0(X), the involution θ∗ on A is justthe original involution on X.

With these preliminaries out of the way, we can now begin to analyze the struc-ture of (separable) real type I C∗-algebras. Some of this information is undoubtedlyknown to experts, but it is surprisingly hard to dig it out of the literature, so wewill try give a complete treatment, without making any claims of great originality.

The one case which is easy to find in the literature concerns finite-dimensionalreal C∗-algebras, which are just semisimple Artinian algebras over R. The interestin this case comes from the real group rings RG of finite groups G, which areprecisely of this type. A convenient reference for the real representation theoryof finite groups is [64, §13.2]. A case which is not much harder is that of realrepresentation theory of compact groups. In this case, the associated real C∗-alge-bra is infinite-dimensional in general, but splits as a (C∗-)direct sum of (finite-dimensional) simple Artinian algebras over R. This case is discussed in great detailin [1, Ch. 3], and is applied to connected compact Lie groups in [1, Ch. 6 and 7].

Recall from Corollary 1.4 that the irreducible representations of a real C∗-alge-bra A are of three types. How does this type classification relate to the involution

of Proposition 1.10 on AC? The answer (which for the finite group case appears in[64, §13.2]) is given as follows:


Theorem 1.11. Let A be a real C∗-algebra and let AC be its complexification.Let π be an irreducible representation of A (on a real Hilbert space H). If π is ofreal type, then we get an irreducible representation πC of AC on HC by complexi-fying, and the class of this irreducible representation πC is fixed by the involutionof Proposition 1.10. If π is of complex type, then H can be made into a complexHilbert space Hc (whose complex dimension is half the real dimension of H) eithervia the action of π(A)′ or via the conjugate of this action, and we get two distinctirreducible representations of AC on Hc which are interchanged under the involu-

tion of Proposition 1.10 on AC. Finally, if π is of quaternionic type, then H canbe made into a quaternionic Hilbert space via the action of π(A)′. After tensoringwith C, we get a complex Hilbert space HC whose complex dimension is twice thequaternionic dimension of H, and we get an irreducible representation πC of AC onHC whose class is fixed by the involution of Proposition 1.10.

Now suppose further that A is separable and type I, so that π(A) contains thecompact operators on H, and in particular, there is an ideal m in A which mapsonto the trace-class operators. Thus π has a well-defined “character” χ on m in thesense of [29], and the representations πC of AC discussed above have characters χC

on mC. When π is of real type, χC restricted to m is just χ (and is real-valued).When π is of quaternionic type, χC restricted to m is χ

2 . When π is of complextype, the two complex irreducible extensions of π have characters on m which arenon-real and which are complex conjugates of each other, and which add up to χ.

Example 1.12. Before giving the proof, it might be instructive to give someexamples. First let A = C∗

R(Z), the free real C∗-algebra on one unitary u. The

trivial representation u �→ 1 is of real type and complexifies to the trivial repre-sentation of AC = C∗(Z). Similarly the sign representation u �→ −1 is of real

type. The representation u �→(

cosφ sinφ− sinφ cosφ

)on R2 (φ not a multiple of π)

is of complex type. Note that this representation is equivalent to the one given

by u �→(cosφ − sinφsinφ cosφ

)since these are conjugate under

(0 11 0

). This repre-

sentation class corresponds to a pair of inequivalent irreducible representations ofAC = C∗(Z) on C, one given by u �→ eiφ and one given by u �→ e−iφ. The involution

on AC sends one of these to the other.Next let A = RQ8, the group ring of the quaternion group of order 8. This

has a standard representation on H ∼= R4 (sending the generators i, j, k ∈ Q8 tothe quaternions with the same name) which is of quaternionic type. Complexifyinggives two copies of the unique 2-dimensional irreducible complex representation ofAC.

Note incidentally that D8, the dihedral group of order 8, and Q8 have thesame complex representation theory. Keeping track of the types of representationsenables us to distinguish the two groups.

Proof of Theorem 1.11. A lot of this is obvious, so we will just concentrateon the parts that are not. If π is of real type, its commutant is R and its complex-ification πC has commutant C and is thus irreducible. The class of πC is fixed bythe involution, since for a ∈ A,

θ∗(πC)(a) = τ ◦ πtC ◦ θ(a) = τ ◦ πt

C(a∗) = τ (πt(a∗)) = (π(a)t)t = π(a).


If π is of complex type, we need to show that we get two distinct irreducible repre-

sentations of AC which are interchanged under the involution on AC. If this werenot the case, then π viewed as an irreducible representation on a complex Hilbertspace Hc (via the identification of π(A)′ with C) would extend to an irreduciblecomplex representation (let’s call it πc) of AC which is isomorphic to θ∗π

c. Now ifa + ib ∈ AC, where a, b ∈ A, then σ(a+ ib) = a − ib and θ(a+ ib) = a∗ + ib∗. So

θ∗(πc)(a+ib) = τ ◦πt

C(a∗+ib∗) = π(a)+iπ(b), since for operators onHc, τ (T ∗) = T ,

the conjugate operator. The complexification of H is canonically identified withHc ⊕Hc, so the complexification πC of π is thus identified with πc ⊕ θ∗(π

c). If thiswere isomorphic to πc ⊕ πc, then its commutant would be isomorphic to M2(C).But the commutant of πC must be the complexification of the commutant of π, sothis is impossible. If π is of quaternionic type, its commutant is isomorphic to H,which complexifies toM2(C). That means the complexification of π has commutantM2(C), and thus the complexification of π is unitarily equivalent to a direct sum oftwo copies of an irreducible representation πC of AC. That the class of πC is fixedby the involution follows as in the real case.

Now let’s consider the part about characters. If π is of real type, πC is its com-plexification and so the characters of πC and of π coincide on m. (Complexificationof operators preserves traces.) If π is of quaternionic type, its complexification isequivalent to two copies of πC, so on m, the character χ of π is the character ofthe complexification of π and so coincides with twice the character χC of πC, whichis thus necessarily real-valued. Finally, suppose π is of complex type. If a ∈ m,then θ∗(πC)(a) = π(a), so we see in particular that the characters of πc and ofθ∗(π

c) (on m) are complex conjugates of one another, and add up to the characterof πC

∼= πc ⊕ θ∗(πc). But complexification of an operator doesn’t change its trace,

so πC and π have the same character on m, and the characters of πc and of θ∗(πc)

add up to χ on m. �

Remark 1.13. One can also phrase the results of Theorem 1.11 in a way morefamiliar from group representation theory. Let A be a real C∗-algebra and let π bean irreducible representation of AC on a complex Hilbert space HC such that theclass of π is fixed under the involution of Proposition 1.10. Then π is associatedto an irreducible representation of A of either real or quaternionic type. To tellwhich, observe that one of two possibilities holds. The first possibility is there is anA-invariant real structure on HC, i.e., HC is the complexification of a real Hilbertspace H which is invariant under A, in which case π is of real type. This condition isequivalent to saying that there is a conjugate-linear map ε : HC → HC commutingwith A and with ε2 = 1. (H is just the +1-eigenspace of ε.)

The second possibility is that there is a conjugate-linear map ε : HC

→ HC commuting with A and with ε2 = −1. In this case if we let i act onHC via the complex structure and let j act by ε, then since ε is conjugate-linear, iand j anticommute, and so we get an A-invariant structure of a quaternionic vectorspace on HC, whose dimension over H is half the complex dimension of HC. Inthis case, π clearly has quaternionic type. This point of view closely follows thepresentation in [10, II, §6].

The books [1] and [10] discuss the question of how one can tell the type (real,complex, or quaternionic) of an irreducible representation of a compact Lie group.In this case, one also has a criterion based on the value of the Frobenius-Schurindicator

∫χ(g2) dg, which is 1 for representations of real type, 0 for representations


of complex type, and −1 for representations of quaternionic type. But since thiscriterion is based on tensor products for representations for groups, it doesn’t seemto generalize to real C∗-algebras in general.

2. Real C∗-algebras of continuous trace

We return now to the structure theory of (separable, say) real C∗-algebras oftype I.

Definition 2.1. Let A be a real C∗-algebra with complexification AC. We sayA has continuous trace if AC has continuous trace in the sense of [13, §4.5], thatis, if elements a ∈ (AC)+ for which π �→ Trπ(a) is finite and continuous on AC aredense in (AC)+.

Theorem 2.2. Any non-zero postliminal real C∗-algebra (this is equivalent tobeing type I, even in the non-separable case — see [49, Ch. 6] or [61, §4.6]) has anon-zero ideal of continuous trace.

Proof. That AC has a non-zero ideal I of continuous trace is [13, Lemma4.4.4]. So we need to show that I can be chosen to be σ-invariant, or equivalently,

to show that I can be chosen invariant under the involution of Proposition 1.10.Simply observe that I+σ(I) is still a closed two-sided ideal and is clearly σ-invariant.Furthermore, it still has continuous trace since if a ∈ I+ and ta : π �→ Trπ(a) isfinite and continuous, then π �→ Trπ(σ(a)) = Trπ(θ(a)) = Tr θ∗(π)(a) = ta ◦ θ∗(π)is also finite and continuous, so that σ(a) is also a continuous-trace element. �

Corollary 2.3. Any non-zero postliminal real C∗-algebra has a compositionseries (possibly transfinite) with subquotients of continuous trace.

Proof. This follows by transfinite induction just as in the complex case. �

Because of Theorem 2.2 and Corollary 2.3, it is reasonable to focus specialattention on real C∗-algebras with continuous trace. To such an algebra A (whichwe will assume is separable to avoid certain pathologies, such as the possibilitythat the spectrum might not be paracompact) is associated a Real space (X, ι) in

the sense of Atiyah [5], that is, a locally compact Hausdorff space X = AC and aninvolution ι onX defined by Proposition 1.10. The problem then arises of classifyingall the real continuous-trace algebras associated to a fixed Real space (X, ι). Thereis always a unique such commutative real C∗-algebra, given by Theorem 1.9.

When one considers noncommutative algebras, ∗-isomorphism is too fine formost purposes, and the most natural equivalence relation turns out to be Moritaequivalence, which works for real C∗-algebras just as it does for complex C∗-alge-bras. Convenient references for the theory of Morita equivalence (in the complexcase) are [53,54]. A Morita equivalence between real C∗-algebras A and B is givenby an A-B bimodule X with A-valued and B-valued inner products, satisfying afew simple axioms:

(1) 〈x, y〉Az = x〈y, z〉B and 〈a · x, y〉B = 〈x, a∗ · y〉B, 〈x · b, y〉A = 〈x, y · b∗〉Afor x, y, z ∈ X and a ∈ A, b ∈ B.

(2) The images of the inner products are dense in A and in B.

(3) ‖〈x, x〉A‖1/2A = ‖〈x, x〉B‖1/2B is a norm on X, X is complete for this norm,and A and B act continuously on X by bounded operators.


The real continuous-trace algebras with spectrum (X, ι) have been completelyclassified by Moutuou [46] up to spectrum-fixing Morita equivalence, at least inthe separable case. (Actually Moutuou worked with graded C∗-algebras. See also[15, §3.3] for a translation into the ungraded case and the language we use here.)

First let us define the fundamental invariants.

Definition 2.4. Let A be a real continuous-trace algebra with spectrum (X, ι).

In other words X = AC, which is Hausdorff since A has continuous trace, and letι be the involution on X defined by Proposition 1.10. The sign choice of A is themap α : Xι → {+,−} attaching a + sign to fixed points of real type and a − signto fixed points of quaternionic type. (Of course, ι acts freely on X �Xι, and theorbits of this action correspond to the pairs of conjugate representations of complextype.)

Note that if we give {+,−} the discrete topology, then it is easy to see that αis continuous2, so it is constant on each connected component of Xι.

Incidentally, the name sign choice for this invariant comes from a physicalapplication we will see in Section 3, where it is related to the signs of O-planes instring theory.

The other invariant of a (separable) real continuous-trace algebra is theDixmier-Douady invariant. For a complex continuous-trace algebra with spectrum X, this isa class in H2(X, T ) (sheaf cohomology), where T is the sheaf of germs of continuousT-valued functions on X. We have a short exact sequence of sheaves

(2.1) 0 → Z → R → T → 1,

where R is the sheaf of germs of continuous real-valued functions, coming from theshort exact sequence of abelian groups

0 → Z → R → T → 1.

Since R is a fine sheaf and thus has no higher cohomology, the long exact sequencein sheaf cohomology coming from (2.1) gives H2(X, T ) ∼= H3(X,Z), and indeed,the Dixmier-Douady invariant is usually presented as a class in H3.

However, for purposes of dealing with real continuous-trace algebras, we needto take the involution ι on X (and on T ) into account. This will have the effect ofgiving a Dixmier-Douady invariant in an equivariant cohomology group H2

ι (X, T )defined by Moutuou [45], who denotes it HR2(X, T ) (with the ι understood). TheHR• groups are similar to, but not identical with, the Z/2-equivariant cohomologygroups H•(X;Z/2,F) as defined in Grothendieck’s famous paper [28, Ch. V]. Theprecise relationship in our situation is as follows:

Theorem 2.5. Let (X, ι) be a second-countable locally compact Real space, i.e.,space with an involution, and let π : X → Y be the quotient map to Y = X/ι. Thenif T is the sheaf of germs of T-valued continuous functions on X, equipped with theinvolution induced by the involution (x, z) �→ (ι(x), z) on X × T, then Moutuou’sHR2(X, T ) coincides with H2(Y, T ι), where T ι is the induced sheaf on Y , i.e.,the sheafification of the presheaf U �→ C(π−1(U),T)ι. By [28, (5.2.6)], there isan edge homomorphism H2

ι (X, T ) → H2(X;Z/2, T ) (which is not necessarily anisomorphism).

2One way to see this is to apply the part of Theorem 1.11 about characters. If e ∈ A+ isa local minimal projection near x ∈ X, then Trπ(e) = 1 if π is close to x and α(x) = + andTrπ(e) = 2 if π is close to x and α(x) = −.


Proof. In order to deal with quite general topological groupoids, Moutuou’sdefinition ofH•

ι (X,F) in [45] uses simplicial spaces and a Cech construction. But inour situation, X and Y are paracompact and the groupoid structure on X is trivial,so by the equivariant analogue of [68, Theorem 1.1] and the isomorphism betweenCech cohomology and sheaf cohomology for paracompact spaces [24, TheoremeII.5.10.1], it reduces here to ordinary sheaf cohomology. �

Grothendieck’s equivariant cohomology groups H•(X;Z/2,F) are the derivedfunctors of the equivariant section functor X �→ Γ(X,F)ι. Moutuou’s groups aregenerally smaller. A few examples will clarify the notion, and also explain thedifference between Grothendieck’s functor and Moutuou’s.

(1) If ι is trivial on X, the involution on T is just complex conjugation, andH•

ι (X, T ) can be identified with H•(X, T ι) = H•(X,Z/2). Note, for ex-ample, that if X is a single point, then Grothendieck’s H•(X;Z/2, T )would be the group cohomology H•(Z/2,T), which is Z/2 in every evendegree, whereas Moutuou’s H•

ι (pt, T ) is just H•(pt,Tι) = Z/2, concen-trated in degree 0.

(2) If ι acts freely, so that π : X → Y is a 2-to-1 covering map, H•ι (X, T ) can

be identified with Grothendieck’s

H•+1(X;Z/2,Z) ∼= H•+1(Y,Z)

for • > 0, via the equivariant version of the long exact sequence associatedto (2.1) and [28, Corollaire 3, p. 205]. Here Z is a locally constant sheafobtained by dividing X × Z by the involution sending (x, n) to (ιx,−n).

Definition 2.6. Now we can explain the definition of the real Dixmier-Douadyinvariant of a separable real C∗-algebra A. Without loss of generality, we can ten-sor A with KR, which doesn’t change the algebra up to spectrum-fixing Moritaequivalence. Then AC becomes stable, and is locally, but not necessarily globally,isomorphic to C0(X,K). By paracompactness (here we use separability of A, whichimplies X is second countable and thus paracompact), there is a locally finite cover-ing {Uj} of X such that AC is trivial over each {Uj}. We can also assume each Uj is

ι-stable. The trivializations of AC over the Uj give a Cech cocycle in H1({Uj},PU),given by the “patching data” over overlaps Uj ∩Uk. Here PU is the sheaf of germsof PU -valued continuous functions, since PU is the automorphism group of K. Theimage of this class in H1(X,PU) ∼= H2(X, T ) is the complex Dixmier-Douady in-variant. Here we use the long exact cohomology sequence associated to the sequenceof sheaves

(2.2) 1 → T → U → PU → 1,

where again the middle sheaf is fine since the infinite unitary group (with the strongor weak operator topology) is contractible.

In our situation, there is a little more structure because AC was obtained bycomplexifying A. So we have the conjugation σ on AC, which induces the involutionι on X and on the sheaves U , PU , and T over X. Furthermore, the cocycle ofthe patching data must be ι-equivariant, and so defines the real Dixmier-Douadyinvariant, which is its coboundary in H2

ι (X, T ).

Theorem 2.7 (Moutuou [46]). The spectrum-fixing Morita equivalence classesof real continuous-trace algebras over (X, ι) form a group (where the group operation


comes from tensor product over X) which is isomorphic to H0(Xι,Z/2)⊕H2ι (X, T )

via the map sending an algebra A to the pair consisting of its sign choice and realDixmier-Douady invariant (in the sense of Definitions 2.4 and 2.6).

Remark 2.8. The formulation of this theorem in [46] looks rather different,for a number of reasons, though it is actually more general. For an explanation ofhow to translate it into this form, see [15, §3.3].

Example 2.9. Here are three examples, that might be relevant for physicalapplications, that show how one computes the Brauer group of Theorem 2.7 inpractice. In all cases we will take X to be a K3-surface (a smooth simply con-nected complex projective algebraic surface with trivial canonical bundle) and theinvolution ι to be holomorphic (algebraic).

(1) Suppose the involution ι is holomorphic and free. In this case the quotientY = X/ι is an Enriques surface (with fundamental group Z/2) and ιreverses the sign of a global holomorphic volume form. (See for example[47, §1].) There is no sign choice invariant since the involution is free, andthus all representations must be of complex type. The Dixmier-Douadyinvariant lives in (twisted) 3-cohomology of the quotient space Y . ByPoincare duality, H3(Y,Z) ∼= H1(Y,Z), but since X is 1-connected, theclassifying map Y → BZ/2 = RP∞ is a 2-equivalence (an isomorphismon π1 and surjection on π2) and induces an isomorphism on twisted H1.So H3(Y,Z) ∼= H1(Y,Z) ∼= H1(BZ/2,Z) ∼= Hgroup

1 (Z/2,Z) = 0. So theDixmier-Douady invariant is always trivial in this case.

(2) If ι is a so-called Nikulin involution (see [44,69]), then Xι consists of 8isolated fixed points. Let Z = (X �Xι)/ι. By transversality, the comple-ment X �Xι of the fixed-point set is still simply connected, so π1(Z) ∼=Z/2 and the map Z → BZ/2 is a 2-equivalence. We have H2

ι,c(X �

Xι, T ) ∼= H3c (Z,Z), and by Poincare duality, H3

c (Z,Z)∼= H1(Z,Z) ∼=

H1(BZ/2,Z) = 0. From the long exact sequence

(2.3) H1(Xι,Z/2) = 0 → H2ι,c(X �Xι, T )

→ H2ι (X, T ) → H2(Xι, T ) = 0,

(see [24, Theoreme II.4.10.1]) we see that H2ι (X, T ) = 0 and the Dixmier-

Douady invariant is always trivial in this case. However, there are manypossibilities for the sign choice since H0(Xι,Z/2) ∼= (Z/2)8.

(3) It is well known that there are K3-surfaces X with a holomorphic mapf : X → CP2 that is a two-to-one covering branched over a curve C ⊂CP2 of degree 6 and genus 10. Such a surface X admits a holomorphicinvolution ι having C as fixed-point set. We want to compute the Brauergroup of real continuous-trace algebras over (X, ι). Since Xι = C isconnected, there are only two possible sign choices, and algebras with signchoice − are obtained from those with sign choice + simply by tensoringwith H. So we may assume the sign choice on the fixed set is a +. Thecalculation of the possible Dixmier-Douady invariants is complicated anduses Theorem 2.5.

Theorem 2.10. Let X be a K3-surface and ι a holomorphic involution on Xwith fixed set Xι = C a smooth projective complex curve of genus 10 and withquotient space Y = X/ι = CP2. Then H2

ι (X, T ) = 0.


Proof. By Theorem 2.5, H2ι (X, T ) ∼= H2(CP2,F), where the sheaf F is Z/2

over C and is locally isomorphic to T over the complement. By [24, TheoremeII.4.10.1], we obtain an exact sequence

(2.4) H1(C,Z/2) → H2ι,c(X � C, T ) → H2

ι (X, T ) → H2(C,Z/2)

→ H3ι,c(X � C, T ) → H3

ι (X, T ) → H3(C,Z/2) = 0.

Note that (X �Xι)/ι ∼= CP2 � C. Thus in (2.4), Hjι,c(X � C, T ) ∼= Hj+1

c (CP2 �

C,Z) ∼= H3−j(CP2 � C,Z). Since C ⊂ CP2 is a curve of degree 6, the map

H2(CP2,Z) → H2(C,Z) induced by the inclusion is multiplication by 6, and wefind from the long exact sequence

H2(CP2,Z)6−→ H2(C,Z) → H3

c (CP2 � C,Z) → 0

that H3c (CP

2 � C,Z) ∼= H1(CP2 � C,Z) ∼= Z/6. This implies that for j ≤ 1,

Hj(CP2�C,Z) will coincide with the Z-homology of a lens space with fundamental

group Z/6, or with Hgroupj (Z/6,Z) = Z/2, j even, and 0, j odd. Hence (2.4) reduces

to

0 → H2ι (X, T ) → Z/2

δ−→ Z/2 → H3ι (X, T ) → 0,

and H2ι (X, T ) is either 0 or Z/2, depending on whether the connecting map δ is

nontrivial or not.To complete the calculation, we use Theorem 2.5. This identifies H2

ι (X, T )

with I2,02 in the spectral sequence

Ip,q2 = Hp(Y,Hq(Z/2, T )) ⇒ Hp+q(X;Z/2, T )

of [28, Theoreme 5.2.1]. We will examine this spectral sequence as well as the otherone in that theorem,

IIp,q2 = Hp(Z/2, Hq(X, T )) ⇒ Hp+q(X;Z/2, T ).

First consider I•,•2 . We have a short exact sequence of sheaves

(2.5) 1 → (T )X�C → T → (T )C → 1,

and ι acts trivially on C and freely on X�C. Thus Hq(Z/2, (T )X�C) = 0 for q > 0[28, Corollaire 3, p. 205]. So from the long exact cohomology sequence derivedfrom (2.5), Hq(Z/2, T ) = Hq(Z/2, (T )C) is supported on C for q > 0. On C, theaction of ι is by complex conjugation, and so one easily sees that Hq(Z/2, T ) =Hq(Z/2, (T )C) = (Z/2)C for q > 0 even, 0 for q odd. So for q > 0, Ip,q2 vanishes forq odd and is Hp(C,Z/2) for q even, which is Z/2 for p = 0 or 2, (Z/2)20 for p = 1,

and 0 for p > 2. In particular, I0,12 = 0, so d2 : I0,12 → I2,02 vanishes and so the edge

homomorphism H2ι (X, T ) → H2(X;Z/2, T ) is injective. Furthermore, I1,12 = 0 and

I0,22 = Z/2, so H2(X;Z/2, T ) is finite and∣∣H2(X;Z/2, T )∣∣ ≤ 2 ·

∣∣H2ι (X, T )

∣∣ .Equality will hold if and only if the map d3 : I0,23 = Z/2 → I3,03 = H3

ι (X, T ) istrivial.

Now consider the other spectral sequence II•,•2 . We have H0(X, T ) = C(X,T),which since X is simply connected fits into an exact sequence

(2.6) 0 → Z → C(X,R) → C(X,T) → 1.

Now in the exact sequence (2.1), the action of Z/2 is via a combination of theinvolution ι on X and complex conjugation, which corresponds to multiplication


by −1 on R and Z. Thus as a Z/2 module, the group on the left in (2.6) is reallyZ, Z with the non-trivial action. On the q = 0 row, we use equation (2.6) and thefact that higher cohomology of a finite group with coefficients in a real vector spacehas to vanish to obtain that

IIp,02 = Hp(Z/2, C(X,T)) ∼= Hp+1(Z/2,Z) ∼={0, p odd,

Z/2, p even > 0.

For q > 0, we know that Hq(X, T ) ∼= Hq+1(X,Z), which will be nonzero (andtorsion-free) only for q = 1 and q = 3. Again, since the action on Z/2 on the con-stant sheaf Z in (2.1) is by multiplication by −1, the action of Z/2 on H3(X, T ) ∼=H4(X,Z) ∼= Z is by multiplication by −1. The case of H1(X, T ) ∼= H2(X,Z) ∼= Z22

is more complicated because we also have the action of ι on H2(X,Z), which hasfixed set of rank 1 [47, p. 595]. So we need to determine the structure of H2(X,Z)as a Z/2-module. The action of ι on H2 has to respect the intersection pairing, withrespect to which H2(X,Z) splits (non-canonically) as E8⊕E8⊕H⊕H⊕H, where

E8 is the E8 lattice and H is a hyperbolic plane (Z2 with form given by

(0 11 0

)).

Since H2(X,Z)ι ∼= Z, one can quickly see that the only possibility is that ι actsby −1 on both E8 summands and on two of the H summands, and interchangesthe generators of the other H summand. Our action here of Z/2 is reversed fromthis, so as Z/2-module, H1(X, T ) ∼= Z20 ⊕ H. Since Hp(Z/2,Z) is non-zero onlyfor p even and Hp(Z/2, H) = 0 for p > 0 (by simple direct calculation, or elseby Shapiro’s Lemma, since H as a Z/2-module is induced from Z as a module for

the trivial group), we find that II1,12 = H1(Z/2,Z20 ⊕ H) = 0. Since we already

computed that II0,22 = 0 and II2,02 = Z/2, we see that |H2(X;Z/2, T )| ≤ 2. It will

be 0 only if d2 : II0,12 → II2,02 is non-zero. Putting everything together, we finallysee that the only possibilities for the two spectral sequences are as in Figures 1 and2. Comparing the (dotted) diagonal lines of total degrees 2 and 3 in the two figures,we conclude that H2

ι (X, T ) and H3ι (X, T ) must both vanish. �

3. K-Theory and Applications

In this last section, we will briefly discuss the (topological) K-theory of real C∗-algebras, and explain some key applications to manifolds of positive scalar curvatureand to orientifold string theories in physics. We should mention that other physicalapplications have appeared in the theory of topological insulators in condensedmatter theory [30,36], though we will not go into this area here. Along the way,connections will show up with representation theory via the real Baum-Connesconjecture.

The (topological) K-theory of real C∗-algebras is of course a special case oftopological K-theory of real Banach algebras. As such it has all the usual proper-ties, such as homotopy invariance and Bott periodicity of period 8. A convenientreference is [63].

A nice feature of the K-theory of real continuous-trace C∗-algebras is that itunifies all the variants of topological K-theory (for spaces) that have appeared inthe literature. This includes of course realK-theoryKO, complexK-theory K, andsymplectic K-theory KSp, but also Atiyah’s “Real” K-theory KR [5], Dupont’ssymplectic analogue of KR [16], sometimes called KH, and the self-conjugate


q

3

��

0 0 0 0

2 Z/2 (Z/2)20 Z/2 0

1 0 0 0 0

0 ? Z20 H2ι (X, T ) H3

ι (X, T )

· 0 1 2 3 �� p

Figure 1. The first Grothendieck spectral sequence I•,•2

q

3

��

0 Z/2 0 Z/2

2 0 0 0 0

1 Z20 0 (Z/2)20 0

0 ? 0 Z/2 0

· 0 1 2 3 �� p

Figure 2. The second Grothendieck spectral sequence II•,•2

K-theory KSC of Anderson and Green [3,27]. KR•(X, τ ) is the topological K-theory of the commutative real C∗-algebra C0(X, τ ) of Theorem 1.9. KSC•(X) isKR•(X×S1), where S1 is given the (free) antipodal involution [5, Proposition 3.5].In addition, the K-theory of a stable real continuous-trace with a sign choice (butvanishing Dixmier-Douady invariant) is “KR-theory with a sign choice” as definedin [14], and the K-theory of a stable real continuous-trace with no sign choice but


a nontrivial Dixmier-Douady invariant is what has generally been called “twistedK-theory” (of either real or complex type, depending on the types of the irreduciblerepresentations of the algebra). See [55,57] for some of the original treatments, aswell as [6,34] for more modern approaches.

3.1. Positive scalar curvature. A first area where real C∗-algebras andtheir K-theory plays a significant role is the classification of manifolds of positivescalar curvature. The first occurrence of real C∗-algebras in this area is implicit inan observation of Hitchin [31], that if M is a compact Riemannian spin manifoldof dimension n with positive scalar curvature, then the KOn-valued index of theDirac operator on M has to vanish. For n divisible by 4, this observation was notnew and goes back to Lichnerowicz [40], but for n ≡ 1, 2 mod 2, a new torsionobstruction shows up that cannot be “seen” without real K-theory.

The present author observed that there is a much more extensive obstructiontheory when M is not simply connected. Take the fundamental group π of M , acountable discrete group. Complete the real group ring Rπ in its greatest C∗-normto get the real group C∗-algebra A = C∗

R(π). (Alternatively, one could use the

reduced real group C∗-algebra Ar = C∗R,r(π), the completion of the group ring for

its left action on L2(π). For present purposes it doesn’t much matter.) Coupling

the Dirac operator on M to the universal flat C∗R(π)-bundle M ×π A over M , one

gets a Dirac index with values in KOn(A), which must vanish if M has positivescalar curvature. Thus we have a new source of obstructions to positive scalarcurvature.

As shown in [56,58], this KOn(A)-valued index obstruction can be computedto be μ ◦ f∗(αM ), where αM ∈ KOn(M) is the “Atiyah orientation” of M , i.e., theKO-fundamental class defined by the spin structure, f : M → Bπ is a classifying

map for the universal cover M → M , and μ : KOn(Bπ) → KOn(A) is the “realassembly map,” closely related to the Baum-Connes assembly map in [7].3

In [58, Theorem 2.5], I showed that for π finite, the image of the reducedassembly map μ (what one gets after pulling out the contribution from the trivialgroup, i.e., the Lichnerowicz and Hitchin obstructions) is precisely the image inKO•(Rπ) of the 2-torsion in KO•(Rπ2), π2 ⊆ π a Sylow 2-subgroup. This livesin degrees 1 and 2 mod 4 and comes from the irreducible representations of π2 ofreal and quaternionic type, in the sense that we explained in Theorem 1.11. So farthe obstructions detected by μ are the only known obstructions to positive scalarcurvature on closed spin manifolds of dimension > 4 with finite fundamental group.

The problem of existence or non-existence of positive scalar curvature on a spinmanifold can be split into two parts, one “stable” and one “nonstable.” Stabilityhere refers to taking the product with enough copies of a “Bott manifold” Bt8, asimply connected closed Ricci-flat 8-manifold representing the generator of Bottperiodicity. Since index obstructions in K-theory of real C∗-algebras live in groupswhich are periodic mod 8, stabilizing the problem by crossing with copies of Bt8

compensates for this by introducing 8-periodicity on the geometric side. Indeed itwas shown in [60] that the stable conjecture (M ×Btk admits a metric of positive

3The relationship is this. Let Eπ denote the universal proper π-space and let Eπ denote theuniversal free π-space. These coincide if and only if π is torsion-free. The Baum-Connes assemblymap is defined on KOπ

• (Eπ) whereas our map is defined on KOπ• (Eπ) = KOn(Bπ). Since Eπ is

a proper π-space, there is a canonical π-map Eπ → Eπ (unique up to equivariant homotopy) andso our μ factors through the Baum-Connes assembly map, and agrees with it if π has no torsion.


scalar curvature for sufficiently large k if and only if μ◦f∗(αM ) vanishes) holds whenπ is finite. Stolz has extended this theorem as follows: for a completely generalclosed spin manifold Mn with fundamental group π, M × Btk admits a metric ofpositive scalar curvature for sufficiently large k if and only if μ ◦ f∗(αM ) vanishes,provided that the real Baum-Connes conjecture (bijectivity of the Baum-Connesassembly map μ : KOπ

• (Eπ) → KO•(C∗R,r)) holds for π. In fact, Baum-Connes can

be weakened here in two ways — one only needs injectivity of μ, not surjectivity,and one can replace C∗

R,r by the full C∗-algebra C∗R. Since the full C∗-algebra

surjects onto the reduced C∗-algebra, injectivity of the assembly map for the fullC∗-algebra is a weaker condition. Sketches of Stolz’s theorem may be found in[65,66], though unfortunately the full proof of this was never written up.

Since the real version of the Baum-Connes conjecture has just been seen toplay a fundamental role here, it is worth remarking that the real and complexversions of the Baum-Connes conjecture are actually equivalent [8,62]. Thus thereal Baum-Connes conjecture holds in the huge number of cases where the complexBaum-Connes conjecture has been verified.

3.2. Representation theory. Since we have already mentioned the realBaum-Connes conjecture, it is worth mentioning that this, as well as the generaltheory of real C∗-algebras, has some relevance to representation theory. SupposeG is a locally compact group (separable, say, but not necessarily discrete). Thereal group C∗-algebra C∗

R(G) is the completion of the real L1-algebra (the convo-

lution algebra of real-valued L1 functions on G) for the maximal C∗-algebra norm.Obviously this defines a canonical real structure on the complex group C∗-algebraC∗(G), and similarly we have C∗

R,r(G) inside the reduced C∗-algebra C∗r (G). Com-

puting the structure of C∗R(G) or of C∗

R,r(G) gives us more information than justcomputing the structure of their complexifications. For instance, it gives us thetype classification of the representations, as we saw in Theorem 1.11 and Example1.12. The real Baum-Connes conjecture, when it’s known to hold, gives us at leastpartial information on the structure of C∗

R,r(G) (its K-theory).

Here are some simple examples (where the real structure is not totally uninter-esting) to illustrate these ideas.

Example 3.1. Let G = SU(2). As is well known, this has (up to equiva-lence) irreducible complex representation of each positive integer dimension. Itis customary to parameterize the representations Vk by the value of the “spin”k = 0, 12 , 1,

32 , · · · (this is the highest weight divided by the unique positive root),

so that dimVk = 2k + 1. The character χk of Vk is given on a maximal torus by

eiθ �→ sin(2k+1)θsin θ , which is real-valued, and thus all the representations must have

real or quaternionic type. In fact, Vk is of real type if k is an integer and is ofquaternionic type if k is a half-integer. (That’s because V1 is the complexificationof the covering map SU(2) → SO(3), while V1/2 acts on the unit quaternions, andall the other representations can be obtained from these by taking tensor productsand decomposing. A tensor product of real representations is real, and a tensorproduct of a real representation with a quaternionic one is quaternionic.) Indeedif one computes the Frobenius-Schur indicator

∫Gχk(g

2) dg for Vk using the Weyl


integration formula, one gets∫G

χk(g2) dg =

1

π

∫ 2π

0

(e4kiθ + e4(k−1)iθ + · · ·+ e−4kiθ

)sin2 θ dθ

=1

4π

∫ 2π

0

(e4kiθ + e4(k−1)iθ + · · ·+ e−4kiθ

)(2− e2iθ − e−2iθ

)dθ.

If k is an integer, we get

1

4π

∫ 2π

0

((· · ·+ e4iθ) + 1 + (e−4iθ + · · · )

)(2− e2iθ − e−2iθ

)dθ = 1,

where the terms in small parentheses are missing if k = 0, while if k is a half-integer,we get

1

4π

∫ 2π

0

(· · ·+ e2iθ + e−2iθ + · · ·

)(2− e2iθ − e−2iθ

)dθ = −1.

This confirms the type classification we gave earlier.Thus C∗

R(G) ∼=

⊕k∈N

M2k+1(R) ⊕⊕

k= 12 ,

32 ,···

Mk+ 12(H). In particular, we see

that KO•(C∗R(G)) ∼= (KO•)

∞⊕ (KSp•)∞, and in degrees 1 and 2 mod 4, this is an

infinite direct sum of copies of Z/2, whereas the torsion-free contributions appearonly in degrees divisible by 4. Conversely, if one had some independent methodof computing KO•(C

∗R(G)), it would immediately tell us that G has no irreducible

representations of complex type, and infinitely many representations of both realand of quaternionic type.

Example 3.2. Let H be the compact group T∪j T, where j is an element withj2 = −1, jzj−1 = z for z ∈ T. This is a nonsplit extension of Gal(C/R) ∼= Z/2by T and is secretly the maximal compact subgroup of WR, the Weil group of the

reals (which splits as R×+ ×H). The induced action of j on T = Z sends n �→ −n.

So the Mackey machine tells us that the irreducible complex representations of Hare the following:

(1) two one-dimensional representations χ±0 which are trivial on T and send

j �→ ±1. These representations are obviously of real type.(2) a family πn = IndHT σn, n ∈ Z � {0} of two-dimensional representations,

where σn(z) = zn, z ∈ T. These representations are all of quaternionictype since they come from complexifying the representation H → H×

given by z �→ zn, j �→ j.

We immediately conclude that C∗R(H) ∼= R ⊕ R ⊕ (H)∞. Thus

KO•(C∗R(H)) is elementary abelian of rank 2 in degrees 1 and 2 mod 8, and is

(Z/2)∞ in degrees 5 and 6 mod 8. Again, if we had an independent way to com-pute KO•(C

∗R(H)), it would tell us about the types of the representations.

Example 3.3. A slightly more interesting example is G = SL(2,C), a sim-ple complex Lie group with K = SU(2) as maximal compact subgroup. Thereduced dual of G is Hausdorff, and the complex reduced C∗-algebra C∗

r (G) isa stable continuous-trace algebra with trivial Dixmier-Douady invariant, i.e., it is

Morita equivalent to C0(Gr). All the irreducible complex representations of C∗r (G)

are principal series representations, and all unitary principal series are irreducible.

Thus we see that C∗r (G) is Morita equivalent to C0(M/W ), where M is a Cartan

subgroup, which we can take to be C×, and W = {±1} is the Weyl group, which


acts on C× ∼= Z × R by −1 (on both factors). So C∗r (G) is Morita equivalent to

C0([0,∞)) ⊕⊕

n≥1 C0(R), with [0,∞) = ({0} × R)/W . (For all of this one can

see [18, V] or [50], for example.) The complex Baum-Connes map gives an iso-

morphism KG• (G/K) ∼= R(K)⊗K•−3

μ−→ K•(C∗r (G)), sending the generator of the

representation ring R(K) associated to Vk (in the notation of Example 3.1) to thegenerator of the Z summand in K0(C

∗r (G)) associated to the principal series with

discrete parameter ±(2k + 1). There is no contribution to K0(C∗r (G)) from the

spherical principal series (corresponding to the fixed point n = 0 of W on Z) sinceR/{±1} ∼= [0,∞) is properly contractible.

Now let’s analyze the real structure. We can start by looking at K-types. Thegroup SL(2,C) is the double cover of the Lorentz group SO(3, 1)0. Representationsthat descend to SO(3, 1)0 must have K types that factor through SU(2) → SO(3),and so have integral spin. All integral spin representations have real type, so theserepresentations are also of real type, at least when restricted to K. The genuinerepresentations of SL(2,C) that do not descend to SO(3, 1)0 must have K-typeswith half-integral spin, and these representations are of quaternionic type, at leastwhen restricted to K. There is one principal series which is obviously of real type,namely the “0-point” of the spherical principal series, since this representation issimply IndGB 1, where B is a Borel subgroup. Since the trivial representation ofB is of real type, we get a real form for the complex induced representation byusing induction with real Hilbert spaces instead. And thus we get an irreduciblereal representation on L2

R(G/B) ∼= L2

R(K/T). In fact the other spherical principal

series can be realized on this same Hilbert space (see [18, p. 261]) so they, too,are of real type. But this method won’t work for other characters of B sincenone of the other one-dimensional unitary characters of C× are of real type. Ifwe look at a principal series representation of G with discrete parameter ±n ∈ Z,its restriction to K can be identified with IndKT χn, which by Frobenius reciprocitycontains Vk with multiplicity equal to the multiplicity of χn in Vk. This is 0 if 2kand n have opposite parity or if |n| > 2k, and is 1 otherwise. So this principal seriesrepresentation has all its K-types of multiplicity 1 and has real (resp., quaternionic)type when restricted to K provided n is even (odd).

We can analyze things in more detail by seeing what the involution (of Propo-

sition 1.10) on G does to the principal series. Clearly it sends IndGB χ to IndGB χ, ifχ is a one-dimensional representation of M viewed as a representation of B. Butsince W = {±1}, χ = w · χ, for w the generator of W , and we get an equivalent

representation. Thus the involution on Gr is trivial. One can also check this veryeasily by observing that all the characters of G are real-valued. (See for example[70, Theorem 5.5.3.1], where again the key fact for us is that w ·χ = χ.) J. Adamshas studied this property in much greater generality and proved:

Theorem 3.4 (Adams [2, Theorem 1.8]). If G is a connected reductive algebraicgroup over R with maximal compact subgroup K, if −1 lies in the Weyl groupof the complexification of G, and if every irreducible representation of K is ofreal or quaternionic type, then every unitary representation of G is also of real orquaternionic type.

Thus we know that the involution on Gr is trivial and that C∗r (G) is a real

C∗-algebra of continuous trace, with spectrum a countable union of contractiblecomponents, all but one of which are homeomorphic to R, with the exceptional


component homeomorphic to [0,∞). The real Dixmier-Douady invariants must all

vanish since H2(Gr,Z/2) = 0. The sign choice invariants are now determined bythe K-types, since all the K-types have multiplicity one and thus a representationof G of real (resp., quaternionic) type must have all its K-types of the same type.Putting everything together, we see that we have proved the following theorem:

Theorem 3.5. The reduced real C∗-algebra of SL(2,C) is a stable real contin-uous-trace algebra, Morita equivalent to

CR

0 ([0,∞))⊕⊕

n>0 even

CR

0 (R)⊕⊕

n>0 odd

CH

0 (R).

Schick’s proof in [62] that the complex Baum-Connes conjecture implies thereal Baum-Connes conjecture is stated only for discrete groups, but it goes overwithout difficulty to general locally compact groups, at least in the case of triv-ial coefficients. Since the complex Baum-Connes conjecture (without coefficients)is known for all connective reductive Lie groups [38, 71], the real Baum-Connes

map is an isomorphism KOG• (G/K)

μ−→ KO•(C∗R,r(G)). This by itself gives some

information on the real structure of the various summands in C∗R,r(G). Since

G/K has a G-invariant spin structure in this case, by the results of [35, §5],KOG

• (G/K) ∼= KOK• (G/K) ∼= KOK

• (p), which by equivariant Bott periodicityis KO•−3(C

∗R(K)), which we computed in Example 3.1. (Here p is the orthogonal

complement to the Lie algebra of K inside the Lie algebra of G. In this case, p isisomorphic as a K-module to the adjoint representation of K.) On the other sideof the isomorphism, we have KO•(C

R0 (R))

∼= KO−•(R) ∼= KO−•−1(pt) ∼= KO•+1,and similarly KO•(C

H0 (R))

∼= KSp−•(R) ∼= KO−•−5(pt) ∼= KO•+5. So the torsion-free summands in KO•(C

∗R,r(G)) are all in degrees 3 mod 4. Since there are no

torsion-free summands in KO•(C∗R,r(G)) in degrees 1 mod 4, we immediately con-

clude that no unitary principal series (except perhaps for the spherical principalseries, which don’t contribute to the K-theory) are of complex type, and that thereare infinitely many lines of principal series of both real and quaternionic type. Thisis a large part of Theorem 3.5, and is not totally trivial to check directly.

One interesting feature of the Baum-Connes isomorphism is the degree shift.Since KOG

• (G/K) ∼= KO•−3(C∗R(K)), while KO•(C

∗R,r(G)) is a sum of copies of

KO•+1, associated to principal series with K-types of integral spin and KO•+5,associated to principal series with K-types of half-integral spin, we see that repre-sentations of K of integral spin on the left match with those of half-integral spin onthe right, and those of half-integral spin on the left match with those of integral spinon the right. This is due to the “ρ-shift” in Dirac induction. If we were to replaceSL(2,C) by the adjoint group G = PSL(2,C), the maximal compact subgroupwould become K = SO(3), with K-types only of integral spin, but on the left,since G/K would no longer have a G-invariant spin structure, KO•(G/K) wouldbe given by genuine representations of the double cover of K, i.e., representationsonly of half-integral spin.

Example 3.6. The techniques we used in Example 3.3 and Theorem 3.5 canalso be used to compute the reduced real C∗-algebras of arbitrary connected com-plex reductive Lie groups. A useful starting point is [50, Proposition 4.1], which

states that for such a group G, Gr is Hausdorff and C∗r (G) ∼= C0(Gr) ⊗ K is a

stable continuous-trace algebra with trivial Dixmier-Douady class. The cases of


SO(4n,C), SO(2n+1,C), and Sp(n,C) are particularly easy (and interesting). By[1, Theorem 7.7] or [10, VI.(5.4)(vi)], all representations of SO(2n+ 1) are of realtype, and by [1, Theorem 7.9] or [10, VI.(5.5)(ix)], all representations of SO(4n)are of real type. Thus we obtain

Theorem 3.7. Let G = SO(2n+ 1,C) or SO(4n,C). Then

C∗r,R(G) ∼= CR

0 (Gr)⊗KR

is a stable real continuous-trace algebra with trivial Dixmier-Douady class.

Proof. By Theorem 3.4, the involution on Gr is trivial, and the sign choiceinvariant has to be constant on each connected component. Let M be a Cartansubgroup of G, let W be its Weyl group, let B = MN be a Borel subgroup, letK be a maximal compact subgroup (an orthogonal group SO(2n+ 1) or SO(4n)),and let T = K ∩M , a maximal torus in K. The irreducible representations in thereduced dual are all principal series IndGB χ, where χ is a character of M extendedto a character of B by taking it to be trivial on N . When restricted to K, this

is the same as IndKT χ|T . If ρ ∈ K is a K-type, then by Frobenius reciprocity, ρappears in this induced representation with multiplicity equal to the multiplicityof χ|T in ρ|T . So given χ, take ρ to have highest weight in the W -orbit of χ|T ,and we see that ρ occurs in IndGB χ with multiplicity 1. Since ρ is real and IndGB χis of either real or quaternionic type, we see that its being of real type is the onlypossibility. (Otherwise the invariant skew-symmetric form on the representationwould restrict to a skew-symmetric invariant form on ρ.) Thus C∗

r,R(G) is a stablereal continuous-trace algebra with all irreducible representations of real type. TheDixmier-Douady invariant has to vanish since as pointed out in [50, p. 277], all

connected components of Gr are contractible. �

In a similar fashion we have

Theorem 3.8. Let G = Sp(n,C), let M be a Cartan subgroup, and let W beits Weyl group. Let K = Sp(n), a maximal compact subgroup, and let T = M ∩K,a maximal torus in K Then C∗

r,R(G) is a stable real continuous-trace algebra which

is Morita equivalent to a direct sum of pieces of the form CR0 (Y ) and CH

0 (Y ). Here

Y ranges over the components of M/W . Infinitely many pieces of each type (real

or quaternionic) occur. If χ ∈ M and χ|T is its “discrete parameter”, then theassociated summand in C∗

r,R(G) is of real type if and only if the representation ofK with highest weight in the W -orbit of χ is of real type.

Proof. This is exactly the same as the proof of Theorem 3.7, the only dif-ference being that “half” of the representations of K are of quaternionic type (see[1, Theorem 7.6] and [10, VI.(5.3)(vi)]). �

3.3. Orientifold string theories. A last area where real C∗-algebras andtheir K-theory seem to play a significant role is in the study of orientifold stringtheories in physics. (See for example [51, §8.8] and [52, Ch. 13].) Such a theory isbased on a spacetime manifold X equipped with an involution ι, and the theory isbased on a “sigma model” where the fundamental “strings” are equivariant mapsfrom a “string worldsheet” Σ (an oriented Riemann surface) to X, equivariant withrespect to the “worldsheet parity operator” Ω, an orientation-reversing involutionon Σ, and ι, the involution on X. Restricting attention only to equivariant strings is


basically what physicists often call GSO projection, after Gliozzi-Scherk-Olive [23],and introduces enough flexibility in the theory to get rid of lots of unwanted states.In order to preserve a reasonable amount of supersymmetry, usually one assumesthat the spacetime manifold (except for a flat Minkowski space factor, which wecan ignore here) is chosen to be a Calabi-Yau manifold, that is, a complex Kahlermanifold X with vanishing first Chern class, and that the involution of X is eitherholomorphic or anti-holomorphic. If we choose X to be compact, then in lowdimensions there are very few possibilities — if X has complex dimension 1, thenit is an elliptic curve, and if X has complex dimension 2, then it is either a complextorus or a K3 surface. In the papers [14, 15], the case of an elliptic curve wastreated in great detail. Example 2.9 and Theorem 2.10 were motivated by the caseof K3 surfaces, which should also be of great physical interest.

Now we need to explain the connection with real C∗-algebras and their K-theory. An orientifold string theory comes with two kinds of important subman-ifolds of the spacetime manifold X: D-branes, which are submanifolds on which“open” strings — really, compact strings with boundary — can begin or end (wherewe specify boundary conditions of Dirichlet or Neumann type), and O-planes, whichare the connected components of the fixed set of the involution ι. There are chargesattached to these two kinds of submanifolds. D-branes have charges in K-theory[43,72], where the kind of K-theory to be used depends on the specific details of thestring theory, and should be a variant of KR-theory for orientifold theories. TheO-planes have ± signs which determine whether the Chan-Paton bundles restrictedto them have real or symplectic type. These sign choices result in “twisting” of theKR-theory, such as appeared above in Definition 2.4. In addition, there is a furthertwisting of the KR-theory due to the “B-field” which appears in the Wess-Zuminoterm in the string action. It would be too complicated to explain the physics in-volved, but mathematically, the B-field gives rise to a class in Moutuou’s H2

ι (X, T ).But in short, the effect of the O-plane charges and the B-field is to make the D-branecharges live in twisted KR-theory, i.e., in the K-theory of a real continuous-tracealgebra determined by the O-plane charges and the B-field. In this way, (type II)orientifold string theories naturally lead to K-theory of real continuous-trace alge-bras, which is most interesting from the point of view of physics when (X, ι) is aCalabi-Yau manifold with a holomorphic or anti-holomorphic involution [14,15].

An important aspect of string theories is the existence of dualities betweenone theory and another. These are cases where two seemingly different theoriespredict the same observable physics, or in other words, are equivalent descriptionsof the same physical system. The most important examples of such dualities areT-duality, or target-space duality, where the target space X of the model is changedby replacing tori by their duals, and the very closely related mirror symmetry ofCalabi-Yau manifolds. These dualities do not have to preserve the type of the theory(IIA or IIB) — in fact, in the case of T-duality in a single circle, the type is reversed— and they frequently change the geometry or topology of the spacetime and/or thetwisting (sign choice and/or Dixmier-Douady class). The possible theories with Xan elliptic curve and ι holomorphic or anti-holomorphic were studied in [14,15,19],and found to be grouped into 3 classes, each containing 3 or 4 different theories. Allof the theories in a single group are related to one another by dualities, and theoriesin two different groups can never be related by dualities. One way to see this is viathe twisted KR-theory classifying the D-brane charges. Theories which are dual to


one another must have twisted KR-groups which agree up to a degree shift, whileif the KR-groups are non-isomorphic (even after a degree shift), the two theoriescannot possibly be equivalent. Thus calculations of twisted KR-theory provide amethodology for testing conjectures about dualities in string theory.

In the case whereX is an elliptic curve and ι is holomorphic or anti-holomorphic,the twisted KR-groups were computed in [14,15]. In one group of three theories (ιthe identity, ι anti-holomorphic with a fixed set with two components and with triv-ial sign choice, and ι holomorphic with four isolated fixed points), the KR-theoryturned out to be KO−•(T 2), up to a degree shift. In the next group (ι holomorphicand free, ι anti-holomorphic and free, ι holomorphic with four fixed points with signchoice (+,+,−,−), and ι anti-holomorphic with a fixed set with two componentsand sign choice (+,−)), the groups turned out to be KSC−• ⊕ KSC−•−1 up toa degree shift. In the last group (ι the identity but the Dixmier-Douady invariant(B-field) nontrivial, ι holomorphic with four isolated fixed points and sign choice(+,+,+,−), and ι anti-holomorphic with fixed set a circle), the KR-theory turnedout to be KO−• ⊕ KO−• ⊕ K−•−1 up to a degree shift. The KR-groups in oneT-duality group are not isomorphic to those in another, so there cannot be anyadditional dualities between theories.

Rather curiously, it turns out (as was shown in [59]) that all of the isomor-phisms of twisted KR-groups associated to dualities of elliptic curve orientifoldtheories arise from the real Baum-Connes isomorphisms for certain solvable groupswith Z or Z2 as a subgroup of finite index. This suggests a rather mysteriousconnection between representation theory and duality for string theories, which weintend to explore further. It will be especially interesting to study dualities be-tween orientifold theories compactified on abelian varieties of dimension 2 or 3 andon K3-surfaces, and ultimately on simply connected Calabi-Yau 3-folds.

References

[1] J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969.MR0252560 (40 #5780)

[2] Jeffrey Adams, The real Chevalley involution, Compos. Math. 150 (2014), no. 12, 2127–2142,DOI 10.1112/S0010437X14007374. MR3292297

[3] D. W. Anderson, The real K-theory of classifying spaces, Proc. Nat. Acad. Sci. U. S. A. 51(1964), no. 4, 634–636.

[4] Richard F. Arens and Irving Kaplansky, Topological representation of algebras, Trans. Amer.Math. Soc. 63 (1948), 457–481. MR0025453 (10,7c)

[5] M. F. Atiyah, K-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367–386.

MR0206940 (34 #6756)[6] Michael Atiyah and Graeme Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–

330; English transl., Ukr. Math. Bull. 1 (2004), no. 3, 291–334. MR2172633 (2006m:55017)[7] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and

K-theory of group C∗-algebras, C∗-algebras: 1943–1993 (San Antonio, TX, 1993), Con-temp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291, DOI10.1090/conm/167/1292018. MR1292018 (96c:46070)

[8] Paul Baum and Max Karoubi, On the Baum-Connes conjecture in the real case, Q. J. Math.55 (2004), no. 3, 231–235, DOI 10.1093/qjmath/55.3.231. MR2082090 (2005d:19005)

[9] Victor M. Bogdan, On Frobenius, Mazur, and Gelfand-Mazur theorems on division algebras,Quaest. Math. 29 (2006), no. 2, 171–209, DOI 10.2989/16073600609486159. MR2233367(2007c:46044)

[10] Theodor Brocker and Tammo tom Dieck, Representations of compact Lie groups, GraduateTexts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the Germanmanuscript; Corrected reprint of the 1985 translation. MR1410059 (97i:22005)


[11] A. Connes, A factor not anti-isomorphic to itself, Bull. London Math. Soc. 7 (1975), 171–174.MR0435864 (55 #8815)

[12] A. Connes, A factor not anti-isomorphic to itself, Ann. Math. (2) 101 (1975), 536–554.MR0370209 (51 #6438)

[13] Jacques Dixmier, C∗-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford,1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library,Vol. 15. MR0458185 (56 #16388)

[14] Charles Doran, Stefan Mendez-Diez, and Jonathan Rosenberg, T-duality for orientifolds andtwisted KR-theory, Lett. Math. Phys. 104 (2014), no. 11, 1333–1364, DOI 10.1007/s11005-014-0715-0. MR3267662

[15] Charles Doran, Stefan Mendez-Diez, and Jonathan Rosenberg, String theory on ellipticcurve orientifolds and KR-theory, Comm. Math. Phys. 335 (2015), no. 2, 955–1001, DOI10.1007/s00220-014-2200-0. MR3316647

[16] Johan L. Dupont, Symplectic bundles and KR-theory, Math. Scand. 24 (1969), 27–30.MR0254839 (40 #8046)

[17] George A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. MR0397420 (53 #1279)

[18] J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280.MR0164248 (29 #1547)

[19] Dongfeng Gao and Kentaro Hori, On the structure of the Chan-Paton factors for D-branesin type II orientifolds, preprint, arXiv:1004.3972, 2010.

[20] T. Giordano, Antiautomorphismes involutifs des facteurs de von Neumann injectifs. I(French), J. Operator Theory 10 (1983), no. 2, 251–287. MR728909 (85h:46087)

[21] T. Giordano, A classification of approximately finite real C∗-algebras, J. Reine Angew. Math.385 (1988), 161–194, DOI 10.1515/crll.1988.385.161. MR931219 (89h:46078)

[22] Thierry Giordano and Vaughan Jones, Antiautomorphismes involutifs du facteur hyperfinide type II1 (French, with English summary), C. R. Acad. Sci. Paris Ser. A-B 290 (1980),no. 1, A29–A31. MR564145 (82b:46079)

[23] F. Gliozzi, Joel Scherk, and David I. Olive, Supersymmetry, supergravity theories and thedual spinor model, Nuclear Phys. B 122 (1977), no. 2, 253–290.

[24] Roger Godement, Topologie algebrique et theorie des faisceaux (French), Hermann, Paris,1973. Troisieme edition revue et corrigee; Publications de l’Institut de Mathematiquede l’Universite de Strasbourg, XIII; Actualites Scientifiques et Industrielles, No. 1252.MR0345092 (49 #9831)

[25] K. R. Goodearl, Notes on real and complex C∗-algebras, Shiva Mathematics Series, vol. 5,Shiva Publishing Ltd., Nantwich, 1982. MR677280 (85d:46079)

[26] K. R. Goodearl and D. E. Handelman, Classification of ring and C∗-algebra direct limitsof finite-dimensional semisimple real algebras, Mem. Amer. Math. Soc. 69 (1987), no. 372,viii+147, DOI 10.1090/memo/0372. MR904013 (88k:46067)

[27] Paul S. Green, A cohomology theory based upon self-conjugacies of complex vector bundles,Bull. Amer. Math. Soc. 70 (1964), 522–524. MR0164347 (29 #1644)

[28] Alexander Grothendieck, Sur quelques points d’algebre homologique (French), Tohoku Math.J. (2) 9 (1957), 119–221. MR0102537 (21 #1328)

[29] Alain Guichardet, Caracteres des algebres de Banach involutives (French), Ann. Inst. Fourier(Grenoble) 13 (1963), 1–81. MR0147925 (26 #5437)

[30] Matthew B. Hastings and Terry A. Loring, Topological insulators and C∗-algebras:theory and numerical practice, Ann. Physics 326 (2011), no. 7, 1699–1759, DOI10.1016/j.aop.2010.12.013. MR2806137 (2012j:82005)

[31] Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR0358873(50 #11332)

[32] Lars Ingelstam, Real Banach algebras, Ark. Mat. 5 (1964), 239–270 (1964). MR0172132(30 #2358)

[33] V. F. R. Jones, A II1 factor anti-isomorphic to itself but without involutory antiautomor-phisms, Math. Scand. 46 (1980), no. 1, 103–117. MR585235 (82a:46075)

[34] Max Karoubi, Twisted K-theory—old and new, K-theory and noncommutative geometry,EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2008, pp. 117–149, DOI 10.4171/060-1/5.MR2513335 (2010h:19010)


[35] G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91(1988), no. 1, 147–201, DOI 10.1007/BF01404917. MR918241 (88j:58123)

[36] Alexei Kitaev, Periodic table for topological insulators and superconductors, Landau Memo-rial Conference, AIP Conf. Proc., vol. 1134, 2009, paper 22, arXiv:0901.2686.

[37] Gottfried Kothe, Stanis�law Mazur’s contributions to functional analysis, Math. Ann. 277(1987), no. 3, 489–528, DOI 10.1007/BF01458329. MR891589 (88i:01112)

[38] Vincent Lafforgue, K-theorie bivariante pour les algebres de Banach et conjecture de Baum-

Connes (French), Invent. Math. 149 (2002), no. 1, 1–95, DOI 10.1007/s002220200213.MR1914617 (2003d:19008)

[39] Bingren Li, Real operator algebras, World Scientific Publishing Co., Inc., River Edge, NJ,2003. MR1995682 (2004k:46100)

[40] Andre Lichnerowicz, Spineurs harmoniques (French), C. R. Acad. Sci. Paris 257 (1963), 7–9.MR0156292 (27 #6218)

[41] Pierre Mazet, La preuve originale de S. Mazur pour son theoreme sur les algebres normees(French), Gaz. Math. 111 (2007), 5–11. MR2289675 (2008d:46001)

[42] S. Mazur, Sur les anneaux lineaires, C. R. Acad. Sci., Paris 207 (1938), 1025–1027 (French).[43] Ruben Minasian and Gregory Moore, K-theory and Ramond-Ramond charge, J. High En-

ergy Phys. 11 (1997), Paper 2, 7 pp. (electronic), DOI 10.1088/1126-6708/1997/11/002.MR1606278 (2000a:81190)

[44] D. R. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), no. 1,105–121, DOI 10.1007/BF01403093. MR728142 (85j:14071)

[45] El-kaıoum M. Moutuou, On groupoids with involutions and their cohomology, New York J.Math. 19 (2013), 729–792. MR3141812

[46] El-kaıoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal.266 (2014), no. 5, 2689–2739, DOI 10.1016/j.jfa.2013.12.019. MR3158706

[47] Viacheslav V. Nikulin and Sachiko Saito, Real K3 surfaces with non-symplectic invo-lution and applications, Proc. London Math. Soc. (3) 90 (2005), no. 3, 591–654, DOI10.1112/S0024611505015212. MR2137825 (2006b:14063)

[48] T. W. Palmer,Real C∗-algebras, Pacific J. Math. 35 (1970), 195–204. MR0270162 (42 #5055)[49] Gert K. Pedersen, C∗-algebras and their automorphism groups, London Mathematical Society

Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR548006 (81e:46037)

[50] M. G. Penington and R. J. Plymen, The Dirac operator and the principal series for com-plex semisimple Lie groups, J. Funct. Anal. 53 (1983), no. 3, 269–286, DOI 10.1016/0022-1236(83)90035-6. MR724030 (85d:22016)

[51] Joseph Polchinski, String theory. Vol. I, Cambridge Monographs on Mathematical Physics,Cambridge University Press, Cambridge, 2005. An introduction to the bosonic string; Reprintof the 2003 edition. MR2151029 (2006j:81149a)

[52] Joseph Polchinski, String theory. Vol. II, Cambridge Monographs on Mathematical Physics,Cambridge University Press, Cambridge, 2005. Superstring theory and beyond; Reprint of2003 edition. MR2151030 (2006j:81149b)

[53] Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C∗-algebras,Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence,RI, 1998. MR1634408 (2000c:46108)

[54] Marc A. Rieffel, Morita equivalence for operator algebras, Operator algebras and applica-tions, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc.,Providence, R.I., 1982, pp. 285–298. MR679708 (84k:46045)

[55] Jonathan Rosenberg, Homological invariants of extensions of C∗-algebras, Operator algebrasand applications, Part 1 (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer.Math. Soc., Providence, RI, 1982, pp. 35–75. MR679694 (85h:46099)

[56] Jonathan Rosenberg, C∗-algebras, positive scalar curvature, and the Novikov conjecture.III, Topology 25 (1986), no. 3, 319–336, DOI 10.1016/0040-9383(86)90047-9. MR842428

(88f:58141)[57] Jonathan Rosenberg, Continuous-trace algebras from the bundle theoretic point of view, J.

Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368–381. MR1018964 (91d:46090)[58] Jonathan Rosenberg, The KO-assembly map and positive scalar curvature, Algebraic topol-

ogy Poznan 1989, Lecture Notes in Math., vol. 1474, Springer, Berlin, 1991, pp. 170–182,DOI 10.1007/BFb0084745. MR1133900 (92m:53060)


[59] Jonathan Rosenberg, Real Baum-Connes assembly and T-duality for torus orientifolds, J.Geom. Phys. 89 (2015), 24–31, DOI 10.1016/j.geomphys.2014.12.004. MR3305978

[60] Jonathan Rosenberg and Stephan Stolz, A “stable” version of the Gromov-Lawson conjecture,

The Cech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Amer. Math. Soc.,Providence, RI, 1995, pp. 405–418, DOI 10.1090/conm/181/02046. MR1321004 (96m:53042)

[61] Shoichiro Sakai, C∗-algebras and W ∗-algebras, Classics in Mathematics, Springer-Verlag,Berlin, 1998. Reprint of the 1971 edition. MR1490835 (98k:46085)

[62] Thomas Schick, Real versus complex K-theory using Kasparov’s bivariant KK-theory,Algebr. Geom. Topol. 4 (2004), 333–346, DOI 10.2140/agt.2004.4.333. MR2077669(2005f:19007)

[63] Herbert Schroder, K-theory for real C∗-algebras and applications, Pitman Research Notes

in Mathematics Series, vol. 290, Longman Scientific & Technical, Harlow; copublished in theUnited States with John Wiley & Sons, Inc., New York, 1993. MR1267059 (95f:19006)

[64] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; GraduateTexts in Mathematics, Vol. 42. MR0450380 (56 #8675)

[65] Stephan Stolz, Positive scalar curvature metrics—existence and classification questions, Pro-ceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), Birkhauser,Basel, 1995, pp. 625–636. MR1403963 (98h:53063)

[66] Stephan Stolz, Manifolds of positive scalar curvature, Topology of high-dimensional mani-folds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes, vol. 9, Abdus Salam Int. Cent. Theoret.Phys., Trieste, 2002, pp. 661–709. MR1937026 (2003m:53059)

[67] Erling Størmer, Real structure in the hyperfinite factor, Duke Math. J. 47 (1980), no. 1,145–153. MR563372 (81g:46088)

[68] Jean-Louis Tu, Groupoid cohomology and extensions, Trans. Amer. Math. Soc. 358(2006), no. 11, 4721–4747 (electronic), DOI 10.1090/S0002-9947-06-03982-1. MR2231869(2007i:22008)

[69] Bert van Geemen and Alessandra Sarti, Nikulin involutions on K3 surfaces, Math. Z. 255(2007), no. 4, 731–753, DOI 10.1007/s00209-006-0047-6. MR2274533 (2007j:14057)

[70] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, NewYork-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188.MR0498999 (58 #16979)

[71] Antony Wassermann, Une demonstration de la conjecture de Connes-Kasparov pour les

groupes de Lie lineaires connexes reductifs (French, with English summary), C. R. Acad.Sci. Paris Ser. I Math. 304 (1987), no. 18, 559–562. MR894996 (89a:22010)

[72] Edward Witten, D-branes and K-theory, J. High Energy Phys. 12 (1998), Paper 19, 41 pp.(electronic), DOI 10.1088/1126-6708/1998/12/019. MR1674715 (2000e:81151)

[73] Wies�law Zelazko, Banach algebras, Elsevier Publishing Co., Amsterdam-London-New York;PWN–Polish Scientific Publishers, Warsaw, 1973. Translated from the Polish by Marcin E.Kuczma. MR0448079 (56 #6389)

Department of Mathematics, University of Maryland, College Park, Maryland

20742-4015

E-mail address: [email protected]: http://www2.math.umd.edu/~jmr/


Separable states, maximally entangled states,and positive maps

Erling Størmer

Dedicated to R. V. Kadison on the occasion of his 90th birthday.

Abstract. Using maximally entangled states we introduce an invariant S(a)for a ∈ Mn⊗Mn, which is useful for distinguishing separable density matricesfrom entangled ones. The results obtained are then applied to study the SPAof a positive map.

1. Introduction

The theory of positive maps on C*-algebras goes back to the pioneering workof Kadison on isometries and the generalized Schwarz inequality around 1950 [8,9].Since then the theory grew gradually until it got a boost when it was realized around1990 that it was very useful in the mathematical formalism of quantum informationtheory. The present paper will be an extension of some of the ideas arising fromthe theory and essentially the paper [13]. For a density operator a ∈ Mn ⊗ Mn,where Mn denotes the complex n × n matrices, we shall introduce an invariantS(a) defined via maximally entangled states, which gives necessary conditions forseparability of a. We then develop some of the theory for S(a) and show that0 ≤ S(a) ≤ 1 for a separable, that S(a) ≤ n for all a, and we show some results forwhen S(a) = 1. In the last section we shall use the invariant to study the structuralphysical approximations, the SPA, of positive maps, and in particular show that theSPA of some optimal maps are not separable, contradicting an earlier conjecture,see [1–3,10]. This was first shown by Ha and Kye [6] shortly before it was donein [13], using essentially the same example, but using different methods. In thispaper we follow the treatment in [13] closely.

The author is indebted to H. Christandl for bringing the paper [7] to his at-tention.

2. The invariant S(a)

Recall, see e.g. [11] Lemma 4.1, that each vector x ∈ Cn ⊗ Cn has a Schmidtdecomposition

x =∑i

cigi ⊗ hi,

where the vectors gi, (respective hi) form an orthonormal basis in Cn, and the norm‖ x ‖2=

∑| ci |2= 1.


259

260 ERLING STØRMER

A pure state ωx on Mn ⊗Mn is called maximally entangled if x can be writtenin the form above with ci = n− 1

2 for all i ≤ n. For simplicity of notation we shallcall a vector x as above with ci = 1 for all i for a maximally entangled vector. Thenn−1ωx is a maximally etangled state, and ‖ x ‖2= n.

Definition 1. Let a be a self-adjoint matrix in Mn ⊗Mn. Then

S(a) = max{(ax, x) : x is a maximally entangled vector in Cn ⊗ Cn}= max{nωy(a) : ωy maximally entangled state}

This invariant has also been considered by M. and P. Horodecki [7]. In thepaper [13] we used the same notation S(a) for (ax, x) when x is the vector

∑ei⊗ei,

where (ei) is the usual orthonormal basis for Cn. It is clear from the definition thatS(a) ≤ n ‖ a ‖. We shall mostly consider S(a) when a is a density matrix. Thenwe have the following description of matrices a with maximal value for S(a). Wedenote by [x] the rank 1 projection onto the 1-dimensional subspace Cx of Cn.

Proposition 2. Let a be a density matrix in Mn ⊗ Mn. Then S(a) = n iffa = [x] for some maximally entangled vector x.

Proof. If a = [x] then by the above

n = n ‖ a ‖≥ S(a) ≥ (ax, x) = (x, x) = n,

so S(a) = n. Conversely suppose S(a) = n. By spectral theory there exist mutuallyorthogonal rank 1 projections pi ∈ Mn ⊗ Mn and λi ≥ 0 with sum 1 such thata =

∑λipi. By compactness of the set of maximally entangled vectors there exists

a maximally entangled vector x such that

n = S(a) = (ax, x) =∑

λi(pix, x) ≤∑

λiS(pi) ≤∑

λin = n.

Thus (pix, x) = n for all i, hence pi = [x], completing the proof of the proposition.Let (gi) and (hi) be orthonormal bases for Cn. Let (eij), respectively (fkl),

be matrix units such that eijgm = δjmgi and similarly for fkl. Then each matrixa ∈ Mn ⊗Mn can be written in the form

(1) a =∑

a(ij)(kl)eij ⊗ fkl.

Lemma 3. If x =∑n

i=1 gi ⊗ hi is a maximally entangled vector, and a ∈Mn ⊗Mn is given by equation (1) then

(ax, x) =∑

a(ij)(ij).

Proof.

(ax, x) =∑ijkl

a(ij)(kl)(eij ⊗ fkl∑r

gr ⊗ hr,∑s

gs ⊗ hs)

=∑

a(ij)(kl)δjrδlr(gi ⊗ hk,∑

gs ⊗ hs)

=∑

a(ij)(kl)δjrδlrδisδks

=∑

a(ij)(ij).

The proof is complete.

Recall that an operator a ∈ Mn ⊗ Mn is separable if a can be written in theform a =

∑bi ⊗ ci with bi, ci ≥ 0.

MAXIMALLY ENTANGLED STATES AND POSITIVE MAPS 261

Theorem 4. Let a be a separable density matrix in Mn⊗Mn.Then 0≤S(a)≤1.

Proof. Let notation be as before Lemma 3. Suppose first a = b ⊗ c with b, c ≥ 0.Rescaling b and c we may assume they are density matrices. Let b = (bij) withrespect to the basis (gi) and c = (ckl) with respect to (hj). Then

a(ij)(kl) = bijckl.

Hence by Lemma 3, if ‖ . ‖2 denotes the Hilbert-Schmidt norm and ‖ . ‖1 thetrace norm, then

(ax, x) =∑

bijcij

≤ (∑

| bij |2∑

| cij |2)12

= ‖ b ‖2‖ c ‖2≤ ‖ b ‖1‖ c ‖1= 1.

Since this holds for all maximally entangled vectors x, S(a) ≤ 1. In the generalcase a =

∑λkbk⊗ck with bk, ck density operators and λk ≥ 0 and

∑λk = 1. Thus

by the above applied to bk ⊗ ck,

(ax, x) =∑

λk(bk ⊗ ckx, x) ≤∑

λk = 1,

proving the theorem.

Corollary 5. Let a = b⊗ c ∈ Mn ⊗Mn with b and c density matrices. ThenS(a) = 1 iff b and c are rank 1 projections.

Proof. If S(a) = 1 it follows from the above proof that ‖ b ‖2=‖ b ‖1= 1 andsimilarly for c. This is possible only if rank b = 1 and rank c = 1 with norms equalto 1, hence that they are rank 1 projections.

Conversely if b = [y] and c = [z] let (yi) respectively (zi) be orthonormal basesfor Cn with y1 = y, z1 = z. Then x =

∑yi ⊗ zi is a maximally entangled vector

such that

1 ≥ S(a) ≥ (ax, x) =((b⊗ c)

(∑yi ⊗ zi

),∑

yi ⊗ zi

)= ((b⊗ c)(y ⊗ z), y ⊗ z) = 1.

so S(a) = 1, proving the corollary.

If in the above corollary (yi) and (zi) are orthonormal bases such that b =[∑

biyi] and c = [∑

cizi], write b = (bij), c = (cij) as in the proof of Theorem 4.Then the transpose of c is the matrix ct = (cij), and we get as in the proof ofTheorem 4

1 =∑

bijcij =< b, ct >2=‖ b ‖2‖ ct ‖2,where <,>2 denotes the Hilbert-Schmidt inner product. Thus b = ct , and S(b ⊗c) = ((b⊗ c)x, x) by the proof of Theorem 4. Using Corollary 5 for the converse wetherefore have

Lemma 6. With the above notation, if x =∑

yi ⊗ zi is a maximally entangledvector, and b = [

∑biyi], c = [

∑cizi] then S(b⊗ c) = 1 = (b⊗ cx, x) iff b = ct.

We shall also need another result on rank 1 projections, namely the following.

262 ERLING STØRMER

Proposition 7. Let q = [y] be a rank 1 projection in Mn ⊗Mn, where y is aunit vector in Cn ⊗ Cn. Then S(q) ≥ 1. Furthermore, S(q) = 1 iff q = e ⊗ f forrank 1 projections e and f .

Proof. The vector y has a Schmidt decomposition

y =i=k∑i=1

λigi ⊗ hi,

with λ ≥ 0, where (gi), (hi) orthonormal systems, see [11], Lemma 4.1. Extend

them to orthonormal bases , so y =∑i=n

i=1 λigi ⊗ hi with λi = 0 for i ≥ k. Thevector x =

∑gi ⊗ hi is maximally entangled. We have

S(q) ≥ (qx, x)

= ((x, y)y, x)

= | (x, y) |2

= | (∑

gi ⊗ hi,∑

λkgk ⊗ hk) |2

= |∑

(gi ⊗ hi, λigi ⊗ hi) |2

= |∑

λi |2

≥ |∑

λ2i |2

= 1,

since y is a unit vector, so∑

λ2i = 1, proving the first part.

If S(q) = 1 the above shows that∑

λi =∑

λ2i , hence, since λi ≥ 0, λi = λ2

i

for all i, hence λi = 1 or 0. Hence there is a unique i such that y = gi ⊗ hi, henceq = [y] = e⊗ f where e = [gi], f = [hi], proving one implication in the proposition.The converse follows from Corollary 5.

It is clear from the definition of S(a) that if a is a sum a =∑

ai of positivematrices, then S(a) ≤

∑S(ai). Since S(ai) is obtained as S(ai) = (aixi, xi) with

xi a maximally entangled vector, and the vectors xi may be different for different ai, it is difficult to conclude much about the relationship between S(a) and

∑S(ai).

Our next result yields information on this problem.

Theorem 8. Let a be a separable density matrix in Mn ⊗Mn. Then S(a) = 1iff there exist a maximally entangled vector x =

∑yi⊗zi ∈ Cn⊗Cn and unit vectors

uk =∑

akiyi ∈ Cn such that if ek = [uk], then a is a convex sum a =∑

λkek ⊗ fk,where fk = etk. Furthermore ((ek ⊗ fk)x, x) = 1 for all k.

Proof. Since a is separable a =∑

λkbk⊗ck with bk, ck density matrices. If S(a) = 1then there is a maximally entangled vector x =

∑yi ⊗ zi such that

1 = (ax, x) =∑

λk((bk ⊗ ck)x, x) ≤∑

λk = 1,

using Theorem 4. Thus ((bk ⊗ ck)x, x) = 1 for all k. By Corollary 5 ek = bk andfk = ck are rank 1 projections. Let uk =

∑akiyi such that ek = [uk]. Then

ek = (akiakj), so by Corollary 6 and its proof fk = etk, and

1 = S(ek ⊗ fk) = ((ek ⊗ fk)x, x)


for the same x for all k. Conversely if the above conditions hold, then

(ax, x) =∑

λk((ek ⊗ fk)x, x) = 1,

so S(a) = 1, completing the proof of the theorem.

Note that we cannot conclude that the projections ek⊗fk are mutually orthog-onal. However, if a =

∑μipi ⊗ qi is a convex sum with pi, qi rank 1 projections,

then S(a) = 1 implies S(pi ⊗ qi) = 1, as follows from the above proof.

Recall that a state ρ on Mn ⊗Mn is called a PPT- state if ρ ◦ (ι⊗ t) is also astate, where ι denotes the identity map and t the transpose map. This means thatif a is a density matrix for ρ, then ι⊗ t(a) is also a density matrix. We next showthat S(a) is preserved under this transformation.

Theorem 9. Let a ∈ Mn⊗Mn be a density matrix such that ι⊗ t(a) ≥ 0, thenS(a) = S(ι⊗ t(a)).

Proof. Let (ei) be the standard basis for Cn. Let

J(∑

xiei) =∑

xiei.

Then bt = Jb∗J for b ∈ Mn. If x =∑

xiei, y =∑

yiei then

(Jx, y) =∑

xiyi =∑

xiyi = (x, Jy) = (Jy, x).

Thus if b, c ∈ Mn, then

(ι⊗ t(b⊗ c)x⊗ y, x⊗ y) = (bx, x)(cty, y)

= (bx, x)(cJy, Jy)

= ((b⊗ c)x⊗ Jy, x⊗ Jy).

Since y ⊥ z iff Jy ⊥ Jz, if x is a maximally entangled vector in Cn ⊗ Cn, thenι ⊗ J(x) is maximally entangled. It thus follows from the above computation andthe definition of S(a) that S(a) = S(ι⊗ t(a)). The proof is complete.

Consequently it does not help that a state is PPT in order to use the invariantS(a) to verify whether a state is separable or not.

We end this section with an open problem, namely; Let a be a density operatorwith S(a) ≤ 1. Is a separable? If not, what if a furthermore satisfies ι⊗ t(a) ≥ 0?

3. Positive maps

In this section we relate the invariant S(a) to positive maps of Mn into itself.We first recall the necessary definitions. Let φ : Mn → Mn be a linear map. Thenφ is positive, written φ ≥ 0, if φ(a) ≥ 0 for all a ≥ 0. φ is completely positive ifφ⊗ ι : Mn⊗Mn → Mn⊗Mn is positive. φ is optimal if φ−ψ ≥ 0 for ψ completelypositive, implies ψ = 0. Let (eij) be a complete set of matrix units for Mn. Thenthe Choi matrix Cφ for φ is the matrix

Cφ =∑

eij ⊗ φ(eij) ∈ Mn ⊗Mn.

Then φ is completely positive iff Cφ ≥ 0, see [12], Theorem 4.1.8, and φ ≥ 0 iffTr ⊗ Tr(Cφa ⊗ b) ≥ 0 for all a, b ≥ 0 in Mn, see [12], Theorem 4.1.11, where Tris the usual trace on Mn. φ is said to be super-positive or entanglement breaking ifTr⊗ Tr(CφCψ) ≥ 0 for all positive maps ψ of Mn into itself. We then have that φis super-positive iff Cφ is separable iff φ =

∑aiωi, where ωi are states on Mn, and

264 ERLING STØRMER

ai ≥ 0 in Mn, as follows from [12], Lemma 4.2.3 and Proposition 5.1.4. If φ is apositive map of Mn into itself then the map

(2) a −→ φ(1)Tr(a) + φ(a)

is super-positive, see [12],Theorem 7.5.4 or [13].For the rest of this section we consider maps of the form φ(a) = AdV (a) =

V ∗aV with V ∈ Mn. By [12], Lemma 4.2.3 and Proposition 5.1.4 it follows thatsuch a map is super-positive iff Cφ is separable, and by definition, iff rankV = 1, see[12], Definition 5.1.2. We next show that these properties are reflected in S(Cφ).

Theorem 10. Let V ∈ Mn and let φ = AdV . Then S(Cφ) ≥‖ V ‖22 . Further-more Cφ is separable iff S(Cφ) =‖ V ‖22 .

Proof. Let (eij) be the usual matrix units corresponding to the usual orthonormalbasis ei for C

n. Let P =∑

eij ⊗ eij . Then P = n[xo], where xo =∑

ei ⊗ ei. Then

Cφ =∑

eij ⊗ V ∗eijV = (1⊗ V )∗P (1⊗ V ),

so Cφ is a positive rank 1 matrix. Its trace is

Tr ⊗ Tr(Cφ) = Tr ⊗ Tr(∑

eij ⊗ V ∗eijV )

= Tr(∑

V ∗eiiV )

= Tr(V ∗V ) =‖ V ‖22,so that ‖ V ‖−2

2 Cφ is a rank 1 projection [y] for a unit vector y. Hence by

Proposition 7 ‖ V ‖−22 S(Cφ) = S([y]) ≥ 1, i.e.

S(Cφ) ≥‖ V ‖22proving the first part of the theorem.

Finally, by Proposition 7 S([y]) = 1 iff [y] = e⊗f for e and f rank 1 projections,hence S(Cφ) =‖ V ‖22 iff Cφ is separable. The proof is complete.

We next consider the other extreme case, i.e. when V is unitary. In this casewe can chose the maximal entangled vector as x = (1⊗V ∗)xo, with xo as above andthus assume V = 1. Then Cφ = P , and (Pxo, xo) = n(xo, xo) = n2. FurthermoreTr ⊗ Tr(Cφ) = Tr ⊗ Tr(P ) = n, hence 1/nCφ is a density matrix such that

(1/nCφxo, xo) = 1/n(nxo, xo) = n.

We have just shown

Proposition 11. If V is a unitary matrix, and φ = AdV then 1/nCφ is adensity matrix, and S(1/nCφ) = n.

4. The SPA of a positive map

In the applications to physics generic positive maps cannot be used since theyare not physical, and then cannot be directly implemented. It is therefore challeng-ing to try to find a physical way to approximate the action of a positive map. Thisis the goal of the structural approximation, SPA, see e.g. [1,10]. The idea is tomix a positive map φ with a simple completely positive map making the mixturecompletely positive. The resulting map can then be realised in the laboratory, andits action characterizes entanglement of the states detected by φ.


We now define and describe a few basic properties of the SPA of φ following[4,13]. Let φ be a unital positive map of Mn into itself, and let W = 1/nCφ. ThenTr ⊗ Tr(W ) = 1. We put for 0 ≤ t ≤ 1

W (t) =1− t

n21⊗ 1 + tW.

Then W (t) has trace 1. Note that CTr = 1⊗ 1, and n−21⊗ 1 is a density matrix.Write W = W+ −W− with W+,W− ≥ 0 and W+W− = 0, and similarly for Cφ.As is easily seen, see e.g.[4], eq. (14).

t∗ = (1 + n2 ‖ W− ‖)−1

is the maximal t for which W (t) ≥ 0. The SPA of φ, SPA(φ) is defined as

SPA(φ) = W (t∗)

and is the Choi matrix of the completely positive map on the line segment betweenφ and the tracial state 1/nTr which is nearest to φ. A straightforward computation,see e.g. [13], page 2202, shows

SPA(φ) = W (t∗) =1

n+ n2 ‖ C−φ ‖

(‖ C−φ ‖ 1⊗ 1 + Cφ).

We now easily obtain the following sufficient condition for separability for SPA(φ).For a special class of maps this was also obtained in [4].

Proposition 12. When φ is a unital positive map then SPA(φ) is the densitymatrix for a state, which is separable if ‖ C−

φ ‖= 1.

Proof. Since ‖ C−φ ‖ 1 ⊗ 1 + Cφ ≥ 0, SPA(φ) ≥ 0, and the above formula implies

Tr⊗Tr(SPA(φ)) = 1, it follows that SPA(φ) is the density matrix for a state. Byequation (2) Tr+φ is super-positive, hence if ‖ C−

φ ‖= 1, then 1⊗ 1+Cφ = CTr+φ

is separable.

We also have a necessary condition for separability of SPA(Cφ). This will bean application of Theorem 4.

Proposition 13. If φ is unital and SPA(φ) separable then

S(Cφ) ≤ n+ n(n− 1) ‖ C−φ ‖ .

Proof. By Proposition 12 SPA(φ) is the density matrix for a state, which by as-sumption is separable. Hence by Theorem 4,

0 ≤ S(SPA(φ)) ≤ 1.

Since S(1 ⊗ 1) = n, it follows from the formula stated just before Proposition 12that

‖ C−φ ‖ n+ S(Cφ) = ‖ C−

φ ‖ S(1⊗ 1) + S(Cφ)

= S(‖ C−φ ‖ 1⊗ 1 + Cφ)

= (n+ n2 ‖ C−φ ‖)S(SPA(φ))

≤ n+ n2 ‖ C−φ ‖ .

Hence S(Cφ) ≤ n+ n(n− 1) ‖ C−φ ‖, proving the proposition.

266 ERLING STØRMER

The definition of SPA(φ) is only of interest when φ is not separable, or evennot majoring a completely positive map, hence one studies SPA(φ) when φ isan optimal map. By checking the standard examples of optimal maps physicistsconjectured that SPA(φ) of an optimal map is always separable, see [2,10]. Thisconjecture was shown to be false by Ha and Kye in [6], and the author shortlyafterwords. We used essentially the same maps, but the profs were quite different,as I used a variant of the invariant S(a).

The maps we consider, are the following extension of the Choi map,see [12], ofM3 into itself. Let a, b, c be non-negative real numbers, −π ≤ θ ≤ π. Define themap φ(a, b, c, θ) : M3 → M3 by

φ(a, b, c, θ)((xij)) =⎛⎝ax11 + bx22 + cx33 −eiθx12 −e−iθx13

−e−iθx21 cx11 + ax22 + bx33 −eiθx23

−eiθx31 −e−iθx32 bx11 + cx22 + ax33

⎞⎠ .

Then the Choi matrix for φ is given by

Cφ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a 0 0 0 −eiθ 0 0 0 −e−iθ

0 c 0 0 0 0 0 0 00 0 b 0 0 0 0 0 00 0 0 b 0 0 0 0 0

−e−iθ 0 0 0 a 0 0 0 −eiθ

0 0 0 0 0 c 0 0 00 0 0 0 0 0 c 0 00 0 0 0 0 0 0 b 0

−eiθ 0 0 0 −e−iθ 0 0 0 a

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.

Let pθ = max{2Reei(θ−2π/3), 2Reeiθ, 2Reei(θ+2π/3)}. Then 1 ≤ pθ ≤ 2. The mainproperties of φ where shown by Ha and Kye [5]. They are summarised in

Theorem 14. Let φ be as above. Then

(i) φ is positive iff a+ b+ c ≥ pθ, and bc ≥ (1− a)2 if a ≤ 1.

(ii) If φ ≥ 0 and 1 < pθ < 2 then φ is optimal if 0 ≤ a < 1, and bc = (1− a)2.

We see from this theorem that we can find optimal maps φ such that if a isclose to 1, b and c are close to 0, and θ is close to π, then if we divide by a+ b+ c,so φ becomes unital, then Cφ is close to the matrix A = (aij) where aij = 0 when ior j is different from 1,5,9, while aij = 1 if both i and j belong to the set {1, 5, 9}.But S(A) = 9, or S(1/3A) = 3 > 1. Thus S(1/3A) has maximal possible value fora density matrix, so by continuity Cφ cannot be separable. To show this we can

also use Proposition 13. Let as before xo =∑i=3

i=1 ei ⊗ ei. Then

(Cφxo, xo) = 3a− 3eiθ − 3e−iθ = 3(a− 2 cos θ).

As θ is close to π, cos θ is close to −1, and ‖ C−φ ‖ is close to 0. Thus (Cφxo, xo)

is close to 3(1 + 2) = 9 > 3 + 6 ‖ C−φ ‖, so by Proposition 13 SPA(φ) cannot be

separable. For more details see [13].


References

[1] R. Augusiak, J. Bae, �L. Czekaj, and M. Lewenstein, On structural physical approximationsand entanglement breaking maps, J. Phys. A 44 (2011), no. 18, 185308, 21, DOI 10.1088/1751-8113/44/18/185308. MR2788732 (2012d:81044)

[2] D. Chruscinski, J. Pytel, and G. Sarbicki, Constructing optimal entanglement witnesses,Phys. Review A 80,(2009), 062314.

[3] D. Chruscinski and G. Sarbicki, Entanglement witnesses: construction, analysis and classi-fication, J. Phys. A 47 (2014), no. 48, 483001, 64, DOI 10.1088/1751-8113/47/48/483001.MR3280004

[4] D. Chruscinski and J. Pytel, Optimal entanglement witnesses from generalized reduc-tion and Robertson maps, J. Phys. A 44 (2011), no. 16, 165304, 15, DOI 10.1088/1751-8113/44/16/165304. MR2787094 (2012g:81028)

[5] K.-C. Ha and S.-H. Kye, Entanglement witnesses arising from Choi type positive linear maps,J. Phys. A 45 (2012), no. 41, 415305, 17, DOI 10.1088/1751-8113/45/41/415305. MR2983336

[6] K.-C. Ha and S.-H. Kye, The structural physical approximations and optimal entanglement

witnesses, J. Math. Phys. 53 (2012), no. 10, 102204, 7, DOI 10.1063/1.4754279. MR3050573[7] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of

distillation protocols Phys. Review A. 59 (1999), 4206.[8] R. V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338.

MR0043392 (13,256a)[9] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator alge-

bras, Ann. of Math. (2) 56 (1952), 494–503. MR0051442 (14,481c)[10] J. K. Korbicz, M. L. Almeida, J. Bae, M. Lewenstein, and A. Acin, Structural approximations

to positive maps and entanglement breaking channels, Phys. Review A. 78 (2008), 062105.[11] D. Petz, Quantum information theory and quantum statistics, Theoretical and Mathematical

Physics, Springer-Verlag, Berlin, 2008. MR2363070 (2009c:81026)[12] E. Størmer, Positive linear maps of operator algebras, Springer Monographs in Mathematics,

Springer, Heidelberg, 2013. MR3012443[13] E. Størmer, Separable states and the structural physical approximation of a positive map, J.

Funct. Anal. 264 (2013), no. 9, 2197–2205, DOI 10.1016/j.jfa.2013.02.015. MR3029151

Department of Mathematics, University of Oslo, 0316 Oslo, Norway


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671 Robert S. Doran and Efton Park, Editors, Operator Algebras and TheirApplications, 2016

667 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, DmitryKhavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 2: Complex Analysis, QuasiconformalMappings, Complex Dynamics, 2016

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663 David Kohel and Igor Shparlinski, Editors, Frobenius Distributions: Lang-Trotterand Sato-Tate Conjectures, 2016

662 Zair Ibragimov, Norman Levenberg, Sergey Pinchuk, and Azimbay Sadullaev,Editors, Topics in Several Complex Variables, 2016

661 Douglas P. Hardin, Doron S. Lubinsky, and Brian Z. Simanek, Editors, ModernTrends in Constructive Function Theory, 2016

660 Habib Ammari, Yves Capdeboscq, Hyeonbae Kang, and Imbo Sim, Editors,Imaging, Multi-scale and High Contrast Partial Differential Equations, 2016

659 Boris S. Mordukhovich, Simeon Reich, and Alexander J. Zaslavski, Editors,Nonlinear Analysis and Optimization, 2016

658 Carlos M. da Fonseca, Dinh Van Huynh, Steve Kirkland, and Vu Kim Tuan,Editors, A Panorama of Mathematics: Pure and Applied, 2016

657 Noe Barcenas, Fernando Galaz-Garcıa, and Monica Moreno Rocha, Editors,Mexican Mathematicians Abroad, 2016

656 Jose A. de la Pena, J. Alfredo Lopez-Mimbela, Miguel Nakamura, and JimmyPetean, Editors, Mathematical Congress of the Americas, 2016

655 A. C. Cojocaru, C. David, and F. Pappalardi, Editors, SCHOLAR—a ScientificCelebration Highlighting Open Lines of Arithmetic Research, 2015

654 Carlo Gasbarri, Steven Lu, Mike Roth, and Yuri Tschinkel, Editors, RationalPoints, Rational Curves, and Entire Holomorphic Curves on Projective Varieties, 2015

653 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, DmitryKhavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 1: PDE, Differential Geometry, RadonTransform, 2015

652 Marina Avitabile, Jorg Feldvoss, and Thomas Weigel, Editors, Lie Algebras andRelated Topics, 2015

651 Anton Dzhamay, Kenichi Maruno, and Christopher M. Ormerod, Editors,Algebraic and Analytic Aspects of Integrable Systems and Painleve Equations, 2015

650 Jens G. Christensen, Susanna Dann, Azita Mayeli, and Gestur Olafsson,Editors, Trends in Harmonic Analysis and Its Applications, 2015

649 Fernando Chamizo, Jordi Guardia, Antonio Rojas-Leon, and Jose MarıaTornero, Editors, Trends in Number Theory, 2015

648 Luis Alvarez-Consul, Jose Ignacio Burgos-Gil, and Kurusch Ebrahimi-Fard,Editors, Feynman Amplitudes, Periods and Motives, 2015

647 Gary Kennedy, Mirel Caibar, Ana-Maria Castravet, and Emanuele Macrı,Editors, Hodge Theory and Classical Algebraic Geometry, 2015

646 Weiping Li and Shihshu Walter Wei, Editors, Geometry and Topology ofSubmanifolds and Currents, 2015

645 Krzysztof Jarosz, Editor, Function Spaces in Analysis, 2015

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume contains the proceedings of the AMS Special Session Operator Algebras andTheir Applications: A Tribute to Richard V. Kadison, held from January 10–11, 2015, inSan Antonio, Texas.

Richard V. Kadison has been a towering figure in the study of operator algebras formore than 65 years. His research and leadership in the field have been fundamental in thedevelopment of the subject, and his influence continues to be felt though his work and thework of his many students, collaborators, and mentees.

Among the topics addressed in this volume are the Kadison-Kaplanksy conjecture, clas-sification of C∗-algebras, connections between operator spaces and parabolic induction,spectral flow, C∗-algebra actions, von Neumann algebras, and applications to mathemati-cal physics.

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