Operator algebras and their applications a tribute to Richard V. Kadison : AMS Special Session Operator Algebras and Their Applications: a Tribute to Richard V. Kadison, January 10-11, 2015, San Antonio, TX 1470419483, 9781470419486, 127-145-175-1

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671

Operator Algebras and Their Applications A Tribute to Richard V. Kadison AMS Special Session Operator Algebras and Their Applications: A Tribute to Richard V. Kadison January 10–11, 2015 San Antonio, TX

Robert S. Doran Efton Park Editors

American Mathematical Society

Operator Algebras and Their Applications A Tribute to Richard V. Kadison AMS Special Session Operator Algebras and Their Applications: A Tribute to Richard V. Kadison January 10–11, 2015 San Antonio, TX

Robert S. Doran Efton Park Editors

Richard V. Kadison

671

Operator Algebras and Their Applications A Tribute to Richard V. Kadison AMS Special Session Operator Algebras and Their Applications: A Tribute to Richard V. Kadison January 10–11, 2015 San Antonio, TX

Robert S. Doran Efton Park Editors

American Mathematical Society Providence, 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: Contemporary 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.) – Classifications of C ∗ -algebras. msc | Functional analysis – Selfadjoint operator algebras (C ∗ -algebras, von Neumann (W ∗ -) algebras, etc.) – Noncommutative dynamical systems. msc | Functional analysis – Selfadjoint operator algebras (C ∗ -algebras, von Neumann (W ∗ -) algebras, etc.) – Noncommutative differential geometry. msc | K-theory – K-theory and operator algebras – Index theory. msc | Topological groups, Lie groups – Lie groups – Representations of Lie and linear algebraic groups over real fields: analytic methods. msc Classification: LCC QA326 .O6522 2016 | DDC 512/.556–dc23 LC record available at http://lccn.loc.gov/2015043280 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/671

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 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 conjecture Paul Baum, Erik Guentner, and Rufus Willett

1

∗

Generalization of C -algebra methods via real positivity for operator and Banach algebras David P. Blecher

35

Higher weak derivatives and reflexive algebras of operators Erik Christensen

69

Parabolic induction, categories of representations and operator spaces Tyrone Crisp and Nigel Higson

85

Spectral multiplicity and odd K-theory-II Ronald G. Douglas and Jerome Kaminker

109

On the classification of simple amenable C*-algebras with finite decomposition rank George A. Elliott and Zhuang Niu 117 Topology of natural numbers and entropy of arithmetic functions Liming Ge

127

Properness conditions for actions and coactions S. Kaliszewski, Magnus B. Landstad, and John Quigg

145

Reflexivity of Murray-von Neumann algebras Zhe Liu

175

Hochschild cohomology for tensor products of factors Florin Pop and Roger R. Smith

185

On the optimal paving over MASAs in von Neumann algebras Sorin Popa and Stefaan Vaes

199

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

209

∗

Structure and applications of real C -algebras Jonathan Rosenberg

vii

235

viii

CONTENTS

Separable states, maximally entangled states, and positive maps Erling Størmer

259

Preface Richard V. Kadison has been a towering figure in the study of operator algebras for more than 65 years. His research and leadership in the field have been fundamental in the development of the subject, and his influence continues to be felt though his work and the work of his many students, collaborators, and mentees. This volume contains the proceedings of an AMS Special Session Operator Algebras and Their Applications: A Tribute to Richard V. Kadison, held on January 10-11, 2015, in San Antonio, Texas. The table of contents reveals contributions by an outstanding group of internationally known mathematicians. Most of the papers are 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 been carefully refereed and will not appear in print elsewhere. All of the contributors are esteemed members of the mathematical community, and for this reason we have elected to simply present the papers in alphabetical order by the first-named author. We the editors thank everyone who participated in both the AMS Special Session and the preparation of this volume. Without the hard work of the authors and the referees, as well as the editorial staff of the American Mathematical Society, this volume would have never seen the light of day. We especially thank Christine M. Thivierge for her invaluable assistance and patience. In addition, we thank Gestur Olafsson 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 of operator 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. Araizu San Jose State University

George Elliott University of Toronto

Joe Ball Virginia Tech University

Adam Fuller University of Nebraska, Lincoln

Paul Baum Pennsylvania State Unversity

Liming Ge University of New Hampshire and Chinese Academy of Sciences

Alex Bearden University of Houston

Elizabeth Gillaspy University of Colorado, Boulder

David P Blecher University of Houston

James Glimm Stony Brook University

R´emi Boutonnet University of California, San Diego

Jan Gregus Abraham Baldwin Agricultural College

Michael Brannan University of Illinois, Urbana-Champaign

Benjamin Hayes Vanderbilt University

Joel Cohen University of Maryland

Nigel Higson Pennsylvania State University

Ken Davidson University of Waterloo

Richard Kadison University of Pennsylvania

Bruce Doran Accenture

David Kerr Texas A&M University

Bob Doran Texas Christian University Ronald Douglas Texas A&M University

Magnus Landstad Norwegian University of Science and Technology

Ken Dykema Texas A&M University

David Larson Texas A&M University

Edward Effros University of California, Los Angeles

Zhe Liu University of Central Florida

Søren Eilers University of Copenhagen

Jireh Loreaux University of Cincinnati xi

xii

LIST OF PARTICIPANTS

Terry Loring University of New Mexico

Mikael Røordam University of Copenhagen

Martino Lupini York University (Canada)

Jonathan Rosenberg University of Maryland

Ellen Maycock American Mathematical Society

Christopher Schafhauser University of Nebraska, Lincoln

Azita Mayeli City University of New York

Mohamed W. Sesay Howard University

Matt McBride University of Oklahoma

Juhhao Shen University of New Hampshire

Niels Meesschaert KU Leuven (Belgium)

Fred Shultz Wellesley College

Ramis Movassagh MIT and Northeastern University

Roger Smith Texas A&M University

Paul Muhly University of Iowa

Baruch Solel Technion (Israel)

Magdalena Musat University of Copenhagen

Myungsin-Sin Song Southern Illinois University

Pieter Naaijkens Leibniz Univerit¨ at Hannover Judith Packer University of Colorado, Boulder Efton Park Texas Christian University Geoffry Price United States Naval Academy Sorin Popa University of California, Los Angeles Ian Putnam University of Victoria Timothy Rainone Texas A&M University and University of Waterloo Kamran Reihani Texas A&M University Marc A Rieffel University of California, Berkeley Min Ro University of Oregon

Erling Størmer University of Oslo Wojciech Szymansk University of Southern Denmark Mark Tomforde University of Houston John Vastola University of Central Florida Henry Warchall National Science Foundation Gary Weiss University of Cincinnati Alan Wiggins University of Michigan, Dearborn Wei Zhang Purdue University

Contemporary Mathematics Volume 671, 2016 http://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 Kadison on the occasion of his ninetieth birthday. Abstract. We survey results connecting exactness in the sense of C ∗ -algebra theory, coarse geometry, geometric group theory, and expander graphs. We summarize 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 and these monsters lead to counterexamples to Baum-Connes type conjectures: the recent work of Osajda allows us to give a more streamlined approach than currently exists elsewhere in the literature. We then summarize our work on reformulating the Baum-Connes conjecture using exotic crossed products, and show that many counterexamples to the old conjecture give confirming examples to the reformulated one; our results in this direction are a little stronger than those in our earlier work. Finally, we give an application of the reformulated Baum-Connes conjecture to a version of the Kadison-Kaplansky conjecture on idempotents in group algebras.

1. Introduction The Baum-Connes conjecture relates, in an important and motivating special case, the topology of a closed, aspherical manifold M to the unitary representations of its fundamental group. Precisely, it asserts that the Baum-Connes assembly map (1.1)

∗ K∗ (M ) → K∗ (Cred (π1 (M ))

is an isomorphism from the K-homology of M to the K-theory of the reduced C ∗ -algebra of its fundamental group. The injectivity and surjectivity of the BaumConnes assembly map have important implications—injectivity implies that the higher signatures of M are oriented homotopy invariants (the Novikov conjecture), and that M (assumed now to be a spin manifold) does not admit a metric of positive scalar curvature (the Gromov-Lawson-Rosenberg conjecture); surjectivity implies that 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 was partially 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 involve some sort of large-scale or coarse geometric hypothesis on the universal cover of the manifold M . A sample result, important in the context of this piece, is that if the universal cover of M is coarsely embeddable in a Hilbert space, then the Baum-Connes assembly map is injective [64, 77], so that the Novikov conjecture holds for M . For some time, it was thought possible that every bounded geometry metric space was coarsely embeddable in Hilbert space. At least there was no counterexample to this statement until Gromov made the following assertions [52]: an expander does not coarsely embed in Hilbert space, and there exists a countable discrete group that ‘contains’ an expander in an appropriate sense. With the appearance of this influential paper, there began a period of rapid progress on counterexamples to the Baum-Connes conjecture [37], and other conjectures. In particular, these so-called Gromov monster groups were found to be counterexamples to the BaumConnes conjecture with coefficients; they were also found to be the first examples of non-exact groups in the sense of C ∗ -algebra theory; and expander graphs were found to be counterexamples to the coarse Baum-Connes conjecture. We mention that, while counterexamples to most variants of the Baum-Connes conjecture have been found, there is still no known counterexample to the conjecture as we have stated 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 intimately related. This point of view is not particularly novel—the counterexamples given by Higson, Lafforgue and Skandalis all have the failure of exactness as their root cause [37]. More recent work has moreover suggested that at least some of the counterexamples can be obviated by forcing exactness [21, 25, 57, 74, 75]. We shall exploit this point of view to reformulate the conjecture by replacing the reduced C ∗ -algebra of the fundamental group, and the associated reduced crossed product that is used when coefficients are allowed, on the right hand side by a new group C ∗ -algebra and crossed product; the new crossed product will by its definition be exact. By doing so, we shall undercut the arguments that have lead in the past to the counterexamples, and indeed, we shall see that some of the former counterexamples are confirming examples for the new, reformulated conjecture. To close this introduction, we give a brief outline of the paper. The first several sections are essentially historical. In Sections 2 and 3 we provide background information on exact groups, crossed products, and the group theoretic and coarse geometric properties relevant for the theory surrounding the Baum-Connes and Novikov conjectures. We discuss the relationships among these properties, and their connection to other areas of C ∗ -algebra theory. Section 4 contains a detailed discussion of expanders, focusing on the aspects of the theory necessary to produce the counterexamples that will appear in later sections. Here, we follow an approach outlined by Higson in a talk at the 2000 Mount Holyoke conference, but updated to a slightly more modern perspective. For the Baum-Connes conjecture itself, expanders provide counterexamples through the theory of Gromov monster groups. In Section 5 we describe the history and recent progress on the existence of these groups, beginning with the original paper of 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 for the Baum-Connes conjecture itself. We begin in Section 6 by recalling the necessary machinery to define the conjecture, focusing on the aspects needed for the subsequent discussion. In Section 7 we describe how Gromov monster groups give counterexamples to the conjecture, by essentially reducing to the discussion of expanders given in Section 4. In Section 8 we describe how to adjust the right hand side of the Baum-Connes conjecture by replacing the reduced C ∗ -algebraic crossed product with a new crossed product functor having better functorial properties and in Section 9 we explain how and why this reformulated conjecture outperforms the original by verifying it in the setting of the counterexamples from Section 7. In Section 10 we give an application of the reformulated conjecture to the KadisonKaplansky 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 makes sense for arbitrary (second countable) locally compact groups, and indeed this is the level of generality we worked at in our original paper [10]. Here, we restrict to the discrete case because it is the most relevant for non-exact groups, and because it simplifies some details. A G-C ∗ -algebra is a C ∗ -algebra equipped with an action α of G by ∗-automorphisms. The natural representations for G-C ∗ -algebras are the covariant representations: these consist of a C ∗ -algebra representation of A as bounded linear operators on a Hilbert space H, together with a unitary representation of G on the same Hilbert space, π : A → B(H)

and

u : G → B(H),

satisfying the covariance condition π(αg (a)) = ug π(a)ug−1 . Essentially, a covariant representation spatially implements the action of G on A. Crossed products of a G-C ∗ -algebra A encode both the algebra A and the Gaction into a single, larger C ∗ -algebra. We introduce the notation A alg G for the algebra of finitely supported A-valued functions on G equipped with the following product and involution:  f1 (h)αg (f2 (h−1 g)) and f ∗ (t) = αg (f (g −1 )∗ ). f1  f2 (g) = h∈G

The algebra A alg G is functorial for G-equivariant ∗-homomorphisms in the obvious way. We shall refer to A alg G as the algebraic crossed product of A by G. Finally, a covariant representation integrates to a ∗-representation of A alg G according to the formula  π(f (g))ug . π  u(f ) = g∈G

Two completions of the algebraic crossed product to a C ∗ -algebra are classically studied: the maximal and reduced crossed products. The maximal crossed product is 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 covariant representation integrates (uniquely) to a representation of A max G; indeed, it

4

PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

is characterized by this property. The reduced crossed product is defined to be the image of the maximal crossed product in the integrated form of a particular covariant representation. Precisely, fix a faithful and non-degenerate representation π of A on a Hilbert space H and define a covariant representation π  : A → B(H ⊗ 2 (G))

λ : G → B(H ⊗ 2 (G))

and

by π (a)(v ⊗ δg ) = π(αg−1 (a))v ⊗ δg

λh (v ⊗ δg ) = v ⊗ δhg .

and

The reduced crossed product A red G is the image of A max G under the integrated form of this covariant representation. In other words, A red G is the completion of A alg G for the norm π  λ(f ). f red =  The reduced crossed product (and its norm) are independent of the choice of faithful and non-degenerate representation of A. Incidentally, one may check that π   λ is injective on A alg G. It follows that the maximal norm is in fact a norm on the algebraic 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 of the maximal and reduced crossed products as a dense ∗-subalgebra. Kirchberg and Wassermann introduced, in their work on continuous fields of C ∗ -algebras, the notion of an exact group [42]. They define a group G to be exact if, for every short exact sequence of G-C ∗ -algebras (2.1)

0

/I

/A

/B

/0

the corresponding sequence of reduced crossed products / I red G

0

/ A red G

/ B red G

/0

is itself short exact. Several remarks are in order here. First, the map to B red G is always surjective, the map from I red G is always injective, and the composition of the two non-trivial maps is always zero. In other words, exactness of the sequence can only fail in that the image of the map into A red G may be properly contained in the kernel of the following map. Second, the sequence obtained by using the maximal crossed product (instead of the reduced) is always exact; this follows essentially from the universal property of the maximal crossed product. There is a parallel theory of exact C ∗ -algebras in which one replaces the reduced crossed products by the spatial tensor products. In particular, a C ∗ -algebra D is exact if for every short exact sequence of C ∗ -algebras (2.1)—now without group action—the corresponding sequence 0

/ I ⊗D

/ A⊗D

/ B⊗D

/0

is exact. Here, we use the spatial tensor product; the analogous sequence defined using the maximal tensor products is always exact, for any D. In the present context, the theories of exact (discrete) groups and exact C ∗ -algebras are related by 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 reduced C ∗ -algebra is exact (as a C ∗ -algebra).

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

5

A central notion for us will be that of a crossed product functor . By this we shall mean, for each G-C ∗ -algebra A a completion A τ G of the algebraic crossed product fitting into a sequence A max G → A τ G → A red G, in which each map is the identity when restricted to Aalg G (a dense ∗-subalgebra of each of the three crossed product C ∗ -algebras). This is equivalent to requiring that the 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 sequence of 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 every group, but the reduced crossed product is exact only for exact groups. We shall see other 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 study of the Baum-Connes conjecture, and for issues related to exactness: a-T-menability of groups and coarse embeddability of metric spaces, and their relation to various forms of amenability. The following definition—due to Gromov [26, Section 7.E]—is fundamental for work on the Baum-Connes conjecture. 3.1. Definition. A countable discrete group G is a-T-menable if it admits an affine isometric action on a Hilbert space H such that the orbit of every v ∈ H tends 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 finite subset 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 the moment 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 their inverses. We define the associated word length by declaring the length of an element g to be the minimal length of such a word; we denote this by |g|. This word length function is a proper length function, meaning that it is a non-negative real valued function 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 general setting, it is well kown (and not difficult to prove) that a countable discrete group admits a proper length function. We now equip G with a proper length function, for example a word length, and define the associated metric on G by d(g, h) = |g −1 h|. This metric has bounded

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PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

geometry, meaning that there is a uniform bound on the number of elements in a ball of fixed radius. It is also left-invariant, meaning that the the left action of G on itself is by isometries. This metric is not intrinsic to G, but depends on the particular length function chosen. Nevertheless, the identity map is a coarse equivalence between any two bounded geometry, left invariant metrics on G. We shall expand on this fact below. Thus, it makes sense to attribute metric properties to 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 (in Hilbert 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 that G acts on a Hilbert space H as in Definition 3.1. Fix v ∈ H and notice that g · v − h · v = g −1 h · v − v ∼ g −1 h · v (where ∼ means differing at most by a universal additive constant). Thus, forgetting the action and recalling that the metric on G has bounded geometry, we see that the orbit map f (g) = g · v is a coarse embedding as in Definition 3.2. 3.4. Remark. Only the metric structure of H enters into the definition of coarse embeddability; the same definition applies equally well to maps from one metric space to another. In this more general setting, a coarse embedding f : X → Y of metric spaces is a coarse equivalence if for some universal constant C every element of Y is a distance at most C from f (X). It is in this sense that the identity map on G is a coarse equivalence between any two bounded geometry, left invariant metrics. The key point here is that the balls centered at the identity in each metric are finite, so that the length function defining each metric is bounded on the balls for the other. To motivate the relevance of these properties for the Baum-Connes and Novikov conjectures suppose, for example, that G acts on a finite dimensional Hilbert space as in Definition 3.1. It is then a discrete subgroup of some Euclidean isometry group Rn  O(n) (at least up to taking a quotient by a finite subgroup). That such groups satisfy the Baum-Connes conjecture follows already from Kasparov’s 1981 conspectus [41, Section 5, Lemma 4], which predates the conjecture itself! The relevance of the general, infinite dimensional version of a-T-menability, and of 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 Bott periodicity theorem of Higson, Kasparov, and Trout [36]. One has the following theorem: 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]. 1 Gromov originally used the terminology uniformly embeddable and uniform embedding; this usage has fallen out of favor since it conflicts with terminology from Banach space theory.

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

7

3.5. Theorem. Let G be a countable discrete group. The Baum-Connes assembly map with coefficients in any G-C ∗ -algebra A K∗top (G; A) → K∗ (A red G) is an isomorphism if G is a-T-menable, and is split injective if G is coarsely embeddable. The class of a-T-menable groups is large: it contains all amenable groups as well as free groups, and classical hyperbolic groups; see [22] for a survey. AT-menability admits many equivalent formulations: the key results are those of Akemann and Walter studying positive definite and negative type functions [1], and of Bekka, Cherix and Valette relating a-T-menabilty as defined above to the properties studied by Akemann and Walter [12] (the latter are usually called the Haagerup property due to their appearance in important work of Haagerup [31] on C ∗ -algebraic approximation results). There are, however, many non a-T-menable groups: the most important examples are those with Kazhdan’s property (T ) such as SL(3, Z): see the monograph [13]. The class of coarsely embeddable groups is huge: as well as all a-T-menable groups, it contains for example all linear groups (over any field) [29] and all Gromov hyperbolic groups [60]. Indeed, for a long time it was unknown whether there existed 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 useful here that coarse emeddability makes sense for arbitrary metric spaces, and not just groups. Before we turn to a discussion of expanders in the next section, we shall describe the close relationship of coarse embeddability to exactness and some other properties of metric spaces, groups and group actions. The key additional idea is that of Property A, which was introduced by Yu to be a relatively easily verified criterion for coarse embeddability [77, Section 2]. Property A was quickly realized to 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 of positive definite kernels, and doing so brings the distinctions among them into sharp focus. Recall that a (normalized) positive definite kernel on a set X is a function 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 

ai aj k(xi , xj ) ≥ 0.

i,j=1

If we are working with a countable discrete group G we may additionally require the kernel to be left invariant, in the sense that k(g1 g, g1 h) = 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 exists a positive definite kernel k : X × X → C such that (i ) |1 − k(x, y)| < whenever d(x, y) < R; (ii ) the set { d(x, y) ∈ [0, ∞) : k(x, y) = 0 } is bounded;

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PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

X is coarsely embeddable if and only if for every R and ε there exists a positive definite kernel k satisfying (i ) above, but instead of (ii ) the following weaker condition: (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 there exists a left invariant positive definite kernel satisfying (i ) and (ii ) above; it is a-T-menable if and only if for every R and ε there exists a left invariant positive definite kernel satisfying (i ) and (ii )’ above. The characterization of Property A for bounded geometry, uniformly discrete metric spaces in this theorem is due to Tu [66]. It is particularly useful for studying C ∗ -algebraic approximation properties, in particular exactness, as it can be used to construct so-called Schur multipliers. The characterization of coarse embeddability can 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 previous theorem, summarizes the properties we have discussed: (3.1)

amenability

+3 Property A ≡ exactness

 a-T-menability

 +3 coarse embeddability .

The class of groups with Property A covers all the examples of coarsely embeddable groups mentioned earlier. Indeed, proving the existence of groups without Property A is as difficult as proving the existence groups that do not coarsely embed. Nonetheless, Osajda has recently shown the existence of coarsely embeddable (and even a-T-menable) groups without Property A [56]. In particular, there are no further implications between any of the properties in the diagram. The following theorem summarizes some of the most important implications relating Property A to C ∗ -algebra theory. 3.7. Theorem. Let G be a countable discrete group. The following are equivalent: (i ) (ii ) (iii ) (iv )

G has Property A; G admits an amenable action on a compact space; G is an exact group; ∗ (G) is an exact C ∗ -algebra. the group C ∗ -algebra Cred

The reader can see the survey [72] or [17, Chapter 5] for proofs of most of these results, as well as the definitions that we have not repeated here. The original references are: [38] for the equivalence of (i) and (ii); [30] (partially) and [58] for the equivalence of (i) and (iv); [43, Theorem 5.2] for (iv) implies (iii) as we already discussed in Section 2; and (iii) implies (iv) is easy. Almost all these implications extend to second countable, locally compact groups with appropriately modified versions of Property A [2,16,24]; the exception is (iv) implies (iii), which is an open question in general. Finally, note that Theorem 3.7 has a natural analog in the setting of discrete metric spaces: see Theorem 4.5 below.

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

9

3.8. Remark. There is an analog of the equivalence of (i) and (ii) for coarsely embeddable groups appearing in [64, Theorem 5.4]: a group is coarsely embeddable if and only if it admits an a-T-menable action on a compact space in the sense of Definition 9.2 below. 4. Expanders In this section, we study expanders: highly connected, sparse graphs. Expanders are the easiest examples of metric spaces that do not coarsely embed. They are also connected to K-theory through the construction of Kazhdan projections; this construction is at the root of the counterexamples to the Baum-Connes conjecture. For our purposes, a graph is a simplicial graph, meaning that we allow neither loops nor multiple edges. More precisely, a graph Y comprises a (finite) set of vertices, 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 containing them, and we write x ∼ y in this case. The number of vertices incident to a given vertex 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   f (y) = f (x) − f (y). Δf (x) = deg(x)f (x) − y : y∼x

y : y∼x

A straightforward calculation shows that Δ is a positive operator; in fact  (4.1)  Δf, f  = |f (x) − f (y)|2 ≥ 0. (x,y) : x∼y

The kernel of the Laplacian on a connected graph is precisely the space of constant functions. Indeed, it follows directly from the definition that constant functions belong to the kernel; conversely, it follows from (4.1) that if Δf = 0 and x ∼ y then f (x) = f (y), so that f is a constant function (using the connectedness). Thus, the second smallest eigenvalue (including multiplicity) of the Laplacian on a connected graph is strictly positive. We shall denote this eigenvalue by λ1 (Y ). An expander is a sequence (Yn ) of finite connected graphs with the following properties: (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 expander is about having both the degree bounded above, and the first eigenvalue bounded away from 0, independent of n. The existence of expanders can be proven with routine counting arguments which in fact show that in an appropriate sense most graphs are expanders: see [47, Section 1.2]. Nevertheless, the explicit construction of expanders was elusive. The first construction was given by Margulis [49]. Shortly thereafter the close connection with Kazhdan’s Property (T) was understood—the collection of finite quotients of a residually finite discrete group with Property (T) are expanders, when equipped with the (Cayley) graph structure coming from a fixed finite generating set of the parent group, and ordered so that

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PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

their cardinalities tend to infinity [47, Section 3.3]. In particular, the congruence quotients of SL(3, Z) are an expander. In the present context, the first counterexamples provided by expanders are to questions in coarse geometry. Given a sequence Yn of graphs comprising an expander, we consider the associated box space, which is a metric space Y with the following 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 possible number of edges on a path connecting them. It is not difficult to construct a box space; one simply declares that the distance from a vertex in Ym to a vertex in the union of the Yn for n < m is sufficiently large. Further, the identity map provides a coarse equivalence (see Remark 3.4) between any two box spaces, so that their coarse geometry is well defined. 4.1. Proposition. A box space associated to an expander sequence is not coarsely embeddable, and hence does not have property A. This proposition was originally stated by Gromov, and proofs were later supplied 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 various types of expanders in various types of Banach space, and other non-positively curved spaces. 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, connected, finite graph Y , we have the projection p onto the constant functions, which is to say, onto the kernel of the Laplacian. This projection can be obtained as a spectral function of the Laplacian; precisely, p = f (Δ) provided that f (0) = 1 and that f ≡ 0 on the remaining eigenvalues of Δ. Now suppose that Y is the box space of a sequence Yn of finite graphs with uniformly bounded vertex degrees. We can then consider the operators

(4.2)

⎛ p1 ⎜0 ⎜ p=⎜0 ⎝ .. .

0 p2 0 .. .

0 0 p3 .. .

⎞ ... . . .⎟ ⎟ . . .⎟ ⎠ .. .

⎛ Δ1 ⎜0 ⎜ and Δ = ⎜ 0 ⎝ .. .

0 Δ2 0 .. .

0 0 Δ3 .. .

⎞ ... . . .⎟ ⎟ . . .⎟ ⎠ .. .

acting on 2 (Y ), identified with the direct sum of the spaces 2 (Yn ). While p will not generally 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 and f ≡ 0 on [c, ∞) then p = f (Δ). We shall refer to p as the Kazhdan projection of the expander. The importance of the Kazhdan projection is difficult to overstate: one can often show that Kazhdan projections are not in the range of Baum-Connes type assembly maps, and are therefore fundamental for counterexamples. This is best understood in the context of metric spaces, and to proceed we need to introduce the coarse geometric analog of the group C ∗ -algebra. For convenience, we consider here

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

11

only the uniform Roe algebra of a (discrete) metric space X. A bounded operator T on 2 (X) has finite propagation if there exists R > 0 such that T cannot propagate signals over a distance greater than R: precisely, for every finitely supported function f on X, the support of T (f ) is contained in the R-neighborhood of the support 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 uniform Roe 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 Kazhdan projection p is not compact, and belongs to C ∗ (Y ). Proof. The Kazhdan projection has infinite rank; it projects onto the space of functions that are constant on each Yn . Further, the Laplacian propagates signals a distance at most 1, so that both Δ and its spectral function p = f (Δ) belong to  the 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 element T ∈ C ∗ (X) whose ‘matrix entries tend to 0 at infinity’; precisely, the suprema sup |Txz |

and

z∈X

sup |Tzx |

z∈X

of matrix entries over the ‘xth row’ and ‘xth column’ tend to zero as x tends to infinity. 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 is compact. The Kazhdan projection in C ∗ (Y ) is a (non-compact!) ghost because the elements pn in its matrix representation (4.2) are ⎛

1 1 ... ⎜ 1 ⎜1 1 . . . pn = ⎜. . card(Yn ) ⎝ .. .. . . . 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 identify its 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, that is, the elements of the Stone-Cech corona β∞ X = βX \ X. A bounded function φ on X has a limit against each ultrafilter. If an ultrafilter corresponds to a point of X this limit is simply the evaluation of φ at that point; if ω is a free ultrafilter, we shall denote the limit by ω-lim(φ). Suppose now we are given a (free) ultrafilter ω ∈ β∞ X. We define a linear functional 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 in the standard basis of 2 (X). We check that Ω is a state on C ∗ (X). Indeed, one checks immediately that Ω(1) = 1, and a simple calculation shows that the diagonal

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PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

entries of T ∗ T are given by (T ∗ T )xx =

(4.3)



|Tzx |2 ;

z ∗ 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 representations of 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 in the 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 have that for every R and S ∈ C ∗ (X) the product RT S is also a ghost. In particular, its on-diagonal matrix entries (RT S)xx tend to zero as x → ∞, so that their limit against every free ultrafilter is also zero. This mean that the norm of T in the GNS representation associated to every free ultrafilter is zero, so that T maps to zero in ∗ (X). C∞ ∗ 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 free ultrafilter, so that they converge to zero in the ordinary sense as x → ∞. Now, according to (4.3) we have  |Tzx |2 ≥ sup |Tzx |2 , (T ∗ T )xx = z

z

so that supz |Tzx |2 → 0 as x → ∞ as well. Applying the same argument to T T ∗ 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 a short sequence (4.4)

0

/K

/ C ∗ (Y )

∗ / C∞ (Y )

/ 0.

The sequence is not exact because the Kazhdan projection belongs to the kernel of the quotient map, although it is not compact. As we shall now show, it is possible to detect the K-theory class of the Kazhdan projection and to see that the sequence is not exact even at the level of K-theory. We shall see in Section 7 that this is the phenomenon underlying the counterexamples to the Baum-Connes conjecture. 4.4. Proposition. The K-theory class of the Kazhdan projection is not in the image of the map K0 (K) → K0 (C ∗ (Y )). Proof. We have, for each ‘block’ Yn a contractive linear map C ∗ (Y ) → B( (Yn )) defined by cutting down by the appropriate projection. These are asymptotically multiplicative on the algebra of finite propagation operators, and we obtain a ∗-homomorphism

B(2 (Yn )) ∗ . (4.5) C (Y ) → n ⊕n B(2 (Yn )) 2

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

13

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

 B(2 (Yn )) → Z. (4.6) K0 n

n

As K1 (⊕n B(2 (Yn )) = 0, the six term exact series in K-theory specializes to an exact sequence   2 n B( (Yn )) /0. / K0 ( B(2 (Yn ))) / K0 K0 (⊕n B(2 (Yn ))) 2 n ⊕n B( (Yn ))

Composing the ‘rank homomorphism’ in line (4.6) with the quotient map n Z →

2 n Z/ ⊕n Z clearly annihilates the image of K0 (⊕n B( (Yn ))), and thus from the sequence above gives rise to a homomorphism



 2 n B( (Yn )) nZ . K0 → ⊕n B(2 (Yn )) ⊕n Z Finally, combining with the K-theory map induced by the ∗-homomorphism in line (4.5) gives a group homomorphism

Z ∗ K0 (C (Y )) → n . ⊕n Z Any K-theory class in the image of the map K0 (K) → K0 (C ∗ (Y )) goes to zero under the map in the line above. On the other hand, the Kazhdan projection restricts to a rank one projection on each 2 (Yn ) and therefore its image is [1, 1, 1, 1, . . . ], and so is non-zero.  The failure of the above sequence (4.4) to be exact is at the base of many failures of exactness and other approximation properties in operator theory and operator algebras. See for example the results of Voiculescu [70] and Wassermann [71]. More recently, the results in the following theorem (an analog of Theorem 3.7 for metric spaces) have been filled in, clarifying the relationship between Property A, ghosts, amenability and exactness. Note that the box space of an expander, or a countable 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 the equivalence of (i) and (ii), and the definition of the coarse groupoid; [62] for the equivalence 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 BaumConnes conjecture began in earnest with the provocative remarks found in the last section of [52]. There, Gromov describes a model for a random presentation

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PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

of a group and asserts that under certain conditions such a random group will almost surely not be coarsely embeddable in Hilbert space, or in any p -space for finite p. The non-embeddable groups arise by randomly labeling the edges of a suitable 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 expander have large girth: the length of the shortest cycle in the nth constituent graph tends to infinity with n. Thus labeled, cycles in the expander graphs give words in the generators which are viewed as relators in an (infinite) presentation of a random group. Gromov then goes on to state that a further refinement of the method would reveal that certain of these random and non-embeddable groups are themselves subgroups of finitely presented groups, which are therefore also non-embeddable. More details appeared in the subsequent paper of Gromov [27], and in the further work of Arzhantseva and Delzant [3]. From the above sketch given by Gromov, it is immediately clear that the original expander graphs Yn would in some sense be ‘contained’ in the Cayley graph of the random group G. And groups ‘containing expanders’ became known as Gromov monsters. As is clearly explained in a recent paper of Osajda [56], it is an inherent limitation of Gromov’s method that the expanders will not themselves be coarsely embedded in the random group. Rather, they will be ‘contained’ in the following weaker sense: there exist constants a, b and cn such that cn is much smaller than the diameter of Yn , and such that for each n there exists a map fn : Yn → G satisfying 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 is nevertheless sufficient for the non-embeddability of G, and for the counterexamples of 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 adopt the following more restricted notion of Gromov monster group. 5.1. Definition. A Gromov monster (or simply monster ) group is a discrete group G, equipped with a fixed finite generating set and which has the following property: there exists a subset Y of G which is isometric to a box space of a large girth, constant degree, expander. Here, it is equivalent to require that each of the individual graphs Yn comprising the expander are isometrically embedded in G; using the isometric action of G on itself, it is straightforward to arrange the Yn (rather, their images in G) into a box space. Building on earlier work with Arzhantseva [5], groups as in this definition were shown to exist by Osajda: see [56, Theorem 3.2]. We recall in rough outline the method. The basic data is a sequence of finite, connected graphs Yn of uniformly finite 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

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

15

will, of course, be taken to be a suitable family of expanders. These conditions are less restrictive than those originally proposed by Gromov, and in his paper Osajda describes an explicit set of expanders that satisfy them. Using a combination of combinatorial and probabilistic arguments, Osajda produces two labelings of the edges in the individual Yn with letters from a finite alphabet: one satisfies a small cancellation condition for pieces from different blocks and the other for pieces from a common block. He then combines these in a straightforward way to obtain a labeling that globally satisfies the C  (1/24) small cancellation condition. The monster group G is the quotient of the free group on the letters used in the final labeling by the normal subgroup generated by the relations read along the cycles of the graphs Yn . It was known from previous work that the C  (1/24) condition implies that the individual Yn will be isometrically embedded in 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 fundamental groups of closed aspherical manifolds, and one may prefer to confine attention to finitely presented groups. Fully realizing Gromov’s original statement, Osajda remarks that a general method developed earlier by Sapir [63] leads to the existence of closed, aspherical manifolds whose universal covers exhibit similar pathologies. 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 contain quasi-isometrically embedded expanders. While groups as in the second statement of this theorem would not qualify as Gromov monster groups under our restricted definition above, their existence is very satisfying. We shall close this section with a more detailed discussion of the relationship between the properties introduced in Section 3. As we mentioned previously, none of the implications in diagram (3.1) is reversible. The most difficult point concerns the existence of discrete groups (or even bounded geometry metric spaces) that are coarsely embeddable but do not have Property A. The first example of such a space was given by Arzhantseva, Guentner and Spakula [4] (non-bounded geometry examples were given earlier by Nowak [54]); their space is the box space in which the blocks are the iterated Z/2-homology covers of the figure-8 space, i.e. the wedge of two circles. In the case of groups, a much more ambitious problem is the existence of a discrete group which is a-T-menable, but does not have Property A. Building on earlier work with Arzhantseva, this problem was recently solved by Osajda [5, 56]. The strategy is similar to the construction of Gromov monsters: use a graphical small cancellation technique to embed large girth graphs with uniformly bounded degree (at least 3) into the Cayley graph of a finitely generated group. Again, under the C  (1/24) hypothesis the graphs will be isometrically embedded. The large girth hypothesis and the assumption that each vertex has degree at least 3 ensure, by a result of Willett [73], that the group will not have Property A. The remaining difficulty is to show that the group constructed is a-T-menable under appropriate hypotheses on the graphs and the labelling. The key idea is due to Wise, who showed that certain finitely presented classical small cancellation

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PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

groups are a-T-menable, by endowing their Cayley graphs with the structure of a space 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 considered it as a higher index map (6.1)

∗ (G)), K∗ (M ) → K∗ (Cred

which takes the K-homology class defined by an elliptic differential operator D on a closed aspherical manifold M with fundamental group G to the higher index of D in the K-theory of the reduced group C ∗ -algebra of G. This point of view is perhaps the most intuitive way to view the conjecture and also leads to some of its most important applications. Here however, we need to get ‘under the hood’ of the Baum-Connes machinery, and give enough definitions so that we can explain our constructions. To formulate the conjecture more generally, and in particular to allow coefficients in a G-C ∗ -algebra, it is usual to use bivariant K-theory and the notion of descent. Even if one is only interested in the classical conjecture of (6.1), the extra generality is useful as it grants access to many powerful tools, and has much better naturality and permanence properties under standard operations on groups. There are two standard bivariant K-theories available: the KK-theory of Kasparov, and the E-theory of Connes and Higson. These two theories have similar formal properties, and for our purposes, it would not make much difference which theory we use (see Remark 8.2 below). However, at the time we wrote our paper [10] it was only clear how to make our constructions work in E-theory, and for the sake of consistency we use E-theory here as well. We continue to work with a countable discrete group G. We shall denote the category whose objects are G-C ∗ -algebras and whose morphisms are equivariant ∗-homomorphisms by GC* ; similarly, C* denotes the category whose objects are C ∗ -algebras and whose morphisms are ∗-homomorphisms. Further, we shall assume that 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 morphism sets. 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 the morphisms sets in EG by E G (A, B). These are abelian groups, and it follows that for a fixed G-C ∗ -algebra B, the assignments A → E G (A, B)

and

A → E G (B, A)

are, respectively, a contravariant and a covariant functor from GC* to the category of abelian groups. Let now EG denote a universal space for proper actions of G; this means that EG is a metrizable space equipped with a proper G-action such that the quotient space is also metrizable, and moreover that any metrizable proper G-space admits a continuous equivariant map into EG, which is unique up to equivariant homotopy. Such spaces always exist [8]. Suppose X ⊆ EG is a G-invariant and cocompact subset; this means that X is closed and that there is a compact subset K ⊆ EG such

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

17

that X ⊆ G · K. Such an X is locally compact (and Hausdorff), and if X ⊆ Y are two 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 Ginvariant and cocompact subsets of EG, becomes a directed set of G-C ∗ -algebras and equivariant ∗-homomorphisms. It follows from this discussion that for any G-C ∗ -algebra A we may form the direct limit K0top (G; A) :=

lim X⊆EG

E G (C0 (X); A),

X cocompact

and similarly for K1 using suspensions. The universal property of EG together with homotopy invariance of the E-theory groups implies that K∗top (G; A) does not depend on the choice of EG up to unique isomorphism. It is called the topological K-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 construction of the equivariant E-theory category to the trivial group gives the Etheory category, which we shall denote by E. The objects in this category are the C ∗ -algebras, and the morphisms from A to B are an abelian group denoted E(A, B). A ∗-homomorphism A → B gives a morphism in this category, and there is a covariant functor from the category of C ∗ -algebras and ∗-homomorphisms to E that 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 is possible to extend this functor to the category EG , so that it becomes defined on the generalized morphisms belonging to EG but not to GC* ; see [28, Theorem 6.22] for a proof. 6.1. Theorem. There is a (maximal ) descent functor max : EG → E which agrees with the usual maximal crossed product functor both on objects and on morphisms in EG coming from equivariant ∗-homomorphisms. To complete the definition of the Baum-Connes assembly map, we need to know that if X is a locally compact, proper and cocompact G-space, then C0 (X) max G contains 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. The K-theory class of the basic projection [pX ] ∈ K0 (C0 (X)max G) = 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 in the statement of the proposition, then the classes [pX ] and [pY ] correspond under the functorially induced map on K-theory.

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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 E G (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 with the 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 is an equivariant inclusion of locally compact, proper, cocompact G-spaces, then the diagram / E(C, A max G) E(C0 (X) max G, 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 appropriate basic projections, and the left hand vertical arrow is composition with the ∗-homomorphism C0 (Y )max G → C0 (X)max G induced by the inclusion X → Y . Hence the assembly maps are compatible with the direct limit defining K0top (G; A), and give a well-defined homomorphism K0top (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 a homomorphism μ : K∗top (G; A) → K∗ (A red G), which is, by definition, the Baum-Connes assembly map. The Baum-Connes conjecture states that this map is an isomorphism. 6.3. Remark. Following through the construction above without passing through the quotient to the reduced crossed product gives the maximal Baum-Connes assembly map μ : K∗top (G; A) → K∗ (A max G). It plays an important role in the theory, but is known not to be an isomorphism in general 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-Connes conjecture 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 way the monster groups constructed by Osajda contain expanders. The existence of counterexamples depends on the following key fact: the left and right hand sides of the Baum-Connes conjecture see short exact sequences of G-C ∗ -algebras differently. To see this, note that the properties of E-theory as discussed in Section 6 imply that the Baum-Connes assembly map is functorial in

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

19

the coefficient algebra: precisely, an equivariant ∗-homomorphism A → B induces a commutative diagram K∗top (G; A)

/ K∗ (A red G)

 K∗top (G; B)

 / K∗ (B red G),

in which the horizontal maps are the Baum-Connes assembly maps, and the vertical maps are induced from the associated morphism A → B in the equivariant E-theory category. The following lemma gives a little more information when the maps come from a short exact sequence. 7.1. Lemma. Let 0

/A

/I

/B

/0

∗

be a short exact sequence of separable G-C -algebras. There is a commutative diagram of Baum-Connes assembly maps K0top (G; I)

/ K top (G; A)

/ K top (G; B)

 K0 (I red G)

 / K0 (A red G)

 / K0 (B red G),

0

0

in which the horizontal arrows are the functorially induced ones. Moreover, the top row is exact in the middle. Proof. The existence and commutativity of the diagram follows from our discussion of E-theory. Exactness of the top row follows from exactness properties of E-theory (see [28, Theorem 6.20]) and the fact that exactness is preserved under direct limits.  The following consequence of the Baum-Connes conjecture with coefficients is immediate 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 conjecture for G with coefficients in all of I, A, and B is true then the corresponding sequence 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 examples where 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 to be 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 action of G on itself. Consider the (non-unital) G-invariant C ∗ -subalgebra of ∞ (G) generated by the functions supported in Y ; as this C ∗ -algebra is commutative, we may

20

PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

as well write it as C0 (W ) where W is its spectrum, a locally compact G-space. Note that as G acts on itself by isometries on the left, the right action of g ∈ G moves elements 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 functions supported within a finite distance from Y . It follows that C0 (W ) contains C0 (G) as an 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 multiplication action of ∞ (G) on 2 (G). Then the pair (M, ρ) is a covariant representation of ∞ (G) for the right G-action. Moreover, it is well-known (compare for example [17, Proposition 5.1.3]) that this pair integrates to a faithful representation of ∞ (G) red G on 2 (G) that takes C0 (G) red G onto the compact operators. As the reduced crossed product preserves inclusions, it makes sense to restrict this representation to C0 (W ) red G, thus giving a faithful representation of C0 (W ) red G on 2 (G). The key facts we need to build our counterexamples are contained in the following lemma. To state it, let C ∗ (Y ) denote the uniform Roe algebra of Y and ∗ C∞ (Y ) the quotient as in Section 4. Represent C ∗ (Y ) on 2 (G) by extending by zero on the orthogonal complement 2 (G \ Y ) of 2 (Y ). 7.3. Lemma. The faithful representations of C0 (W )red G 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 element of C0 (W ). Our first goal is to identify the C ∗ -algebras in the top row of the diagram with the corners of those in the botom row corresponding to the projection χ. We begin with the C ∗ -algebra C0 (W ) red G, which is generated by operators of the form f ρg , where f ∈ C0 (W ) and g ∈ G. The compression of such an operator χ(f ρg )χ : 2 (Y ) → 2 (Y )

(7.1) has matrix coefficients (7.2)

 f (x), y = xg  δx , χf ρg χ(δy )  =  δx , f ρg (δy )  = (f δyg−1 )(x) = 0, else,

for x, y ∈ Y . As discussed above d(x, xg) = |g|, so that the operator in line (7.1) has finite propagation (at most |g|). Hence the corner χ(C0 (W ) red G)χ is contained in C ∗ (Y ). Conversely, suppose T is a finite propagation operator on 2 (Y ). For each g ∈ G define a complex valued function fg on G by   δx , T δxg , x, xg ∈ Y fg (x) = 0, else.

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

21

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

In particular, T belongs to the corner χ(C0 (W )red G)χ. Since this corner contains all the finite propagation operators on Y , we see that it contains C ∗ (Y ) as well. The C ∗ -algebra C0 (∂W ) red G is handled by analogous computations, regarding χ as an element of C0 (∂W ). Finally, under the identification of C0 (G) red G with 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 corners of those in the bottom row corresponding to the projection χ it remains to see that these corners are full. Again, we begin with the C ∗ -algebra C0 (W ) red G. This crossed product is generated by operators of the form f ρg where f is a bounded function with support in the set Y h, for some h ∈ G. Thus, it suffices to show that each such operator belongs to the ideal of C0 (W ) red G generated by χ. Now, the characteristic function of Y h, viewed as an operator on 2 (G), is ρ∗h χρh . It follows that f ρg = f (ρ∗h χρh )ρg = (f ρh−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. Finally, any non-zero projection on 2 (G), and in particular χ, is a full multiplier of C0 (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 Proposition 4.4 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 on K-theory, and so the bottom line is also not exact in the middle: again, the failure of exactness is detected by the class of the Kazhdan projection. Unfortunately, we cannot appeal directly to Corollary 7.2 to show that BaumConnes with coefficients fails for G, as the C ∗ -algebras C0 (W ) and C0 (∂W ) are not separable. To get separable C ∗ -algebras with similar properties, let C0 (Z) be any G-invariant C ∗ -subalgebra of C0 (W ) that contains C0 (G); it follows that Z contains G as a dense open subset, and writing ∂Z = Z \ G gives a short exact sequence of G-C ∗ -algebras. 0

/ C0 (G)

/ C0 (Z)

/ C0 (∂Z)

/0

We want to guarantee that the crossed product C0 (Z) red G contains the Kazhdan projection. There is a straightforward way to do this: our efforts in this section culminate in the following theorem.

22

PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

7.4. Theorem. With notation as above, let C0 (Z) denote any separable Ginvariant C ∗ -subalgebra of C0 (W ) that contains C0 (G) and the characteristic function χ of the expander Y . Then (i ) the crossed product C0 (Z) red G contains the Kazhdan projection associated 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 Laplacian on Y (compare line (4.2) above) is then given by  Δ = dχ − χρg χ g∈G, |g|=1

and is thus in C0 (Z) red G. As both Δ and χ are elements of C0 (Z) red G, the Kazhdan projection p is as well, by the functional calculus. Part (ii) follows from part (i), our discussion of C0 (W ) above, and the commutative 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) is immediate 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 with coefficients in C0 (G) and a chase of the diagram from Lemma 7.1 shows that either surjectivity fails for C0 (Z), or injectivity fails for C0 (∂Z). A more detailed analysis in Theorem 9.7 below shows that in fact the assembly map is an isomorphism with coefficients 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 take the counterexamples from Section 7 into account. The counterexamples to the conjecture stem from analytic properties of the reduced crossed product: a natural way 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 an arbitrary crossed product functor τ . Define the τ -Baum-Connes assembly map to be the composition K∗top (G; A) → K∗ (A max G) → K∗ (A τ G) of the maximal assembly map, and the map induced on K-theory by the quotient map A max G → A τ G; it follows from the discussion in Section 6 that this is the usual Baum-Connes assembly map when τ is the reduced crossed product. And

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

23

one may hope that the τ -Baum-Connes assembly map is an isomorphism under favorable conditions for well behaved τ . One certainly cannot expect all of these ‘τ -Baum-Connes assembly maps’ to be isomorphisms, however: indeed, we have already observed that isomorphism fails for some groups when τ is the reduced crossed product. There are even examples of aT-menable groups and associated crossed products τ for which the τ -Baum-Connes assembly map (with trivial coefficients) is not an isomorphism: see [10, Appendix A]. Considering these examples as well as naturality issues, one is led to the following 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 sequence 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 stably isomorphic if A⊗KG is equivariantly ∗-isomorphic to B ⊗KG . Here, KG denotes 2 the compact operators on the direct sum ⊕∞ 1  (G), equipped with the conjugation action arising from the direct sum of copies of the regular representation. It follows directly from the definition of E-theory that the domain of the Baum-Connes assembly map cannot detect the difference between equivariantly stably isomorphic coefficient algebras. Therefore we would like our crossed product to have the same property: see [10, Definition 3.2] for the precise condition we use. This is a manifestation of Morita invariance. Indeed, separable G-C ∗ -algebras are equivariantly stably isomorphic if and only if they are equivariantly Morita equivalent, as follows from results in [23] and [53], which leads to a general Morita invariance 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 important for proving the conjecture: indeed, following the paradigm established by Kasparov [40], the most powerful known approaches to the Baum-Connes conjecture proceed by proving that certain identities hold in EG (or in the KK G -theory category, or some related more versatile setting as in Lafforgue’s work [45]), and then using descent to deduce consequences for crossed products. Consistency with property (T ). The three properties above hold for the maximal crossed product. However, it is well-known that the maximal crossed product is not the right thing to use for the Baum-Connes conjecture: the Kazhdan projections ∗ (see [67] or [39, Section 3.7]) in Cmax (G) = C max G are not in the image of the top maximal assembly map K∗ (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 the quotient map C max G → C τ G.

24

PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

Summarizing the above discussion, any crossed product that ‘fixes’ the BaumConnes 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 coming from A σ G. The following theorem is one of the main results of [10]; the part dealing 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 product automatically also has properties (iii ) and (iv ). Summarizing, our reformulation of the Baum-Connes conjecture is that the E-Baum-Connes assembly map K∗top (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 among all the crossed products with properties (i) to (iv) above and, for exact groups it is the reduced crossed product. Consequently, for exact groups the reformulated conjecture is nothing other than the original Baum-Connes conjecture. 8.2. Remark. As mentioned above, we chose to work with E-theory, instead of the more common KK-theory to formulate the Baum-Connes conjecture. It is natural to ask whether the development above can be carried out in KK-theory, and in particular whether E admits a descent functor E : KK G → KK. The answer is yes, as long as we restrict as usual to countable groups and separable G-C ∗ -algebras [18]. 9. The counterexamples and the reformulated conjecture In this section we shall revisit the counterexamples presented in Section 7 to the original Baum-Connes conjecture, and study them from the point of view of the reformulated conjecture of Section 8. In particular, we shall continue with the notation 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 ∗ -subalgebra of C0 (W ) containing both C0 (G) and the characteristic function of Y , then the original Baum-Connes conjecture fails for G with coefficients in at least one of C0 (Z) and C0 (∂Z), where again ∂Z = Z \ G. The key point was the failure of exactness 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 .

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

25

In the case of the reformulated conjecture, however, we would be working with the analogous sequence involving the E-crossed product, which is exact (even at the level of C ∗ -algebras). Thus, at this point we know that the proof of Theorem 7.4 will not apply to show the existence of counterexamples to the E-Baum-Connes conjecture. 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 for G with coefficients in both C0 (Z) and C0 (∂Z)! This is a stronger result than in our original paper [10], where we just showed that there exists some Z with the property above. There are two key-ingredients. First, we need a ‘two out of three’ lemma, which will allow us to deduce the E-Baum-Connes conjecture for C0 (Z) from the E-BaumConnes conjecture for C0 (G) and C0 (∂Z). Second, we need to show that the action of G on ∂Z is always a-T-menable: this implies via work of Tu [65] that a strong form 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 in C0 (∂Z). The crucial geometric assumption needed for the second step is that the expander Y has large girth, and therefore looks ‘locally like a tree’. The first ingredient is summarized in the following lemma, which is a more precise 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 conjecture using the reduced crossed product, and the additional assumption that G is exact on the level of K-theory. 9.1. Lemma. Let /A

/I

0

/B

/0

∗

be a short exact sequence of separable G-C -algebras. There is a commutative diagram of six term sequences top / K0top (G; B) / K0top (G; A) K0 (G; I) O NNN NNN NNN NNN NNN NNN NNN NNN NNN NN& NN& NN& / K0 (A E G) / K0 (B E G) K0 (I E G) O top

o NNN NNN NNN NN&

top

K1 (G; B)

K1 (B E G)

o NNN NNN NNN NN&

top

o

K1 (G; A)

K1 (A E G)

 NNN NNN NNN NN&

o

K1 (G; I)



K1 (I E G),

in which the front and back rectangular six term sequences are exact, and the maps from the back sequence to the front are E-Baum-Connes assembly maps. In particular, if the Baum-Connes conjecture holds with coefficients in two out of three of I, 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 action of 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 locally compact space X by homeomorphisms. The action is a-T-menable if there is a continuous function h : X × G → [0, ∞)

26

PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

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 anyfinite subset {g1 , . . . , gn } of G, any finite subset {t1 , . . . , tn } of R n such that i=1 ti = 0, and any x ∈ X, we have that n 

ti tj h(xgi , gi−1 gj ) ≤ 0.

i,j=1

(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 embedded expander Y , and let W and ∂W be as in Section 7 (and as explained at the beginning of this section). Let π : G → Y be any function such that d(x, π(x)) = d(x, Y ) for all x ∈ G. Define a function h : G × G → [0, ∞) by h(x, g) = d(π(x), π(xg)). Then h extends by continuity to a function h : W × G → [0, ∞), and the restriction h : ∂W × G → [0, ∞) has all the properties in Definition 9.2. In particular, the action 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’ on larger and larger sets. One can then use the negative type property of the distance function on a tree to prove that h has the right properties. Now, let C0 (Z) be any separable G-invariant subalgebra of C0 (W ) containing C0 (G) and the characteristic function χ of Y . Set ∂Z = Z \ G. We would like to show 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, ∞) extend to h : Z × G → [0, ∞) (at least for some choice of function π : G → Y with the properties 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 translate 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 is thus compact and open as f is an element of C0 (Z). Finally, note that finitely supported 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 so that the function N → R defined by n → |gn | is non-decreasing. For each x ∈ G, let n(x) be the smallest integer such that xgn(x) is in Y , and define a map π : G → Y by setting π(x) = xgn(x) . Note that d(π(x), x) = d(x, Y ) for all x ∈ G.

EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE

27

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 to the 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 ) and is thus in C0 (Z) by Lemma 9.4. On the other hand, if we let Fm = {x ∈ Nr (Y ) | π(x) = xgm }   m−1 then the characteristic function of Fm equals χEm 1 − i=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  Fm = {x ∈ Nr (Y ) | π(xg) = xgm },  then the characteristic function of Fm is in C0 (Z). For each (k, l) ∈ {1, . . . , n} × {1, . . . , n }, let χk,l denote the characteristic function of Fk ∩ Fl , which is in C0 (Z) by the above discussion. Note that the restriction of hr,g to Fk ∩ Fl sends x ∈ Nr (Y ) to

d(π(x), π(xg)) = d(xgk , xggl ) = |gk−1 ggl |. Hence



hr,g =

n  n 

|gk−1 ggl |χk,l ,

k=1 l=1



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 ∂Z is a-T-menable. Proof. For each fixed g ∈ G, Lemma 9.5 implies that the restriction of the function 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 cover of Z, it follows that hg extends to a continuous function on all of Z. Hence the function 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 from Theorem 9.3 as ∂Z is a quotient of ∂W .  The following corollary is the culmination of our efforts in this section. First, it gives us more information about what goes wrong with the Baum-Connes conjecture than Section 7 does. More importantly for our current work, it shows that the EBaum-Connes conjecture is true for this counterexample: we thus have a concrete class of example where our reformulated conjecture ‘out-performs’ the original one.

28

PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT

9.7. Theorem. Let G be a Gromov monster group with isometrically embedded expander Y . Equip ∞ (G) with the action induced by the right translation action of G on itself, and let C0 (W ) denote the G-invariant C ∗ -subalgebra of ∞ (G) generated by ∞ (Y ). Let C0 (Z) be any separable G-invariant C ∗ -subalgebra of C0 (W ) that contains C0 (G) and the characteristic function χ of Y . Then: (i ) the usual Baum-Connes assembly map for G with coefficients in C0 (Z) is injective; (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 an isomorphism. Proof. The essential point is that work of Tu [65] shows that a-T-menability of the action of G on ∂Z implies that a strong version of the Baum-Connes conjecture for G with coefficients in C0 (∂Z) holds in the equivariant E-theory category EG . This in turn implies the τ -Baum-Connes conjecture for G with coefficients in C0 (∂Z) for any crossed product τ that admits a descent functor. See [10, Theorem 6.2] for more details. The result follows from this, Lemma 9.1, and the fact that the Baum-Connes conjecture is true for any crossed product with coefficients in a proper G-algebra  like C0 (G). 10. The Kadison-Kaplansky conjecture for 1 (G) The Kadison-Kaplansky conjecture states that for a torsion free discrete group ∗ G, there are no idempotents in Cred (G) other than the ‘trivial’ examples given by 0 and 1. It is well-known that the usual Baum-Connes conjecture implies the ∗ (G), the KadisonKadison-Kaplansky conjecture. As 1 (G) is a subalgebra of Cred 1 Kaplansky conjecture implies that  (G) contains no idempotents other than 0 or 1. In this section, we show that the E-Baum-Connes conjecture, and in fact any ‘exotic’ 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-Kaplansky conjecture. Compare [14, Corollary 1.6] for a similar result in the context of the Bost conjecture. 10.1. Theorem. Let G be a countable torsion free group and let σ be a crossed product functor for G. If the σ-Baum-Connes conjecture holds for G with trivial coefficients then the only idempotents in the Banach algebra 1 (G) are zero and the identity. Recall that there is a canonical tracial state ∗ τ : Cred (G) → C,

τ (a) =  δe , aδe .

The trace τ is well known to be faithful in the sense that a non-zero positive element ∗ (G) has strictly positive trace: see, for example [17, Proposition 2.5.3]. One of Cred has the following standard C ∗ -algebraic lemma. ∗ 10.2. Lemma. Let e ∈ Cred (G) be an idempotent. If τ (e) is an integer then e = 0 or e = 1. ∗ Proof. The idempotent e is similar in Cred (G) to a projection p [15, Proposition 4.6.2] so that τ (p) = τ (e) ∈ Z. Positivity of τ and the operator inequality

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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 (Cred (G)) → R which sends the ∗ K-theory class of an idempotent e ∈ Cred (G) to τ (e) [15, Section 6.9]. The key point in the proof of Theorem 10.1 is the following result concerning the image under τ∗ of elements in the range of the Baum-Connes assembly map. ∗ (G)) 10.3. Proposition. Let G be a countable torsion free group. If x ∈ K0 (Cred 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 theorem to the Baum-Connes conjecture, and thus to prove the proposition, is via the BaumDouglas geometric model for K-homology [9, 11]: this is explained in [69, Section 6.3] or [7, Proposition 6.1]. See also [48], particularly Theorem 0.3, for a slightly different approach (and a more general statement that takes into account the case when G has torsion).  Proof of Theorem 10.1. Let e be an idempotent in 1 (G). Since 1 (G) is a ∗ subalgebra of both Cred (G) and Cσ∗ (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) ∗ (G), Baum-Connes assembly map factors through the quotient map Cσ∗ (G) → Cred 1 and this quotient map is the identity on  (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 of the 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 is unchanged, that is, is the same as in the original conjecture as stated by Baum and Connes [8]. At first glance, it may seem surprising that in the reformulated conjecture 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 with coefficients in a G-C ∗ -algebra A factors as K∗top (G, A) → K∗ (1 (G, A)) → K∗ (A red G), where 1 (G, A) is the Banach algebra crossed product. According to the Bost conjecture, the first arrow in this display is an isomorphism; the second arrow is induced by the inclusion 1 (G, A) → A red G. The Bost conjecture is known to hold in a great many cases, in particular, for fundamental groups of Riemannian locally symmetric spaces; see [45]. In these cases, the Baum-Connes conjecture is equivalent to the assertion that the K-theory of the Banach algebra 1 (G, A) is isomorphic to the K-theory of the C ∗ -algebra Ared G, and in fact that the inclusion 1 (G, A) → Ared G induces an isomorphism. While the Bost conjecture may seem more natural because it has the appropriate functoriality in the group G, it does not have the important implications for geometry and topology that the Baum-Connes conjecture does. In particular, it is not known to imply either the Novikov higher signature conjecture or the stable

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Gromov-Lawson-Rosenberg conjecture about existence of positive scalar curvature metrics on spin manifolds. As described here, our reformulation, which involves an appropriate C ∗ -algebra completion of 1 (G, A) retains these implications. From this point of view, an attempt to reformulate the Baum-Connes conjecture should involve finding a pre-C ∗ -norm on 1 (G, A) with the property that the K-theory of the Banach algebra 1 (G, A) equals the K-theory of its C ∗ -algebra completion. This problem involves in a fundamental way the harmonic analysis of the group, and this paper can be viewed as indicating a possible solution.

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Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13502

Generalization of C ∗ -algebra methods via real positivity for 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 notion of (real) positivity in operator algebras, with an eye to extending certain C ∗ algebraic results and theories to more general algebras. As motivation note that the ‘completely’ real positive maps on C ∗ -algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C ∗ -algebra it generates. We have continued this work together with Read, and also with Matthew Neal. Recently with Narutaka Ozawa we have investigated the parts of the theory that generalize further to Banach algebras. In the present paper we describe some of this work, and also give some new updates and complementary results, which are connected with generalizing 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 of this volume, and also discusses some of the origins of the theory of positivity in our sense in the setting of algebras, which the later parts of our paper developes further. The most recent work will be emphasized.

1. Introduction This is a much expanded version of our talk given at the AMS Special Session “Tribute to Richard V. Kadison” in January 2015. We survey some of our work on a new notion of (real) positivity in operator algebras (by which we mean closed subalgebras of B(H) for a Hilbert space H), unital operator spaces, and Banach algebras, focusing on generalizing various C ∗ -algebraic techniques initiated by Richard V. Kadison. In particular Section 2 is in part a tribute to Kadison in keeping with the occasion of this volume, and we will describe a small part of his opus relevant to our setting. This section also discusses some of the origins of the theory of positivity in our sense in the setting of algebras, which the later parts of 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, approximate identity, accretive operators, state space, quasi-state, hereditary subalgebra, Banach algebra, ideal structure. Supported by NSF grant DMS 1201506. c 2016 American Mathematical Society

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real-positivity theory by showing how it relates to these results of Kadison, and also give some small extensions and additional details for our recent paper with Ozawa [21], and for [20] with Neal. With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C ∗ -algebraic results and theories to more general algebras. As motivation note that the ‘completely’ real positive maps on C ∗ -algebras or operator systems are precisely the completely positive maps in the usual sense (see Theorem 3.2 below); however with real positivity one may develop an order theory for more general spaces and algebras that is useful at least for some purposes. We have continued this work together with Read, and also with Matthew Neal; giving many applications. (See papers with these coauthors referenced in the bibliography below.) Recently with Narutaka Ozawa we have investigated the parts of the theory that generalize further to Banach algebras. In all of this, our main goal is to generalize certain nice C ∗ -algebraic results, and certain function space or function algebra results, which use positivity or 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 work of Kadison. The most recent work will be emphasized, particularly parts of the Banach-algebraic paper [21]. One reason for this emphasis is that less background is needed here (for example, we shall avoid discussion of noncommutative topology, and our work on noncommutative peak sets and peak interpolation, which we have surveyed recently in [12] although we have since made more progress in [25]). Another reason is that we welcome this opportunity to add some additional details and complements to [21] (and to [20]). In particular we will prove some facts that were stated there without proof. A subsidiary goal of Sections 6 and 7 is to go through versions for general Banach algebras of results in Sections 3, 4, and 7 of [21] stated for Banach algebras with approximate identities. We will also pose several open questions. The drawback of course with this focus is that the Banach algebra case is sometimes less impressive and clean than the operator algebra case, there usually being a price to be paid for generality. Of course an operator algebra or function algebra A may have no positive elements in the usual sense. However we see e.g. in Theorem 5.2 below that an operator algebra A has a contractive approximate identity iff there is a great abundance of real-positive elements; for example, iff A is spanned by its real-positive elements. Below Theorem 5.2 we will point out that this is also true for certain classes of Banach algebras. Of course in the theory of C ∗ -algebras, positivity and the existence of positive approximate identities are crucial. Some form of our ‘positive cone’ already appeared in papers of Kadison and Kelley and Vaught in the early 1950’s, and in retrospect it is a natural idea to attempt to use such a cone to generalize various parts of C ∗ -algebra theory involving positivity and the existence of positive approximate identities. However nobody seems to have pursued this until now. In practice, some things are much harder than the C ∗ -algebra case. And many things simply do not generalize beyond the C ∗ -theory; that is, our approach is effective at generalizing some parts of C ∗ -algebra theory, but not others. The worst 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 in many cases if one looks a little closer something works, or an interesting conjecture is raised. Successful applications so far include for example noncommutative

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topology (eg. noncommutative Urysohn and Tietze theorems for general operator algebras, and the theory of open, closed and compact projections in the bidual), lifting problems, the structure of completely contractive idempotent maps on an operator algebra (described in Section 3 below), noncommutative peak sets, peak interpolation, and some other noncommutative function theory, comparison theory, the structure of operator algebras, new relationships between an operator algebra and the C ∗ -algebra it generates, approximate identities, etc. We refer the reader to 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 properties of self-adjoint operators individually and with respect to containing operator algebras”. This was from the paper [49], which appeared in 1950. In the early 1950s the war was over, John von Neumann was editor of the Annals of Mathematics and was talking to anybody who was interested about ‘rings of operators’, Kadison was in Chicago and the IAS, and all was well with the world. In 1950, von Neumann wrote a 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 your manuscript on ”Projections in Banach Algebras”. I am very glad that you are submitting it for THE ANNALS, and I will immediately 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 “purely algebraical” rather than “vectorial-spatial” foundation for theories of operator-algebras or operatorlike-algebras. To be more precise: It always seemed to me that there were three successive levels of abstraction - first, and lowest, the vectorial-spatial, in which the Hilbert space and its elements are actually used; second, the purely algebraical, where only the operators or their abstract equivalents are used; third, the highest, the approach when only linear spaces or their abstract equivalents (i.e. operatorially speaking, the projections) are used. [. . . ] After Murray and I had reached somewhat rounded results on the first level, I neglected to make a real effort on the second one, because I was tempted to try immediately the third one. This led to the theory of continuous geometries. In studying this, the third level, I realized that one is led there to the theory of “finite” dimensions only. The discrepancy between what might be considered the “natural” ranges for the first and the third level led me to doubt whether I could guess the correct degree of generality for the second one. . . ”. It is remarkable here to recall that von Neumann invented the abstract definition of Hilbert spaces, the theory of unbounded operators (as well as much of the bounded theory), ergodic theory, the mathematical formulation of quantum mechanics, many fundamental concepts associated with groups (like amenability),

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and several other fields of analysis. Even today, teaching a course in functional analysis 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 what he calls the ‘second level’ of ‘operator algebra’, and at the time of this letter this was ripe and timely for exploration. Happily, about the time the above letter was written, Richard Vincent Kadison 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 operator algebras. Some part of his early work was concerned with positive cones and their properties. We will now briefly describe a few of these and spend much of the remainder of our article showing how they can be generalized to nonselfadjoint operator algebras and Banach algebras. The following comprises just a tiny part of Kadison’s opus; but nonetheless is still foundational and seminal. Indeed much of C ∗ -algebra theory would disappear without this work. At the start of this section we already mentioned his first paper, devoted to ‘order properties of self-adjoint operators 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 generalized the “Banach-Stone” theorem, characterizing surjective isometries between C ∗ -algebras. This result has inspired very many functional analysts and innumerable papers. See for example [38] for a collection of such results, together with their 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 on C ∗ -algebras. His student Størmer continued this in a very long (and still continuing) series of deep papers. Later this Kadison–Schwarz work was connected to completely positive maps, Stinespring’s theorem and Arveson’s extension theorem (see the next paragraph and e.g. [68]), conditional expectations, operator systems and operator spaces, quantum information theory, etc. A related enduring interest of Kadison’s is projections and conditional expectations on C ∗ -algebras and von Neumann algebras. A search of his collected works finds very many contributions to this topic (e.g. [53]). In 1960, Kadison together with I. M. Singer [57] initiated the study of nonselfadjoint operator algebras on a Hilbert space (henceforth simply called operator algebras). Five years later or so, the late Bill Arveson in his thesis continued the study of nonselfadjoint operator algebras, using heavily the Kadison-Fuglede determinant of [54] and positivity properties of conditional expectations. This work was published in [4]; it developes a von Neumann algebraic theory of noncommutative Hardy spaces. We mention in passing that we continued Arveson’s work from [4] in a series of papers with Labuschagne, again using the Kadison-Fuglede determinant of [54] as a main tool (see e.g. the survey [14]), as well as positive conditional expectations and the Kadison–Schwarz inequality. This is another example of using C ∗ -algebraic methods, and in particular tools originating in seminal work of Kadison, in a more general (noncommutative function theoretic) setting. However since this lies in a different direction to the rest of the present article we will say no more about this. In the decade after [4], Arveson went on to write many other seminal papers on nonselfadjoint operator algebras, perhaps most notably [5], in

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which completely positive maps and the Kadison–Schwarz inequality play a decisive role, and which may be considered a source of the later theory of operator spaces and operator systems. Another example: in 1968 Kadison and Aarnes, his first student at Penn, introduced strictly positive elements in a C ∗ -algebra A, namely x ∈ A which satisfy f (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 equivalent: (1) (2) (3) (4)

A has a strictly positive element. A has a countable increasing contractive approximate identity. A = zAz for some positive z ∈ A. 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 1 taken to be (z n ) 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 1 since s(z) is the weak* limit of z n , and the converse is easy. This result is related to the theory of hereditary subalgebras, comparison theory in C ∗ -algebras, etc. In fact much of modern C ∗ -algebra theory would collapse without basic results like this. For example, the Aarnes–Kadison theorem implies the beautiful characterization due to Prosser [71] of closed one-sided ideals in a separable C ∗ -algebra A as the ‘topologically principal (one-sided) ideals’ (we are indebted to the referee for pointing out that Prosser was a student of Kelley). The latter is equivalent to the characterization of hereditary subalgebras of such A as the subalgebras of form zAz. (We recall that a hereditary subalgebra, or HSA for short, is a closed selfadjoint subalgebra D satisfying DAD ⊂ D.) These results are used in many modern theories such as that of the Cuntz semigroup. Or, as another example, the Aarnes–Kadison theorem is used in the important stable isomorphism theorem for Morita equivalence of C ∗ -algebras (see e.g. [10, 28]). Indeed in some sense the Aarnes–Kadison theorem is equivalent to the first assertion of the following: Theorem 2.2. A HSA (resp. closed right ideal) in a C ∗ -algebra A is (topologically) principal, that is of the form zAz (resp. zA) for some z ∈ A iff it has a countable (resp. countable left) contractive approximate identity. Every closed right 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 Gert Pedersen [55], which amongst other things initiates the development of a comparison theory for elements in C ∗ -algebras generalizing the von Neumann equivalence of projections. Again positivity and properties of the positive cone are key to that work. 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 such results and theories, in which positivity is the common theme, to not necessarily

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selfadjoint operator algebras (or perhaps even Banach algebras). In fact one often can, as we have shown in joint work with Charles Read, Matt Neal, Narutaka Ozawa, and others. In the Banach algebra literature of course there are many generalizations 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 mention Sinclair’s generalization from [74] of part of the Aarnes–Kadison theorem: Theorem 2.3 (Sinclair). A separable Banach algebra A with a bounded approximate identity has a commuting bounded approximate identity. If A has a countable bounded approximate identity then Sinclair and others show results like A = xA = Ay for some x, y ∈ A. In part of our work we follow Sinclair in using variants of the proof of the Cohen factorization method to achieve such results but with ‘positivity’. We now explain one of the main ideas. Returning to the early 1950s: it was only then becoming perfectly clear what a C ∗ -algebra was; a few fundamental facts about the positive cone were still being proved. We recall that an unpublished result 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 John Kelley and Vaught [58] based on a 1950 ICM talk by those authors. These sources are referenced in almost every C ∗ -algebra book. The paper of Kelley and Vaught was titled “The positive cone in Banach algebras”, and in the first section of the paper they discuss precisely that. The following is not an important part of their paper, but as in Kadison’s paper a year earlier they have a small discussion on how to make sense of the notion of a positive cone in a Banach algebra, and they prove some basic results here. Both Kadison and Kelley and Vaught have some use for the 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 take 1 to be the identity of a unitization of A. In [22], Charles Read and the author began a study of not necessarily selfadjoint operator algebras on a Hilbert space H; henceforth operator algebras. In this work, FA above plays a pivotal role, and also the cone R+ FA . In [23] we looked at the slightly larger cone rA of so called accretive elements (this is a non-proper cone or ‘wedge’). In an operator algebra these are the elements with positive real part; in a general Banach algebra they are the elements x with Re ϕ(x) ≥ 0 for every state ϕ on a unitization of A. We recall that 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 are the elements with numerical range in the closed right half-plane. We sometimes also 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, Lemma 2.1]. 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 are equivalent: (1) x ∈ rA , that is, x has numerical range in the closed right half-plane. (2) 1 − tx ≤ 1 + t2 x2 for all t > 0.

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(3)  exp(−tx) ≤ 1 for all t > 0. (4) (t + x)−1  ≤ 1t for all t > 0. (5) 1 − tx ≤ 1 − t2 x2  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 Re ϕ(x) ≥ 0) (see [63, Lemma 2.1]). Given (4) with t replaced by 1t , we have 1 − tx = (1 + tx)−1 (1 + tx)(1 − tx) ≤ 1 − t2 x2 . 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 > 0 there exists a t > 0 with 1 − tx < 1 + t. See e.g. [27, p. 30].) Real positive elements, and the smaller set FA above, will play the role for us of positive elements in a C ∗ -algebra. While they are not the same, real positivity is very compatible with the usual definition of positivity in a C ∗ -algebra, as will be seen 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 unital operator space below we mean a subspace of B(H) or a unital C ∗ -algebra containing the identity. We gave abstract characterizations of these objects with Matthew Neal in [16, 19], and have studied them elsewhere. Definition 3.1. A linear map T : A → B between operator algebras or unital operator spaces is real positive if T (rA ) ⊂ rB . It is real completely positive, or RCP for 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 matching material 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 approximate 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 able to 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. In particular, if one is trying to generalize results where completely positive maps and the 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 example in the remaining results in this section.

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We will not say more about unital operator spaces in the present article, except to say that it is easy to see that completely contractive unital maps on a unital operator space are RCP. We give two or three applications from [20] of Theorem 3.3. The first is related to 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 is a 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, we mean an idempotent linear map P : A → A. We say that P is a conditional expectation 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 completely contractive map (resp. projection) on the unitization A1 . Much earlier, we studied completely contractive projections P and conditional expectations 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, Corollary 4.2.9]) that P is a conditional expectation. This is the operator algebra variant of Tomiyama’s theorem for C ∗ -algebras. A well known result of Choi and Effros states that the range of a completely positive projection P : B → B on a C ∗ -algebra B, is again a C ∗ -algebra with product P (xy). The analogous result for unital completely contractive projections on unital operator algebras is true too, and is implicit in the proof of our generalization of Tomiyama’s theorem above. Unfortunately, there is no analogous result for (nonunital) completely contractive projections on possibly nonunital operator algebras without adding extra hypotheses on P . However if we add the condition that P is also ‘real completely positive’, then the question does make good sense and one can easily deduce from the unital case and Proposition 3.5 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 with product P (xy) and cai (P (et )) for some cai (et ) of A, iff P is real completely positive. Proof. For the ‘forward direction’ note that P ∗∗ is a unital complete contraction, 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 originally a remark in [20]. By passing to the bidual we may assume that A is unital. If P (P (1)x) = P (xP (1)) = x for all x ∈ Ran(P ) then we are done by the abstract characterization of operator algebras from [15, Section 2.3], since then P (xy) defines a bilinear completely contractive product on Ran(P ) with ‘unit’ P (1). Let I(A) be the injective envelope of A. We may extend P to a completely positive completely contractive map Pˆ : I(A) → I(A), by [9, Theorem 2.6] and injectivity of I(A). We will abusively sometimes write P for Pˆ , and also for its second adjoint on I(A)∗∗ . The latter is also completely positive and completely contractive. Then 1

1

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

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Hence these quantities are equal. In the limit, P (s(P (1))) = P (1), if s(P (1)) is the support projection of P (1). Hence P (z) = 0 where z = s(P (1)) − P (1). If y ∈ 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(Pˆ ). 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 (z 2 ) ≤ 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 projections and conditional expectations, and in particular the ‘symmetric projection problem’ and the ‘bicontractive projection problem’, in the category of operator algebras, attempting to find operator algebra generalizations of certain deep results of Størmer, Friedman and Russo, Effros and Størmer, Robertson and Youngson, and others (see papers of these authors referenced in the bibliography below), concerning projections and their ranges, assuming in addition that our projections are real completely positive. We say that an idempotent linear P : X → X is completely symmetric (resp. completely bicontractive) if I − 2P is completely contractive (resp. if P and I − P are completely contractive). ‘Completely symmetric’ implies ‘completely bicontractive’. The two problems mentioned at the start of this paragraph concern 1) Characterizing such projections P ; or 2) characterizing the range of such projections. On a unital C ∗ -algebra B the work of some of the authors mentioned at the start of this paragraph establish that unital positive bicontractive projections are also symmetric, and are precisely 12 (I + θ), for a period 2 ∗-automorphism θ : B → B. The possibly nonunital positive bicontractive projections P are of a similar form, and then q = P (1) is a central projection in M (B) with respect to which P decomposes into a direct sum of 0 and a projection of the above form 1 2 (I + θ), for a period 2 ∗-automorphism θ of qB. Conversely, a map P of the latter form is automatically completely bicontractive, and the range of P , which is the set 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 (approximately unital) operator algebras A? The first thing to note is that now ‘completely bicontractive’ is no longer the same as ‘completely symmetric’. The following is our solution to the symmetric projection problem, and it uses Kadison’s Banach–Stone theorem for C ∗ -algebras [50], and our variant of the latter for approximately unital operator algebras (see e.g. [15, Theorem 4.5.13]): Theorem 3.7. [20] Let A be an approximately unital operator algebra, and P : A → A a completely symmetric real completely positive projection. Then the range of P is an approximately unital subalgebra of A. Moreover, P ∗∗ (1) = q is a projection 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 a period 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

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D → D. Also, P is the zero map on q ⊥ A + Aq ⊥ + q ⊥ Aq ⊥ , and 1 P = (I + θ) on D. 2 In fact 1 P (a) = (a + θ(a)(2q − 1)) , a ∈ A. 2 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 a homomorphism, one obtains some interesting algebraic formulae involving q, q ⊥ , A and θ|qAq . For the more general class of completely bicontractive projections, a first look is disappointing–most of the last paragraph no longer works in general. One does not always get an associated completely isometric automorphism θ such that P = 12 (I + θ), and q = P (1) need not be a central projection. However, as also seems to be sometimes the case when attempting to generalize a given C ∗ -algebra fact to more general algebras, a closer look at the result, and at examples, does uncover an interesting question. Namely, given an approximately unital operator algebra A and a real completely positive projection P : A → A which is completely bicontractive, when is the range of P a subalgebra of A and P a conditional expectation? This seems to be the right version of the ‘bicontractive projection problem’ in the operator algebra category. We give in [20] a sequence of three reductions that reduce the question. The first reduction is that by passing to the bidual we may assume that the algebra A is unital. The second reduction is that by cutting down to qAq, where q = P (1) (which one can show is a projection), we may further assume that P (1) = 1 (one can show P is zero on q ⊥ A + Aq ⊥ ). The third reduction is by restricting attention to the closed algebra generated by P , we may further assume that P (A) generates A as an operator algebra. We call this the ‘standard position’ for the bicontractive projection problem. It turns out that when 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 show that A decomposes as A = C ⊕ B, where 1A ∈ B = P (A), C = (I − P )(A), and we have the relations C 2 ⊂ 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 (indeed an automorphism) on A iff P (A) is a subalgebra of A, and we have, similarly to Theorem 3.7: Corollary 3.8. If P : A → A is a unital idempotent on a unital operator algebra then P is completely bicontractive iff there is a period 2 linear surjection θ : A → A such that I ± θcb ≤ 2 and P = 12 (I + θ). The range of P is a subalgebra iff θ is also a homomorphism, and then the range of P is the set of fixed points of this automorphism θ. Also, P is completely symmetric iff θ is completely contractive. 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 complete solution to the bicontractive projection problem.

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Theorem 3.9. Let P : A → A be a real positive bicontractive projection on a (unital or approximately unital) uniform algebra. Then P is symmetric, and so of course by Theorem 3.7 we have that P (A) is a subalgebra of A, and P is a conditional expectation. Proof. We sketch the idea, found in a conversation with Joel Feinstein. By the first two reductions described above we can assume that A and P are unital. We also know that B = P (A) is a subalgebra, since if it were not then the third reduction described above would yield nonzero nilpotents, which cannot exist in a function algebra. Thus by the discussion above the theorem, the map θ(x+y) = x−y there is an algebra automorphism of A, hence an isometric isomorphism (since norm equals  spectral radius). So P = 12 (I + θ) is symmetric. The same three step reduction shows that we can also solve the problem in the affirmative for real completely positive completely bicontractive projections P on a unital 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 to the 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 interest is the cone consisting of elements of A which are ‘sectorial’ of angle θ < π2 . For the purposes of this paper being sectorial of angle θ will mean that the numerical range in A (or in a unitization of A if A is nonunital) is contained in the sector Sθ consisting of complex numbers reiρ with r ≥ 0 and |ρ| ≤ θ. This third cone is a dense subset of the second cone R+ FA if A is an operator algebra [25, Lemma 2.15]. We remark that there exists a well established functional calculus for sectorial operators (see e.g. [43]). Indeed the advantages of this cone and the last one seems to be mainly that these have better functional calculi. For the cone R+ FA , if A is an operator algebra, one could use the functional calculus coming from von Neumann’s inequality. Indeed if I − x ≤ 1 then f → f (I − x) is a contractive homomorphism on the disk algebra. If x is real positive in an operator algebra, one could also use Crouzeix’s remarkable functional calculus on the numerical range of x (see e.g. [31]). If x is sectorial in a Banach algebra, one may use the functional calculus for sectorial operators [43]. A final notion of positivity which we introduced in the work with Read, which is slightly more esoteric, but which is a close approximation to the usual C ∗ -algebraic notion of positivity: In the theorems below we will sometimes say that an element x is nearly positive; this means that in the statement of that result, given > 0 one can also choose the element in that statement to be real positive and within

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 > 0, to choose x to also be a contraction with numerical range within a thin ‘cigar’ centered on the line segment [0, 1] of height < . That is, x has sectorial angle < arcsin . In an operator algebra any contraction x with such a sectorial angle is accretive and satisfies x − Re x ≤ , so x is within of an operator which is positive in the usual sense. Indeed if a is an accretive element in an operator algebra then (principal)

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π , and hence n-th roots of a have spectrum and numerical radius within a sector S 2n are as close as we like (for n sufficiently large) to an operator which is positive in the usual sense (see Section 6). Thus one obtains ‘nearly positive elements’ by taking 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 algebras if roots (or rth powers for 0 < r < 1) of accretive elements are in R+ FA , or if the third cone in the last paragraph is contained in the second cone. We note that the 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’, which at least in an operator algebra are each successively stronger than the last. It is convenient to mentally picture each of these notions by sketching the region containing 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 cone R+ FA , as a dense cone in the right half plane composed of closed disks center a and radius a, for all a > 0. The third cone is pictured as increasing sectors Sθ in C, for increasing θ < π2 . And the ‘nearly positive’ elements are pictured by the thin ‘cigar’ mentioned a paragraph or so back, centered on the line segment [0, 1] of height < , and contained in the closed disk center 12 of radius 12 . We now list some more of our notation and general facts: We write Ball(X) for the set {x ∈ X : x ≤ 1}. For us Banach algebras satisfy xy ≤ xy. If x ∈ A for 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 we usually use on A∗∗ is the ‘second Arens product’ ( in the notation of [32]). This is weak* continuous in the second variable. If A is a nonunital, not necessarily Arens regular, Banach algebra with a bounded approximate identity (bai), then A∗∗ has a so-called ‘mixed identity’ [32, 34, 67], which we will again write as e. This is a right identity for the first Arens product, and a left identity for the second Arens product. A mixed identity need not be unique, indeed mixed identities are just the weak* limit points of bai’s for A. See the book of Doran and Wichmann [34] for a compendium of results about approximate identities and related topics. If A is an approximately unital Banach algebra, then the left regular representation embeds A isometrically in B(A). We will 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 identity of A1 , if A is not already unital. If A is a nonunital, approximately unital Banach algebra then the multiplier unitization A1 may also be identified isometrically with the 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 ⊥⊥ is complemented in X ∗∗ via a contractive projection P so that X ∗∗ = E ⊥⊥ ⊕∞ Ker(P ). In this case there is a unique contractive projection onto E ⊥⊥ . This concept was invented by Alfsen and Effros, and [44] is the basic text for their beautiful and powerful theory. By an M -approximately unital Banach algebra we mean a Banach algebra 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 − xA∗∗ for all x ∈ A∗∗ ,

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unless the last quantity is < 1 in which case 1 − x(A1 )∗∗ = 1. Here e is the identity for A∗∗ if it has one, otherwise it is a mixed identity of norm 1. A result of 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 Banach algebras are M -approximately unital. We use states a lot in our work. However for an approximately unital Banach algebra A with cai (et ), the definition of ‘state’ is problematic. Although we have not noticed this discussed in the literature, there are several natural notions, and which is best seems to depend on the situation. For example: (i) a contractive functional ϕ on A with ϕ(et ) → 1 for some fixed cai (et ) for A, (ii) a contractive functional ϕ on A with ϕ(et ) → 1 for all cai (et ) for A, and (iii) a norm 1 functional on 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, Lemma 2.2], but this is not true in general. The M -approximately unital Banach algebras in 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 usual theory of numerical range and accretive operators, we will take (iii) above as the definition 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) to be 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 with the latter notation here and in [21] in the (generally rare) case that A∗∗ is unital. It should be stressed that in these papers rA∗∗ should not, if A∗∗ is unital, be confused with 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 in particular if A is an approximately unital operator algebra. It is easy to see that A∗∗ ∩ r(A1 )∗∗ is contained in the accretive elements in A∗∗ if A∗∗ is unital, but the other direction seems unclear in general. Of course in the theory of C ∗ -algebras, positivity and the existence of positive approximate identities are crucial. How does one get a ‘positive cai’ in an algebra with cai? We have several ways to do this. First, for approximately unital operator algebras and for a large class of approximately unital Banach algebras (eg. the scaled Banach algebras defined in the next section; and we do not possess an example of a Banach algebra that is not scaled yet) we have a ‘Kaplansky density’ result: w∗ Ball(A) ∩ rA = Ball(A∗∗ ) ∩ rA∗∗ . See Theorem 5.8 below. (We remark that although it seems not to be well known, the most common variants of the usual Kaplansky 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 real positive mixed identity e of norm 1, then one can then get a real positive cai by approximating e by elements of Ball(A)∩rA . See Corollary 5.9. A similar argument allows one to deduce the second assertion in the following result from the first (one also uses the fact that in an M -approximately unital Banach algebra 1 − 2e ≤ 1 for 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 has a cai in 12 FA .

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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 theorem from [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 relax the M -approximately unital hypothesis to the following much milder condition: A is approximately unital and given > 0 there exists a δ > 0 such that if y ∈ A with 1 − y < 1 + δ then there is a z ∈ A with 1 − z = 1 and y − z < . 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 = . Another approach to finding a ‘real positive cai’ under a countability condition from [21, Section 2] uses a slight variant of the ‘real positive’ definition. Namely for a fixed cai e = (et ) for A define Se (A) = {ϕ ∈ Ball(A∗ ) : limt ϕ(et ) = 1} (a subset of S(A)). Define reA = {x ∈ A : Re ϕ(x) ≥ 0 for all ϕ ∈ Se (A)}. If we multiply these states by numbers in [0, 1], we get the associated quasistate space Qe (A). Note that reA contains rA . On the other hand, [21, Theorem 6.5] (or a minor variant of the proof of it) shows that if A∗∗ is unital then reA is never contained in rA∗∗ (or in the accretive 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 for the first step. Suppose that K is a compact space and (fn ) is a bounded sequence in C(K, R), such that limn fn (x) exists for every x ∈ K and is non-negative. Claim: for every > 0, there is a function f ∈ conv{fn } such that f ≥ − on K. Indeed if this were not true, then there exists an > 0 such that for all f ∈ conv{fn } there is a point x in K with f (x) < − . Moreover, for all g ∈ conv{fn }, if f ∈ conv{fn } with f − g < 4 , there is a point x in K with g(x) < − 3 4 . So A = conv{fn } and C = C(K)+ are clearly disjoint. Moreover, it is well known that convex sets E, C in an LCTVS can be strictly separated iff 0 ∈ / E − C, and this is clearly the case for us here. So there is a continuous functional ψ on C(K, R) and scalars M, N with ψ(g) ≤ M < N ≤ ψ(h) for all g ∈ A and h ∈ C. Since C is a cone we may take N = 0. Bythe Riesz–Markov theorem there is a Borel probability measure m such that supn K fn dm < 0. This is a contradiction and proves the Claim, since limn K fn dm ≥ 0 by Lebesgue’s dominated convergence theorem. Now let K = Qe (A) and fn (ψ) = Re ψ(en ) where e = (en ). We have limn fn ≥ 0 pointwise on K, so by the last paragraph for any > 0 a convex combination of the fn is always ≥ − on K. By a standard geometric series type argument we can replace with 0 here, so that we have a real positive element, and with more care this 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 ‘Corollary 3.4’ of the present paper). Corollary 4.4. If A is an approximately unital Banach algebra with a cai e such that S(A) = Se (A), and such that the quasistate space Q(A) is weak* closed, then A has a cai in rA .

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We remark that we have no example of an approximately unital Banach algebra where Q(A) is not weak* closed. In particular, we have found that commonly encountered 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 well known order-theoretic properties of the unit ball of a C ∗ -algebra and its dual. Some of these results also may be viewed as new relations between an operator algebra and a C ∗ -algebra that it generates. There are interesting connections to the classical theory of ordered linear spaces (due to Krein, Ando, Alfsen, etc) as found e.g. in the first chapters of [6]. In addition to striking parallels, some of this classical theory can be applied directly. Indeed several results from [21] (some of which are mentioned below, see e.g. the proof of Theorem 5.4) are proved by appealing to results in that theory. See also [25] for more connections if the algebras are in addition operator algebras. The ordering induced by rA is obviously b  a iff a−b is accretive (i.e. numerical range 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 a C ∗ -algebra B, then A is order cofinal in B. That is, given b ∈ B+ there exists a ∈ A with b  a. Indeed one can do this with b  a  b + . Indeed one can do this with b  Cet  b + , for a nearly positive cai (et ) for A and a constant C > 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 ∗ -algebra B, 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 ∈ 12 FA and nearly positive. (3) For any pair x, y ∈ UA there exist nearly positive a ∈ 12 FA with x  a and y  a. (4) For any b ∈ UA there exist nearly positive a ∈ 12 FA with −a  b  a. (5) For any b ∈ UA there exist x, y ∈ 12 FA 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 a HSA 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 properties in items (3)–(6) in Theorem 5.2. See [21, Section 6] for various variants on this theme. The following is a particularly clean case: Theorem 5.3. [21, Section 6] Let A be an M -approximately unital Banach algebra. Then (1) For any pair x, y ∈ UA there exist a ∈ 12 FA with x  a and y  a. (2) For any b ∈ UA there exist a ∈ 12 FA with −a  b  a. (3) For any b ∈ UA there exist x, y ∈ UA ∩ 12 FA with b = x − y. (4) CA is a generating cone (that is, A = CA − CA ).

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During the writing of the present paper we saw the following improvement of part of Corollaries 6.7 and 6.8, and on some of 6.10 in the submitted version of the paper [21]. At the galleys stage of that paper we incorporated those advances, but unfortunately 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 satisfies that Qe (A) is weak* closed, then (1)–(4) in the last theorem hold, with 12 FA replaced by Ball(A) ∩ reA , and  replaced by the linear ordering defined by the cone reA , and CA replaced by reA . One may drop the three superscript e’s in the last line if in addition 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 their real parts, the norm on the real dual cone is additive. This is known to imply, by the theory of ordered linear spaces [6, Corollary 3.6, Chapter 2], that the open ball of A is a directed set. So for any pair x, y ∈ UA there exist z ∈ UA with x e z and y e z. Applying this again to z, −z there exists w ∈ UA with ±z e w. This w+z e implies that w±z 2 ∈ rA , and z e a where a = 2 . This proves (1). Applying (1) a+b a−b to b, −b we get (2). Setting x = 2 , y = 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 a directed set, and indeed is a net which is a positive cai for B. The first part of this statement is generalized by Theorems 5.2 (3), 5.3 (1), and 5.4 (1). The following generalizes the second part of the statement to operator algebras: Corollary 5.5. [25] If A is an approximately unital operator algebra, then UA ∩ 12 FA is a directed set in the  ordering, and with this ordering UA ∩ 12 FA is an increasing cai for A. We do not know if the second part of the last result is true for any other classes of Banach algebras. We say a Banach algebra A is scaled if every real positive linear map into the scalars 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 approximately unital Banach algebra that is not scaled, and certainly all commonly encountered Banach algebras seem to be scaled. Unital Banach algebras are scaled by e.g. an argument in the proof of [63, Theorem 2.2]. Theorem 5.6. [21, 25] If A is an approximately unital operator algebra, or more 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 algebra, 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.

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

If these hold then Q(A) = S(A)

, 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 approximately unital 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 done earlier with Read. Corollary 5.9. If A is a scaled approximately unital Banach algebra then A has a cai in rA iff A∗∗ has a mixed identity e of norm 1 in rA∗∗ , or equivalently with 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] (see  Lemma 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 a cai 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 ), hence ϕ(e) ≥ 0 by the last line and Proposition 5.7. So e ∈ A∗∗ ∩ r(A1 )∗∗ = rA∗∗ , and so the result follows from our Kaplansky density type theorem in the form of its Corollary 5.9.  The class of algebras A in the last Corollary is the same as the class in the last line of the statement of Theorem 5.4. Thus for such algebras, (1)–(4) in Theorem 5.3 hold, with 12 FA 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 generalizes some parts of the earlier work) connected to the work of Kadison summarized in Section 2, but we will also restate the results from several sections of [21] in the more general setting of Banach algebras with no kind of approximate identity. Also we will give a detailed discussion of roots (fractional powers) in relation to our positivity (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, for example any unitization of A, we define FB A = {a ∈ A : 1B − a ≤ 1}, and write rB A for the set of a ∈ A whose numerical range in B is contained in the right half plane. These sets are closed and convex. Also we define FA = ∪B FB A,

rA = ∪B rB A,

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the unions taken over all unital Banach algebras B containing A. Unfortunately it is not clear to us that FA and rA are always convex, which is needed in most of Section 7 below (indeed we often need them closed too there), and so we will have B to use FB A and rA there instead. Of course we could fix this problem by defining B FA = ∩B FA and rA = ∩B rB A , the intersections taken over all B as above. If we did this then we could remove the superscript B in all results in Section 7 below; this would 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 A B0 0 such that FB A = FA (resp. rA = rA ). This happens if A is an operator algebra because 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 with product (a, λ)(b, μ) = (ab + μa + λb, λμ), and satisfying (a, 0) = aA 0 for a ∈ A. Let B0 be A ⊕ C with this smallest norm. Then FB A = FA and B0 rA = 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.) Define B0 to be the span in B(A) of IA and the isometrically embedded copy of 0 A. This has the smallest norm of any unitization of A. Hence FB A = FA B0 and rA = rA . Proof. If B is any unital Banach algebra containing A, and a ∈ FB A then B0 B0 0 a ∈ FB . So F = F . A similar argument shows that r = r , using Lemma 2.4 A A A A A (2), namely that 2 2 rB A = {a ∈ A : 1B − ta ≤ 1 + t a 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)xA : x ∈ Ball(A)} = sup{(a + λ1B )xB : 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 algebra B B B B, and if C is a subalgebra of A, then FB C = C ∩ FA and rC = C ∩ rA . We now discuss roots (that is, r’th powers for r ∈ [0, 1]) in a subalgebra A of a unital Banach algebra B. Actually, we only discuss the principal root (or power); we recall that the principal rth power, for 0 < r < 1, is the one whose spectrum is contained in a sector Sθ of angle θ < 2rπ. There are several ways to define these that we are aware of. We will review these and show that they are the same. As far as we know, Kelley and Vaught [58] were the first to define the square root of elements of FA , but their argument works for r’th powers for r ∈ [0, 1]. If 1 − x ≤ 1, define ∞    r r > 0. xr = (−1)k (1 − x)k , k k=0

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r     k For k ≥ 1 the sign of kr (−1)k is always negative, and ∞ k=1 k (−1) = −1. Thus the series above converges absolutely, hence converges in A. Indeed it is now easy to see that the series given for xr is a norm limit of polynomials in x with no constant term. Using the Cauchy product formula in Banach algebras in a standard way, 1 one deduces that (x n )n = x for any positive integer n. Proposition 6.3 (Esterle). If A is a Banach algebra then FA is closed under r’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 kr (−1)k (1B − x)k , which is a convex combination in r  Ball(B). So x ∈ 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 the spectrum of x, since such r’th powers are badly behaved at 0. However if 0 is in r r the spectrum of x, and x ∈ rB A , one may define x = lim→0+ (x + 1B ) where the latter is the r’th power according to the Riesz functional calculus. We will soon see that this limit exists and lies in A, and then it follows that it is independent of the particular unital algebra B containing A as a subalgebra (since all unitization norms 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 operator case from the Russian literature from the 50’s [62]. Namely, suppose that B is a unital Banach algebra containing A as a subalgebra, and x ∈ A with numerical range 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 in the complex plane with vertex at 0) of angle ≤ π. Since this is the case we are interested in, we will assume that the numerical range of x is in the closed right half plane. (This is usually not really any loss of generality, since x and hence the just mentioned cone can be ‘rotated’ to ensure this.) Thus the numerical range of x is contained inside a semicircle, namely the one containing the right half of the circle 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 is symmetric about the x-axis, and is composed of an arc of the circle slightly bigger that the right half of the circle, and two line segments which connect zero with the arc. Let Γ be Γ but with points removed that are distance less than to the origin. 1 λt (λ1B − x)−1 dλ. The latter One defines xt to be the limit as → 0 of 2πi Γ integral lies in A + C 1B , by the usual facts about such integrals. If A is nonunital and χ0 is the character on A + C 1B annihilating A then χ0 (xt ) is the limit of  t 1 1 t −1 dλ, which is 2πi Γ λ dλ = 0. So xt ∈ A. Note that xt is 2πi Γ λ (λ1 − χ0 (x)) independent of the particular unitization B used, using the fact that all unitization norms are equivalent. If in addition x is invertible then 0 ∈ / SpB (x), so that we can replace Γ by a curve that stays to one side of 0, so that xr is the rth power of x as given by the Riesz functional calculus. In fact it is shown in [61, Proposition 3.1.9] that xr = lim→0+ (x + 1B )r for t > 0, giving the equivalence with the definition at the start of this discussion. In addition we now see, as we discussed earlier, that the latter limit exists, lies in A, and is independent of B. By [61, Corollary 1.3], the rth power function is continuous on rB A , for any r ∈ (0, 1). Principal nth roots

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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 the functional calculus for sectorial operators [43] (see also e.g. [79, IX, Section 11] for some of the origins of this approach). Namely, if B is a unitization of A (or a unital Banach algebra containing A as a subalgebra) and x ∈ rB A , view x as an operator on B by left multiplication. This is sectorial of angle ≤ π2 , and so we can use the theory of roots (fractional powers) from e.g. [43, Section 3.1] (see also [78]). More generally, if x has no strictly negative numbers in its numerical range with respect to B, then the formula of Stampfli and Williams [76, Lemma 1] and some basic trigonometry 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 properties of such powers include: xs xt = xs+t and (cx)t = ct xt , for positive scalars c, s, t, and t → xt is continuous. There are very many more in e.g. [43]. Also [43, Proposition 3.1.9] shows that xr = lim→0+ (x + I)r for r > 0, the latter power with respect to the usual Riesz functional calculus. It is easy to see from the last fact that the definitions of xr given in this paragraph and in the last paragraph coincide if x ∈ rA and r > 0; so that again xr is in (the copy inside B(B) of) A. Another  ∞ r−1 s (s + x)−1 x ds, the formula we have occasionally found useful is xr = sin(tπ) π 0 Balakrishnan formula (see e.g. [43, 79]). Finally, we mention that there are some lovely iterative descriptions of the square root that we discuss in a forthcoming paper [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 paragraphs and in Proposition 6.3 coincide, if r > 0. We may assume that 0 < r ≤ 1 and 1 (x+ 1 ). Then 1B −y < 1, work in a unital algebra B containing A. Let y = 1+ ∞ rB r and so y as defined in the last paragraphs equals k=0 k (−1)k (1B − y)k since both are easily seen rth power of y as given by the Riesz cal the r  ∞ functional  to equal r k k k (−1) (−1) (1 − y) converges uniformly to (1 − culus. However ∞ k=0 k k=0 k x)k , as → 0+ , since the norm of the difference of these two series is dominated by ∞    r 1

− 1) (1 − x)k  ≤ → 0, (−1)k ( k 1+

1+

k=1   using the fact that for k ≥ 1 the sign of kr (−1)k is always negative. Also, with the 1 r definition of powers in the last paragraphs we have y r = ( 1+ ) = (x + 1B )r → xr + r as → 0 . Thus the definitions of x given in the last paragraphs and in Proposition 6.3 coincide in this case. If A is a subalgebra of a unital Banach algebra B then we define the F-transform on A to be F(x) = x(1B + x)−1 = 1B − (1B + x)−1 for x ∈ rA . This is a relative of the well known Cayley transform in operator theory. Note that F(x) ∈ ba(x) by the basic theory of Banach algebras, and it does not depend on B, again because all unitization norms for A 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 . For Banach 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 are generalizations to general Banach algebras of results from [21]. The main results here in the operator algebra case were proved earlier by the author and Read (some are much sharper in that setting).

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B Lemma 6.4. If A is a subalgebra of a unital Banach algebra B then F(rB A ) ⊂ FA 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 > 0 then 2 x + 1 ∈ CFA where C = + x  . Proof. We have 1 − C −1 (x + 1) = C −1 (C − )1 − x = C −1 By Lemma 2.4 (2), this is dominated by C −1 x  (1 + 2

x2 1 − x.

x2

2 x2 )

= 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 + B R+ FB A = rA and R FA = rA . −1 ∈ FB Proof. If x ∈ rB A and t ≥ 0, then tx(1B + tx) A by Lemma 6.4. −1 → 1B as t  0. So x = By elementary Banach algebra theory, (1B + tx) limt→0+ 1t tx(1B + tx)−1 , from which the results are clear. 

Remark. There is a numerical range lifting result that works in quotients of Banach spaces with ‘identity’ or of approximately unital Banach algebras, if one 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 last paragraph of Section 8 in [21]. As a consequence one can lift a real positive element in such a quotient A/J to a real positive in A. This again is a generalization of a well known C ∗ -algebraic positivity results since as pointed out by Alfsen and Effros (and Effros and Ruan), M -ideals in a C ∗ -algebras (or, for that matter, in an approximately unital operator algebra) are just the two-sided closed ideals (with a cai). See e.g. [15, Theorem 4.8.5]. sin(tπ) xt Lemma 6.7. Let A be a Banach algebra. If x ∈ rA , then ||xt || ≤ 2πt(1−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 any unital Banach algebra containing A, and this case was done in [21]. In the operator algebra case a recent paper of Drury [35] is a little more careful with the estimates for the integral in the Balakrishnan formula mentioned above for xt , and obtains Γ( t ) Γ( 1−t 2 ) xt  ≤ √ 2 2 πΓ(t)Γ(1 − t) if 0 < t < 1 and x ≤ 1. Drury states this for (strictly) positive semidefinite matrices x, and seems to have a typo in the proof, but the proof is easily seen to work for accretive operators on Hilbert space.

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Lemma 6.8. There is a nonnegative sequence (cn ) in c0 such that for any 1 Banach algebra A, and x ∈ FA or x ∈ Ball(A) ∩ rA , we have x n x − 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 1 1 nonzero then lim supn x n  ≤ 1 is the same as saying limn x n  = 1. For 1

1

1

x ≤ x n x − x + x n x ≤ cn + x n x,

n ∈ N,

where (cn ) ∈ c0 as in Lemma 6.8. This property holds if A is an operator algebra by the last assertion of Lemma 6.7. Corollary 6.10. A Banach algebra A with a left bai (resp. right bai, bai) in rA has a left bai (resp. right bai, bai) in FA . And a similar statement holds with rA B and FA replaced by rB A and FA 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 1

the proof in [21, Corollary 3.9], (btn ) 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 one constructed in FA . We imagine that if a Banach algebra has a cai in rA then under mild conditions it 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 that 1 there is a sequence (dn ) of scalars with limit 1 such that x n  ≤ dn for all n ∈ N and x ∈ FA (this is the case for operator algebras by Lemma 6.7). If A has a left bai (resp. right bai, bai) in rA then A has a left cai (resp. right cai, cai) in FA . And B a similar statement holds with rA and FA replaced by rB A and FA for any unital Banach algebra B containing A as a subalgebra. 1

Proof. For the first case, let (fs )s∈Λ = (btn ) be the left bai in FA from Corollary 6.10. Note that fs  ≤ dn and so it is easy to see that fs  → 1 by the Remark after Lemma 6.8. If there is a contractive subnet of (fs ) we are done, so assume that 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 that Λ0 is directed, and that (fs )s∈Λ0 is a subset of (fs )t∈Λ which is a left bai in FA . Then ( f1s  fs )s∈Λ0 is in FA since fs  > 1. So ( f1s  fs )s∈Λ0 is 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 for this reference). Proposition 6.13. Let A be a Banach algebra such that A∗∗ is unital and A has a real positive cai, or more generally suppose that there exists a real positive cai for 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 6.10 since 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

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of contractions, in which case this subnet is a cai in FA , or Λ0 = {s ∈ Λ : fs  ≥ 1} is a directed set and ( f1s  fs )s∈Λ0 is a cai in FA . Next, suppose that (et ) is a cai in rA , and (fs ) is a bai in FA and they have the same weak* limit f . By a re-indexing argument, we can assume that they are indexed 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 , et x1 − x1 , x1 et − x1 , ft x1 − x1 , x1 ft − x1 , et x2 − x2 , · · · , x1 ft − 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 to see that 0 is in the weak closure of Fs,E (since et − ft → 0 weakly and et xk → 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 (where K may depend on n, s, E),  n,s,E with sum 1, such that if rn,s,E = K etk and positive scalars (αkn,s,E )K k=1 k=1 αk K n,s,E and wn,s,E = k=1 αk ftk , then rn,s,E xk − xk , xk rn,s,E − xk , xk wn,s,E − xk , wn,s,E xk − 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 . Since rn,s,E − wn,s,E → 0 with n, it follows that wn,s,E  → 1 with (n, E). So as in the last 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 AarnesKadison type theorem (see the lines after Theorem 2.3; alternatively one may use our Aarnes-Kadison type theorem 7.13 below) that A = xAx for some x ∈ A. Given a cai (ft ) in FA , choose t1 < t2 < · · · with ftk x − x + xftk − x < 2−k . Then it is clear that (ftk ) is a sequential cai in FA .  Remark 6.14. It follows that under the conditions of the last result, one may improve [21, Corollary 6.10] in the way described after that result (using the fact in the remark after [21, Corollary 2.10]). Corollary 6.15. If A is a Banach algebra then rA is closed under rth powers for any r ∈ [0, 1]. So is rB A for any unital Banach algebra B isometrically containing A as a subalgebra. Proof. We saw in the proof of Proposition 6.6 that if x ∈ rB A then x = . Thus by [61, Corollary 1.3] we limt→0+ 1t tx(1 + tx)−1 , and tx(1 + tx)−1 ∈ FB A have that xr = limt→0+ t1r (tx(1 + tx)−1 )r for 0 < r < 1. By Proposition 6.3 and + B r B its proof, the latter powers are in R+ FB A , so that x ∈ R FA = rA ⊂ rA .



In an operator algebra, much is known about the numerical range of fractional powers (see e.g. Section 2 in [26]). In particular, if x is sectorial of angle θ ≤ π2 then xt has sectorial angle ≤ tθ. Indeed this is what allows us to produce ‘nearly positive elements’, as discussed in Section 4. The following fact, which we have not seen in the literature, is the best one has in a general Banach algebra, and this disappointment means that some of the theory from [22, 23, 25] will not generalize to Banach algebras. In particular, one can have accretive elements whose nth roots all 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 ≤ then xt has sectorial angle ≤ tθ + (1 − t) π2 .

π 2

in a unital Banach algebra

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Proof. This is Corollary 6.15 if θ = π2 . Suppose that WB (x) ⊂ Sθ . Then π π e±i( 2 −θ) x is accretive. Hence (wx)t is accretive where w = e±i( 2 −θ) . By [43, Lemma 3.1.4] with f (z) = wz we have (wx)t = wt xt . So wt xt is accretive. Reversing the argument above we see that WB (x) ⊂ ei( 2 −θ) S π2 ∩ e−i( 2 −θ) S π2 = Stθ+(1−t) π2 π

π



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 particular unital Banach algebra containing A. Thus this result follows from the unital case considered in [21, Proposition 3.11].  Lemma 6.18. If p is an idempotent in a Banach algebra A then p ∈ FA iff p ∈ 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∗∗ . 1 1 Indeed (x n ) is a bai for ba(x) in rA , and (F(x) n ) is a bai for ba(x) in FA . Proof. If x ∈ rB A then the proof of [21, Proposition 3.17] shows that ba(x) has a bai in FB , and hence any weak* limit point of this bai is a mixed identity A ∗∗ 1 B ∗∗ ⊂ F  residing in FB ∗∗ A . Indeed (x n ) is a bai for ba(x) in rA , A The following new observation is a simple consequence of the above which we will need later. Corollary 6.20. If A is a nonunital Banach algebra and if E and F are subsets of rA then EA = EB, AF = BF , and EAF = EBF , where B is any unitization 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 the Aarnes–Kadison theorem 2.1. In an operator algebra or Arens regular Banach algebra things are cleaner (see [9, 22, 23]). For a Banach algebra A and x ∈ rA , 1 we write s(x) for the weak* Banach limit of (x n ) in A∗∗ . That is s(x)(f ) = 1 LIMn f (x n ) 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 identity of ba(x)∗∗ , and is idempotent. By the Hahn–Banach theorem it is easy to see that 1

w∗

by an argument after [21, Proposis(x) ∈ conv({x n : n ∈ N}) . In x ∈ rB A then ∗∗ ⊂ FA∗∗ . If ba(x) is Arens regular then tion 3.17] we have s(x) ∈ FB ∗∗ ∩ A∗∗ = FB ∗∗ A s(x) will be the identity of ba(x)∗∗ . We call s(x) above a support idempotent of x, or a (left) support idempotent of xA (or a (right) support idempotent of Ax). The reason for this name is the following result.

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Corollary 6.21. If A is a Banach algebra, and x ∈ rA then xA has a left bai in FA and x ∈ xA = s(x)A∗∗ ∩ A and (xA)⊥⊥ = s(x)A∗∗ . (These products are with respect to the second Arens product.) Proof. The proof of [21, Corollary 3.18] works, and gives that xA has a left B  bai in FB A if x ∈ rA . 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 iff s(y)s(x) = s(x). In this case xA = A iff s(x) is a left identity for A∗∗ . (These products 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. If x ∈ A ∩ rB , then the support projection of x computed in A∗∗ is the same, via the canonical embedding A∗∗ ∼ = A⊥⊥ ⊂ B ∗∗ , as the support projection of x computed ∗∗ in B . In Section 2 we mentioned the paper of Kadison and Pedersen [55] initiating the development of a comparison theory for elements in C ∗ -algebras generalizing the von Neumann equivalence of projections. Again positivity and properties of the positive cone are key to that work. Admittedly their algebras were monotone complete, but many later authors have taken up this theme, with various versions of equivalence or subequivalence of elements in general C ∗ -algebras (see for example [10] or [3, 66] and references therein). Indeed recently the study of Cuntz equivalence and subequivalence within the context of the Elliott program has become one of the most important areas of C ∗ -algebra theory. In [18] Neal and the author began a program of generalizing basic parts of the theory of comparison, equivalence, and subequivalence, to the setting of general operator algebras. In that paper we focused 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 follow the 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 form xA studied below. There is a lot more work needed to be done here, our paper was simply the first steps. Also, we have not tried to see if any of this generalizes to larger classes of Banach algebras. Much of our theory in [18] depends on facts for nth roots of real positive elements. Thus we would expect that a certain portion of this theory generalizes to Banach algebras using the facts about roots summarized in 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 there exists y ∈ A with xyx = x. The following result (which is the non-approximately unital case of [21, Theorem 3.21]) should be compared with the C ∗ -algebraic version of the result due to Harte and Mbekhta [45, 46], and to the earlier version of the result in the operator algebra case (see particularly [22, Section 3], and [25, Subsection 2.4] and [24]).

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Theorem 7.1. Let A be a Banach algebra, and x ∈ rA . The following are equivalent: (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 the assertions regarding (vi), since there we may assume A is unital by definition of spectrum 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 did in [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, and if x ∈ rB A , 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 of the 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 algebra A has no left identity, then xA = A for all x ∈ rA . In [13] we generalized the concept of hereditary subalgebra (HSA), an important tool in C ∗ -algebra theory, to operator algebras, and established that the basics of the 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 the latter term we mean a subalgebra D with DAD ⊂ D. Generalizing Theorem 2.2 above, we showed in [22, 23] that HSA’s and right ideals with left cais in operator algebras are manifestations 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 study the relationship between HSA’s and one-sided ideals with one-sided approximate identities. Some aspects of this relationship is problematic for general Banach algebras (see [21, Section 4]), but it works much better in separable algebras. As we said around Theorem 2.3, our work is closely related to the results of Sinclair and others on the Cohen factorization method (see e.g. [37,74]), which does include some similar sounding but different results. We define a right F-ideal (resp. left F-ideal) in a Banach algebra A to be a closed right (resp. left) ideal with a left (resp. right) bai in FA (or equivalently, by Corollary 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, by Corollary 6.10, in rA ). Perhaps these should be called F-HSA’s to avoid confusion with the notation of [13, 22] where one uses cai’s instead of bai’s, but for brevity we

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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 of right 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 left F-ideals mentioned above. At this point we have to jettison FA and rA as defined at the start of Section 6, if A is not approximately unital, because the remaining results are endangered if FA and rA are not closed and convex. Indeed most results in Sections 4 and 7 of B [21] would seem to need FA and rA to be replaced by FB A and rA for a fixed unital Banach algebra B containing A as a subalgebra. That is, we need to fix a particular unitization of A, not consider all unitizations simultaneously. Of course if A is an operator algebra then there is a unique unitization, hence all this is redundant. (As we said early in Section 6, we could also fix this problem by redefining FA = ∩B FB A and rA = ∩B rB A , the intersections taken over all B as above. Everything below would 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. B B right) bai in FB D (or equivalently, by Corollary 6.10, in rD ). Note that one-sided F B ideals in A are exactly subalgebras of A which are one-sided F -ideals in A + C 1B in the sense of [21, Section 4]. We define a B-hereditary subalgebra (or B-HSA for short) of A to an inner ideal B D in A with a two-sided bai in FB D (or equivalently, by Corollary 6.10, in rD ). Note that B-HSA’s in A are exactly subalgebras of A which are HSA’s in A + C 1B in the sense of [21, Section 4]. Again a B-HSA D induces a pair of right and left FB -ideals J = DA and K = 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 -ideals mentioned above. Because of the facts at the end of the second last paragraph, and because of Corollary 6.20, in the following four results we can assume that A is unital, in which case the proofs are in [21]. These results are all stated for a Banach algebra with unitization B, but they could equally well be stated for a closed subalgebra of a unital Banach algebra B. Lemma 7.6. Suppose that J is a right FB -ideal in a Banach algebra with unitization 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, taking dn K = { n1 en } ∪ {0}, where (en ) is the left bai. Taking K = { nd } ∪ {0} where {dn } n is a countable dense set gives the second. Corollary 7.7. Let A be a Banach algebra with unitization B. The closed right ideals of A with a countable left bai in rB A are precisely the (topologically) B ‘principal right ideals’ zA for some z ∈ FA which is also in the ideal. Every separable right FB -ideal is of this form.

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Corollary 7.8. Let A be a Banach algebra with unitization B. The right FB -ideals in A are precisely the closures of increasing unions of right ideals in A of 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 xk ak for some ak ∈ A. Of course any algebraically finitely generated is algebraically countably generated. 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 = qA for an idempotent q ∈ FB A . This is also equivalent to J being algebraically countably generated 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 unital operator 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 those references. Proof. By Corollary 7.8 a right FB -ideal is the closure of an increasing union of right ideals in A of the form zA for z ∈ FA . However this is the characterization of r-ideals from [22]. Similarly for the left ideal case. A similar argument works for the HSA case using Corollary 7.14; alternatively, if D is a B-HSA then D has  a 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 the right FB -ideals in A are precisely the sets of form EA for a subset E ⊂ FB A (or equivalently, E ⊂ rB ). One direction of this is obvious: just take E to be the bai A B in FB (resp. r ). However the other direction is false in general Banach algebras, A A although it does hold in operator algebras [22] and commutative Banach algebras. (Another characterization of closed ideals with bai’s in commutative Banach algebras may be found in [60].) That EA is a right FB -ideal in A if A is a commutative Banach algebra and E ⊂ FB A , follows from Theorem 7.1 in [21] after noting that by Corollary 6.20 we may replace A by A + C 1B . The key part of the proof of Theorem 7.1 in [21] is to show that for any finite subset G of E there exists an element zG ∈ FB A ∩ EA with GA = zG A. Indeed one can take zG to be the average of the elements in G. 1 Then the net (zGn ), indexed by the finite subsets G of E and n ∈ N, is easily seen to be a left bai in EA from FB A . An application of this: for such subsets E of an operator algebra or commutative Banach algebra A, the Banach algebra generated by E has a bai in FB A . This follows from the argument above since the zG above 1

are in the convex hull of E, hence the bai (zGn ) is in the Banach algebra generated by E. In particular, if A is generated as a Banach algebra by rB A , then A has a bai, and this bai may be taken from rB . (The present paragraph is a summary of the A results in [21, Section 7], and a generalization of these results to the case that A is not approximately unital.) Unless explicitly said to the contrary, all the remaining results in this section are again generalizations to general Banach algebras of results from [21]. Some

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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 matching results in [21]. Lemma 7.11. Let A be a Banach algebra with unitization B. Let D be a closed subalgebra of A. If D has a bai from FB A , then for every compact subset K ⊂ D, there is an x ∈ FB such that K ⊂ xDx ⊂ xAx. D 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 be an inner ideal in A. Then D has a countable bai from FB D (or equivalently, from B rB ) iff there exists an element z ∈ F with D = zAz. Thus D is of the form in D D 1 n Lemma 7.5, and such D has a countable commuting bai from FB D , namely (z ). Any separable inner ideal in A with a bai from rB D 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 do not vanish on A. In [22, 25] we generalized some basic aspects of strictly positive elements 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 algebra then 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 ∈ rB A iff . (iv) A has a sequential bai from rB A The approximate identity in (iv) may be taken to be commuting, indeed it may be 1 taken to be (x n ) for the last mentioned element x. If A is separable and has a bai B in rA then A satisfies (iv) and hence all of the above. Moreover if A is an operator algebra 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 1 [22]. In the operator algebra case, if (ii) holds with x ∈ FA then (( 21 x) n ) is a cai for A in 12 FA 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 strictly real positive element then it is explained in the long discussion before [25, Lemma 3.2] 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 our earlier Corollary 7.10. Corollary 7.14. The B-HSA’s in a Banach algebra A with unitization B are exactly the closures of increasing unions of HSA’s of the form zAz for z ∈ FB A.

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Acknowledgements. We thank the referee for his comments. Tragically, during the revision stage of this article we learned that our dear friend, brother, and coauthor Charles Read had passed away while jogging in Canada, just over 48 hours after 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 truly amazing lecture). See you on the other side of the river, Charles!

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Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13503

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 a bounded 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 eitD xe−itD ξ, η is n times differentiable. We give several characterizations of n times weak differentiability, among which, one is original. These results are used to show that for a von Neumann algebra M on H the algebra of n times weakly D−differentiable operators in M has a natural representation as a reflexive subalgebra of B(H ⊗ C(n+1) ).

1. Introduction Let D be a self-adjoint, usually unbounded, operator on a Hilbert space H and x a bounded operator on H, then Quantum Mechanics, [7] Operator Algebra [5] and Noncommutative Geometry [3] offer plenty of reasons why we should be interested in operators that are formed as commutators [D, x] = Dx−xD. In noncommutative geometry we want to find a set-up such that classical smooth structures may be described in a language based on operators on a Hilbert space. A derivative is described in terms of a commutator [D, x] and a higher derivative via an iterated commutator [D, [D, . . . , [D, x] . . . ]], so a basic question is to determine the set of operators for which such an iterated commutator makes sense. It is not clear when a commutator such as [D, x] is densely defined and bounded on its domain of definition, and for two bounded operators x, y such that [D, x] and [D, y] are bounded 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 a derivation on a subalgebra of B(H) in a canonical way. In the article [2] we realized that the concept we named weak D−differentiability provides a set-up, which may be used to decide for which bounded operators x the commutator [D, x] should be defined. We say that a bounded operator x on H is weakly D−differentiable if for each pair of vectors ξ, η in H the function eitD xe−itD ξ, η is differentiable. For a weakly D−differentiable operator x the commutator [D, x] is then defined and bounded on all of the domain of D, so it is possible to define a derivation δw from the algebra of weakly D−differentiable operators into B(H). We were later informed that the concept of weak D−differentiability, the algebra property of the 2010 Mathematics Subject Classification. Primary 46L55, 58B34; Secondary; 37A55, 47D06, 81S05. c 2016 American Mathematical Society

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weakly differentiable operators and the derivation δw are well known by researchers in mathematical physics [1] and [4], so according to the notation of the book [1], see page 192, the mentioned algebra is C 1 (D, H). We will adopt this notation but 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 A is then denoted C 1 (A, D). First of all we will like to study the algebra of higher weak derivatives which with the notation of [1] is the algebra C n (D, H) and the algebra of n times weakly D−differentiable operators inside a C*-algebra A on H is C n (A, D) := C n (D, H) ∩ A. In section 4 we give several characterizations of those operators that are n times weakly D−differentiable, and we would like to mention here, that a bounded operator x is n times weakly D−differentiable if and only if for any k in {1, . . . , n} the k th commutator [D, [D, . . . , [D, x] . . . ]] is defined and bounded on dom(Dk ). This is known to many mathematicians, but we could not find a reference where the details are easy to follow, so we have included a proof here. This characterization of n times weak D−differentiability will be crucial for the results of section 5 on reflexive algebras. We also give a characterization of higher weak differentiability based on an embedding of the higher commutators [D, [D, . . . [D, a] . . . ]] into a linear space consisting of infinite matrices of bounded operators. This set-up is original, and we hope that it will turn out to be a useful frame inside which some operator theoretical questions can be dealt with in a way which avoids the tiresome considerations of the validity of products and sums of 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 by B(H)  x → eitD xe−itD ∈ B(H) is actually a so-called adjoint semigroup on a dual Banach space. Adjoint semigroups were first studied in [8], and [6] contains a survey of the general theory of adjoint semigroups. Our usage of the general theory is limited, but several things could have been presented in an easier way in [2], if we 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 been presented in an optimal way in [2]. The Definition 1.1 of [2] defines a bounded operator x to be weakly D−differentiable if there exists a bounded operator b on H such that for any pair of vectors ξ, η in H we have   eitD xe−itD − x − b ξ, η = 0. lim | t→0 t This definition implies that for any ξ, η the function t → eitD xe−itD ξ, η is differentiable at t = 0, and it is stated, but not explicitly proven that this latter property implies weak D−differentiability as defined via a weak derivative b. It is quite easy to see that the two sorts of weak D−differentiability are equivalent and all the arguments are presented in [2], but the consequences are not made sufficiently clear. The right formal definition of weak D−differentiability then becomes as follows. Definition 2.1. A bounded operator x on H is weakly D−differentiable if for any pair of vectors ξ, η in H the function t → eitD xe−itD ξ, η is differentiable at t = 0.

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To see that our present definition of weak D−differentiability implies the existence of a weak derivative, i.e. a bounded operator b such that Definition 1.1 of [2] is satisfied, we refer the reader to the proof of (ii) ⇒ (iii) in Theorem 3.8 of [2]. That step is the crucial part of the proof, and it is based on the uniform boundedness principle applied to all the operators {

eitD xe−itD − x : t = 0}. t

This set is bounded because any function such as t → eitD xe−itD ξ, η is differentiable at t = 0, and hence the set of values eitD xe−itD − x ξ, η : t = 0} t is bounded and the principle applies. The existence of b then follows from the rest of Theorem 3.8 of [2]. We will quote that theorem below and define the higher weak derivatives, 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 function t → eitD xe−itD is differentiable at t = 0, with respect to the norm topology on B(H). In analogy with the definition of weak D−differentiability we say that x is strongly D−differentiable if for each vector ξ in H the function t → eitD xe−itD ξ is differentiable at t = 0 with respect to the norm topology on H. It follows from [2] that weak and strong D−differentiability are equivalent but uniform D−differentiability is in general a stronger property. The book [1] studies strong D−differentiability in its Chapter 5, and it mentions that this concept is equivalent to weak D−differentiability, which we prefer to work with, because it seems to be closer to the classical concepts involving differentiable functions on R. Anyway we already have adopted the notation from [1], but modified it 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 selfadjoint operator on H. Then the algebra of n times weakly D−differentiable operators in A is denoted C n (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) := eitD xe−itD . For a weakly D−differentiable operator x in B(H) it then follows, that δw (x) is the d αt (x)|t=0 , but there is also the possibility of having a weak operator derivative dt norm 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] the following result, which tells that higher uniform derivatives are closely related to weak derivatives. Theorem 2.3. Let x be a bounded operator on H and n ≥ 2. If x is n times weakly 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 language used. Not all of the results below may be generalized to higher derivatives and for those properties, which can be extended, we will give the necessary precise definitions, when needed.

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Theorem 2.4. Let x be a bounded operator on H. The following properties are equivalent: (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 (eitD xe−itD − x)ξ, η =δw (x)ξ, η t→0 t  x domD ⊆ domD and δw (x)dom(D) =i(Dx − xD)

∀ξ, η ∈ H

lim

∀t ∈ R :

αt (x) − x ≤δw (x)|t|.

The properties (iii) and (iv) from the theorem just above have no simple generalizations to higher derivatives and will not be discussed here at all. The remaining five properties all suggest natural extensions to the setting of higher weak derivatives 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 to make the following observation explicit. The reason being, that although most people 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 for any pair of vectors ξ, η in H the function αt (x)ξ, η is differentiable on R and d αt (x)ξ, η = αt (δw (x))ξ, η. dt Proof. By definition the equality holds for t = 0, and arguments similar to the ones given in the proof of Lemma 2.1 of [2] show that the identity may be translated from t = 0 to any other real t.  This lemma has an immediate consequence, which we formulate as a proposition, since it is important, although its proof is trivial. Proposition 2.6. A bounded operator x on H is n times weakly D−differenn tiable if and only if x is in dom(δw ) and if and only if for any pair ξ, η in H the n function αt (x)ξ, η is in C (R). If x is n times weakly differentiable then dn n αt (x)ξ, η = αt (δw (x))ξ, η. dtn Proof. Follows from Lemma 2.5 by induction. We will end this section by introducing a norm on C n (D, H). Definition 2.7. For any x in C n (D, H) the norm xn is defined by xn =

n  1 j δw (x). j! j=0



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This is not the same norm as the one defined in [1] Definition 5.1.1 at page 195, to that norm, and it follows from [2] Proposition 3.10 that   n but it is equivalent C (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 its powers is sufficiently well established for most purposes, but it is not. The problem is that we do not know how to relate higher weak derivatives to expressions involving iterated commutators with iD. If x is in dom(δw ) then it follows from Theorem 2.4 that iDx − ixD is defined on all of dom(D) and δw (x) is the closure of iDx − ixD. 2 ), then it is natural to look at the second iD commutator If x is in dom(δw iD(iDx − ixD) − (iDx − ixD)(iD), but we know nothing about its domain of definition, possible boundedness and closure. In this section we will show that the properties of the higher commutators are as nice as we can possibly hope for. We will show that for a bounded n times weakly differentiable operator x, the n times iterated commutator between iD and n (x). We will base x is defined on dom(Dn ) and the closure of this operator equals δw the proof of this on the results of Theorem 2.4. In order to simplify the writings below 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 domain of z which are mapped into the domain of y by z, and a sum is defined on 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 following observation. Lemma 3.2. Let x be a bounded operator in B(H) and n a natural number. If x is n times weakly differentiable then for any k in {1, . . . , n} : k−1 δw (x) : dom(D) → dom(D), k k−1 k−1 (x)|dom(D) = i[D, δw (x)] = d(δw (x)). δw

Proof. If x is n times weakly differentiable, then for any k in {1, . . . , n} we k−1 (x) is in dom(δw ). Then Theorem 2.4 item (vii) presents the claimed have δw k−1 (x).  properties of δw k (x) is the closure of the commutator The statements in Lemma 3.2 show that δw k−1 k−1 [iD, δw (x)], but if k > 1 then δw (x) is defined as a closure of the commutator k−2 k−1 (x)], so we have no direct control over the operator [iD, δw (x)]. This [iD, δw is not sufficient for our purpose, so we want to look at the restriction of such a commutator to dom(Dk ), and then show that on this domain the higher weak derivative may be computed without any closure operations, as a higher commutator, k (x). and that the closure of this algebraically defined commutator equals δw

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Proposition 3.3. Let x be an n times weakly differentiable bounded operator on H, then for k in {1, . . . , n} k−1 (i) δw (x)dom(D) (ii) x dom(Dk ) (iii) dom(dk (x)) k (iv) δw (x)|dom(Dk ) k (v) δw (x)

⊆ dom(D) ⊆ dom(Dk ), = dom(Dk ) = dk (x) = closure(dk (x)).

Proof. The item (i) follows from Lemma 3.2. The following four items are related and we show them by induction on k. For k = 1 the results follow again from item (iv) of Theorem 2.4. Then suppose 1 < k ≤ n and that the statements are true for natural numbers in the set {1, . . . , k − 1}. We start by proving (iii), so we will k−1 (x)ξ and choose a vector ξ in dom(Dk ), then ξ is in dom(Dk−1 ) so dk−1 (x)ξ = δw k−1 k−1 (x)ξ is in dom(D), and finally ξ is in dom(iDd (x)). By assumpby item (i) d 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)k xDk . 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 ) and k−1  k−1 D dom(Dk ). By (i) and the induction hypotheses Dδw (x) is dk−1 (x)D = δw k k−1 (x) on that domain. Hence item (iv) follows. defined on dom(D ) and equals Dd 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 Theorem k k−1 (x) is the closure of the commutator d(δw (x))|dom(Dk ), but 3.8 of [2]. Then δw k the latter equals d (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 a vector ξ in dom(Dk ) shows that k−1  k − 1 k−1 k−1 (x)ξ = (i) (3.1) d (−1)j Dk−1−j xDj ξ j j=0 For j > 0 we have Dj ξ is in dom(Dk−j ) and by assumption xDj ξ is in dom(Dk−j ) so Dk−1−j xDj ξ is in dom(D). Then for j = 0 we find that Dk−1 xξ may be written as a difference of two vectors in dom(D) and hence xξ is a vector in dom(Dk ), and item (ii) is proven.  4. Equivalent Properties In analogy with the results of Theorem 2.4 we want to show that higher order weak differentiability may be characterized in several different ways. Some of the properties we find are expressed in terms of infinite matrices of operators, so we will include a short description of this set-up here. In [2] we defined a sequence of pairwise orthogonal projections with sum I in B(H) by letting en denote the spectral projection for D corresponding to the interval ]n − 1, n]. Then we defined M to be all matrices (yrc ) with r and c integers and yrc an operator in er B(H)ec . Any bounded operator x on H induces an element

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m(x) in M which is defined as m(x)rc := er xec . The operator D has a representation m(D) in M too, and it is defined as a diagonal matrix m(D)rc = 0, if r = c and diagonal elements dr := m(D)rr := Der . Then for any element y = (yrc ) in M, the commutator i[m(D), y] makes sense in M by i[m(D), y]rc := i(dr yrc − yrc dc ), and we may define a linear mapping dM : M → M by ∀y = (yrc ) ∈ M : dM (y)rc := idr yrc − iyrc dc . By the computations above we get that the powers dnM are given as n    n n n . (4.1) ∀n ∈ N ∀y = (yrc ) ∈ M : dM (y)rc = i (−1)n−k dkr yrc dn−k c k k=0

We can now formulate our result on characterizations of higher weak differentiability. Theorem 4.1. Let x be a bounded operator on H and n a natural number. The following properties are equivalent: (i) (ii) (iii) (iv)

n x is in dom(δw ). x is n times weakly D−differentiable. x is n times strongly D−differentiable. ∀k ∈ {1, . . . , n}

x : dom(Dk ) → dom(Dk ) k (x). dk (x) is defined and bounded on dom(Dk ) with closure δw

(v) For k in {1, . . . , n} the infinite matrix dkM (m(x))) represents a bounded operator. (vi) There exists a core F for D such that for any k in {1, . . . , n} the operator dk (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 strongly k (x) is the k th strong derivative by Theorem 2.4, and since differentiable, then δw k (x) is strongly this operator is weakly differentiable, the same theorem shows that δw k+1 differentiable with strong derivative δw (x). (iii) ⇒ (ii): Follows from the Cauchy-Schwarz inequality. (ii) ⇒ (iv): This follows from Proposition 3.3 (iv) ⇒ (v): k (x) exists and is a bounded operator such Let 1 ≤ k ≤ n, then we are given that δw k k k that δw (x)|dom(D ) = d (x). Let c be an integer then ec H ⊆ dom(Dk ) so for any

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integer r we get k er δw (x)ec = er dk (x)ec = ik

k    k j=0

j

(−1)k−j er Dj xDj ec = 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 zk on H such that for any pair of integers r, c we have er zk ec = dkM (x)rc . The case k = 1 is covered by Theorem 2.4. The proof may be found in [2], but we recall the main step, because we will use it repeatedly below. For any vector ξ from ec H we showed that xξ is in dom(D). It then follows that for any integer r and a vector ξ in ec H we have xξ is in dom(D) and er z1 ξ = i(dr er xec − er xec dc )ξ = ier (Dx − xD)ξ. and we concluded that x is weakly differentiable with δw (x) = z1 . We may now j (x) = zj assume that 1 < k ≤ n and x is weakly differentiable of order k − 1 with δw k−1 for 1 ≤ j ≤ k − 1. Then for ξ in ec H we get δw (x)ξ is in dom(D) so we have k−1 k−1 (x)ec − er dM (x)ec dc )ξ er zk ξ =i(dr er dM k−1 k−1 (x) − δw (x)D)ξ. =ier (Dδw k−1 k Hence δw (x) is weakly differentiable and δw (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 = 1 first. Then (iD)x − x(iD) is defined and bounded on the core F for D, so by Theorem 2.4 item (vi) x is in dom(δw ) and y1 = δw (x). Let us now suppose that j (x), for 1 ≤ j ≤ k − 1, then for any ξ in F we 1 < k ≤ n and we know that yj = δw can 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   yk ξ = lim dk (x)ξn = lim (iD)dk−1 (x)ξn − dk−1 (x)(iD)ξn n→∞ n→∞   k−1 k−1 = lim (iD)δw (x)ξn − δw (x)(iD)ξn . n→∞

Since the last part of these equations forms a convergent sequence we find that k−1 lim (iD)δw (x)ξn exists and

n→∞

k−1 k−1 (x)ξn = yk ξ + δw (x)(iD)ξ. lim (iD)δw

n→∞

k−1 Hence δw (x)ξ is in dom(D) and k−1 k−1 (x)ξ − δw (x)(iD)ξ. yk ξ = (iD)δw k−1 k By Theorem 2.4 we get that δw (x) is weakly differentiable and δw (x) = yk , so x is n times weakly differentiable, and the theorem follows. 

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5. Reflexive representations of the algebras of higher weakly differentiable operators in a von Neumann algebra In this section we will consider the case where we are dealing with a von Neumann algebra M on H and study aspects of the algebras C n (M, H) of n times higher weakly D−differentiable elements inside M, but unlike the case in noncommutative geometry we will not assume that C n (M, D) is dense in M in any ordinary topology. The prototype of a von Neumann algebra, or rather the commutative example which may give inspiration for general results on von Neumann algebras is the algebra of measurable essentially bounded functions on the unit circle, L∞ (T, dθ), and in this setting C n (M, D) is nothing but the n times d −differentiable functions, so we find here that for n ≥ 1 we have weakly D := −i dθ n ∞ C (L (T, dθ), D) = C n (C(T), D), and we may wonder if the von Neumann algebra property plays a role at all ? We have no answer, but this might be because our understanding of the relations between noncommutative and commutative geometry is still quite limited. Below we will describe the property called reflexivity of an algebra of bounded operators, but for the moment just say, that a von Neumann algebra M is reflexive and that property is partly inherited by C n (M, D), in the sense that this algebra has a representation as a reflexive algebra of bounded operators 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 for an algebra of operators to have. This follows from von Neumann’s bicommutant theorem which shows that if an algebra of bounded operators on H is self-adjoint and reflexive then it is a von Neumann algebra. The algebras we will study are not self-adjoint, but sub-algebras of the upper triangular matrices in Mn+1 (B(H)), so von Neumann’s theorem does not apply directly in our situation. We will remind you of the definition of reflexivity as it was defined by Halmos and 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) is the lattice of closed subspaces of H which are left invariant by each of the operators in S. (ii) Let G denote a collection of closed subspaces of H then Alg(G) is the algebra of bounded operators on H which leave each of the subspaces in G 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 on a Hilbert space K is that it is an ultraweakly closed subspace of B(K) and then it 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 quotient of the predual of B(K). In the set-up for the classical commutative differential geometry such kinds of dualities are well known and widely used. The reflexivity is actually stronger than this duality property, but so far we have not been able to single out a property which solely depends on the reflexivity of a certain representation of the algebra C n (M, D). We will now formulate the result: Theorem 5.2. Let M be a von Neumann algebra on a Hilbert space H, D a self-adjoint operator on H and n a non-negative integer. There exists a unital

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injective algebraic homomorphism Φn : C n (M, D) → B(H ⊗ Cn+1 ) such that the image Rn (M, D) := Φn (C n (M, D)) is a reflexive algebra on H ⊗ Cn+1 . 1 For any x in C n (M, D) : n+1 xn ≤ Φn (x) ≤ xn . Proof. For n = 0 we have C 0 (M, D) = M, and then C 0 (M, D) is a reflexive subalgebra of B(H), so we define Φ0 := idC 0 (M, D), and R0 := M. For n > 0 we will construct a representation Φn of C n (M, D) into the upper triangular matrices with constant diagonals inside the (n + 1) × (n + 1) matrices over B(H) such that for an x in C n (M, D) the representation is given by ⎛ ⎞ 1 n 2 (x) . . . x δw (x) 12 δw n! δw (x) 1 n−1 ⎜0 ⎟ x δw (x) . . . (n−1)! δw (x)⎟ ⎜ ⎜. ⎟ . . . . . . ⎜ ⎟ ⎜ ⎟ Φn (x) := ⎜ . . . . . . . ⎟ 1 2 ⎜0 ⎟ . . . . δw (x) ⎜ ⎟ 2 δw (x) ⎝0 ⎠ . . . . x δw (x) 0 . . . . 0 x j (x). and the element in the j’th upper diagonal is j!1 δw n We define Rn := Φn (C (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 does not apply right away. The short proof of the homomorphism property is then that the results of Theorem 4.1 show that the algebraic arguments are still valid when restricted to take place on the domain dom(Dn ) only. We will like to show this with some more details because these arguments will be needed, when we want to show 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: ⎛ ⎞ 0 1 0 . 0 ⎜0 0 1 . 0⎟ ⎜ ⎟ ⎟ Bn := ⎜ ⎜. . . . .⎟ . ⎝0 . . 0 1⎠ 0 . . 0 0 (n+1)

Then Bn is nilpotent and satisfies Bn = 0, which will be very useful in the computations to come. First we can describe Φn (x) inside the tensor product B(H) ⊗ Mn+1 (C) as Φn (x) = x ⊗ I +

n  1 j δw (x) ⊗ Bnj , j! j=1

and we see from Theorem 4.1 that all the elements in the sum are defined as elementary operator theoretical products or sums of such products on the space dom(Dn ) ⊗ C(n+1) . We will then define Dn = dom(Dn ) ⊗ C(n+1) , and the coming computations will all take place on this dense subspace of H ⊗ C(n+1) . We will work with 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

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by 0 ≤ j ≤ n. Then the matrix elements are indexed by {ij} with i, j ∈ {0, . . . , n} too. For x in C n (M, D) the Theorem 4.1 shows that for any j in {1, . . . , n} we have 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 C n (M, D) and any vector ξ in dom(Dn ), the vector ξ will be in the domain of definition for x0 Dj1 x1 . . . Djk xk . We will lift this product to the matrices, and in order to do so we introduce the canonical amplification ι(x) of B(H) into M(n+1) (B(H)) by ι(x) := x ⊗ I. Then for x0 , x1 , . . . , xk in C n (M, D) we can define a product of operators which always will be defined on Dn by the following convention:  ι(x0 )Snj1 ι(x1 ) . . . Snjk ι(xk )Dn   if j1 + · · · + jk > n 0Dn  :=  (j +···+jk )  j1 Dn if j1 + · · · + jk ≤ n. x0 (iD) x1 . . . (iD)jk xk ⊗ Bn 1 This means that Sn and ι(C n (M, D)) generate an algebra with this special product. The product is a bit more complicated than just the product of the restrictions to Dn of each of the factors. This is because the operator D does not map dom(Dn ) into dom(Dn ), so Sn does not map Dn into Dn but anyway all the products mentioned make sense by first making the standard operator product and then restricting the outcome to Dn . We can then define Tn as the algebra of matrices defined on Dn with this product and generated by Sn and ι(C n (M, D)) The point of this is that we may now use standard algebra on this associative unital algebra and we define elements Tn and its inverse Tn−1 by exponentiating Sn . The nil-potency of Sn gives us the following formulas inside this algebra: n n   1 j 1 Sn (−Sn )j . Tn := exp(Sn ) = I ⊗ I + Tn−1 := exp(−Sn ) = I ⊗ I + j! j! j=1 j=1 In order to relate Sn and Tn to the unital algebra Rn we remind you that in any unital associative algebra C with a nilpotent element s we may  study  the derivation 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 C n (M, D) we have   j  δw (x) ⊗ Bnj Dn if j ≤ n j  ad(Sn ) (ι(x))Dn = 0Dn if j > n, so the equalities above yield the following identities in the algebra Tn . (5.1)

Tn ι(x)Tn−1 = exp(Sn )ι(x) exp(−Sn ) = exp(ad(Sn ))(ι(x)) n    1 j δw (x) ⊗ Bnj Dn = ι(x) + j! j=1  = 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 will be established. The proof is made by induction and for the case of n = 0 the family L0 is just the set Lat(M) of closed subspaces which are invariant under any element in M, and and it follows from von Neumann’s bicommutant theorem that

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C 0 (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 induction argument it is convenient to think of the Hilbert spaces H ⊗ C(n+1) , as a nested family 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 K onto Hn . For each n we will also identify Hn with the abstract tensor product H ⊗ C(n+1) in the way that an expression ξ ⊗ ej which appears in both spaces are identified. In this way Rn may be identified with some upper triangular matrices whose entries are 0 whenever any index is bigger than n, or described as a subspace of the bounded operators on K which satisfies X = En XEn . We then see that for 0 ≤ j ≤ n the subspace Hj is invariant for Rn , and we also note that for any closed subspace 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 it is practical to redefine ι(x) to act on H ⊗ 2 (N0 ) by ι(x) := x ⊗ I2 (N0 and also redefine Bn as the canonical image of Bn in B(K) under the embedding of Rn into En B(K)En . For a natural number n we define a subspace Pn of Hn by (5.2) n  1 (iD)j ξ⊗en−j : ξ ∈ dom(Dn ) }, Pn := {Tn (ξ⊗en ) : ξ ∈ dom(Dn )} = {ξ⊗en + j! j=1 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, so Pn is a closed subspace of Hn and by the relation (5.1) we get that for any x in C n (M, D) and any ξ in dom(Dn ) we have

(5.3)

xξ ∈ dom(Dn ) since x ∈ C n (M, D), Φn (x)Tn ξ ⊗ en = Tn ι(x)ξ ⊗ en = Tn (xξ) ⊗ en ,

so Pn is an invariant subspace for Rn acting on Hn . If j < n then for any x D) ⊆ C j (M, D) we see by the construction of Φj (x) and Φn (x) that in C n (M,  Φn (x)Hj = Φj (x) Hence for any j < n we also have that Pj is a closed invariant subspace for Rn .  := To construct the last invariant subspace we remind you that if we define D it   D + I then D is also a self-adjoint operator, and since exp(itD) = e exp(itD) the corresponding automorphism groups α t and αt are identical so for any n in N0 we n n  = C n (M, D) and δw  n = Rn so a closed = δw . In particular R have C n (M, D)  n is also invariant for Rn . We may then subspace of Hn , which is invariant for R repeat the construction made for Pn but now based on D +I to obtain the invariant

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subspace Qn which is obtained via the equations below: n  1 n : = I ⊗ En + T (i(D + I))j ⊗ Bnj j! j=1 (5.4)

Qn : = {Tn (ξ ⊗ en ) : ξ ∈ dom(Dn )} = {ξ ⊗ en +

n  1 (i(D + I))j ξ ⊗ en−j : ξ ∈ dom((D + I)n ) }. j! j=1

We need to remark that dom((D + I)n ) equals dom(Dn ), and that statement follows 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 ) : En XEn = X.} The case n = 0 is already established, so let us assume that n > 0 and the statement is true for n − 1, and let X be an operator in Alg(Ln ) such that En XEn = X. Then H(n−1) is an invariant subspace for X so XE(n−1) = E(n−1) XE(n−1) and we find immediately 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) such that XE(n−1) = Φ(n−1) (x). Unfortunately we do not know that the operator x is in C n (M, D) too, but we will show it now and then prove that X = Φn (x). We know 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 ), Tn ξ ⊗ en and Tn ξ ⊗ en be the corresponding vectors in Pn and Qn respectively. The invariance of Pn under X has as its first consequence that for the operator entry xnn of X we get xnn ξ is in dom(Dn ). If we look at the (n − 1)’st coordinate of the vector XTn ξ ⊗ en the invariance of Pn under X implies the equation (5.5)

x(iD)ξ + x(n−1)n ξ = (iD)xnn ξ.

By analogy we get a similar equation based on the invariance of Qn under X so we get (5.6)

x(i(D + I))ξ + x(n−1)n ξ = (i(D + I))xnn ξ.

By subtraction of those equations we get (5.7)

∀ξ ∈ dom(Dn ) :

xξ = xnn ξ,

so since both operators are bounded we have xnn = x. The equation (5.5) may then be applied to show that x(n−1)n = δw (x) and it is possible to continue along this line to show that X = Φn (x), but we will instead address the first element, of the vector XTn ξ ⊗ 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 then define operators L and R on B by left and right multiplications by s, so Lb := sb

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and Rb := bs. then by the binomial formula we get, since L and R commute that for any b in B (5.8) n n   1 1 1 1 1 1 ad(s)j (b)s(n−j) = (L − R)j R(n−j) b = Ln b = sn b. (n − j)! j! (n − j)! j! n! n! j=0 j=0 We will recall the commutator mapping d which we defined in Definition 3.1 as d(x) := [iD, x]. We know from above that x is in C (n−1) (M, D) so by Theorem 4.1 for any j in {1, . . . , n − 1} we have xdom(Dj ) ⊆ dom(Dj ), from equation ( 5.7) we have 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 1 j 1 d (x)(iD)(n−j) dom(Dn ) = (iD)n xdom(Dn ). (n − j)! j! n!

Since x is in C (n−1) (M, D) we have   j dj (x)dom(Dn ) = δw (x)dom(Dn ), for 0 ≤ j ≤ n − 1. On the other hand the invariance of Pn shows that (n−1)   (5.10) (

  1 j 1 1 δw (x)(iD)(n−j) ) + x0n dom(Dn ) = (iD)n xdom(Dn ). (n − j)! j! n! j=0   1 n d (x)dom(Dn ) = x0n dom(Dn ), so By elementary algebra we then get that n! 1 n by Theorem 4.1 we find that x is in C n (M, D) and that n! δw (x) = x0n , as expected. Recall that by (5.3) Pn is invariant under the elements in Rn , and with this in mind we get that Pn must be invariant under Y := (X − Φn (x)), which is a column matrix such that yij = 0 whenever j = n, and also satisfies ynn = 0, which is crucial for the next argument. Given any vector ξ in dom(Dn ) with corresponding vector Tn (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 the graph of an operator defined on the last coordinate, which here vanishes. On the n other hand, for the given ξ in dom(Dn ) we get 0 = Y Tn (ξ ⊗ en ) = i=0 (yin ξ) ⊗ ei . Hence for 0 ≤ i ≤ n we get yin = 0 and then X = Φn (x) for an x in C n (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 and spectral theory of N -body Hamiltonians, Progress in Mathematics, vol. 135, Birkh¨ auser 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. G´ erard, and J. S. Møller, Commutators, C0 -semigroups and resolvent estimates, 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)

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[7] J. von Neumann, Mathematische Grundlagen der Quantenmechanik (German), Unver¨ anderter Nachdruck 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 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13504

Parabolic induction, categories of representations and operator 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. We explain how Frobenius reciprocity fits naturally within the context of operator modules, and examine the prospects for an operator algebraic formulation of Bernstein’s reciprocity theorem (his second adjoint theorem).

1. Introduction Harish-Chandra famously decomposed the regular representation of a real reductive group G into an explicit integral of its isotypical parts. His program to do so had two parts: (a) the classification of the so-called cuspidal representations of G and its Levi subgroups; and (b) the construction, by the process of parabolic induction, of further representations, sufficiently many in number to decompose the regular representation. The cuspidal representations of a real reductive group G are the irreducible and unitary representations of G that are square-integrable, modulo center. Their classification fits well with ideas from C ∗ -algebra K-theory and noncommutative geometry. Indeed the classification was an important source of inspiration for the formulation of the Baum-Connes conjecture. But the functor of parabolic induction has received less attention from operator algebras and noncommutative geometry. Our purpose 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 fits into the theory of Hilbert C ∗ -modules and bimodules in a way that is very similar to Marc Rieffel’s well known treatment of ordinary induction [Rie74]. In joint work with Clare [CCH16a, CCH16b] we studied parabolic induction as a functor between categories of Hilbert C ∗ -modules. Using a considerable amount of representation 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. c 2016 American Mathematical Society

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an adjoint Hilbert C ∗ -bimodule, used it to define a functor of parabolic restriction between categories of Hilbert C ∗ -modules, and proved that parabolic induction and restriction are two-sided “local” adjoints of one another. A drawback of our work was that the concept of local adjunction is significantly weaker than the standard category-theoretic notion of adjunction. This shortcoming 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-for adjunctions are even not properly defined at the Hilbert C ∗ -module level. The purpose of this article is to examine the extent to which the shortcomings of the Hilbert C ∗ -module theory can be remedied by adjusting the context a little. To this end we shall study new categories consisting of group or C ∗ -algebra representations on operator spaces. We shall prove that the new categories have only the familiar irreducible objects, so that they present a plausible context for representation theory. Then we shall formulate and prove a simple theorem about adjoint pairs of functors between the new categories of operator space modules over C ∗ -algebras (as opposed to Hilbert C ∗ -modules). As we shall explain, this implies in a very simple way (that does not require any sophisticated representation theory) a Frobenius reciprocity theorem for parabolic induction (the theorem is that the functor 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 restriction at the level of Harish-Chandra’s Schwartz algebra in the particular case where G = SL(2, R). We shall summarize the tempered representation theory of G in the form of a Morita equivalence between the Harish-Chandra algebra and a simpler and more accessible algebra. Our reason for doing this is to formulate and prove a “second adjoint theorem” for tempered representations in this case, along the lines of Bernstein’s fundamental second adjoint theorem (that parabolic induction also has an explicit right adjoint) in the smooth representation theory of reductive p-adic groups [Ber87, Ber92]. We shall say a good deal more elsewhere [CH] about our second adjoint theorem for tempered representations. Our reason for introducing the result here is to use it as a test for measuring the potential usefulness of new operator-algebraic contexts for representation theory. To this end, we shall conclude by using the explicit formulas obtained for the Harish-Chandra algebra to explore the prospects for an elaboration of the operator space module Frobenius reciprocity relation analyzed in Section 2 so as to include Bernstein’s second adjunction. We shall give one concrete suggestion about how this might be achieved in Section 4.3. Note added by NH. It is an immense pleasure to contribute to this volume celebrating Dick Kadison’s 90th birthday. The whole operator algebra community owes Dick a great debt of gratitude for his decades-long leadership, but in my case the debt is not only mathematical, but personal too. Dick gave me my first job and guided me through my first years as a mathematician. His support and encouragement helped me launch my career, and the advice he gave me then helps me still. Thank you, Dick. 2. Categories of Operator Space Modules We shall study operator space modules over C ∗ -algebras and, later on, over operator algebras. For the most part we shall refer to the monograph [BLM04] for

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background on operator spaces, but we shall repeat some of the basic definitions here. Before we start, let us explain our point of view. In the representation theory of real reductive groups there is broad agreement about the concepts of irreducible representation that are appropriate for study, along with the associated concepts of equivalence among irreducible representations. But representations that lie well beyond 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 used to formulate the Baum-Connes conjecture, involves representations that are far from irreducible. So it is of interest to explore some of the potentially convenient categories of representations that operator algebra theory provides. As we mentioned in the introduction, our immediate concern here is not Ktheory but parabolic induction, together with adjunction theorems such as Frobenius reciprocity. But here, too, the choice of a category of representations matters. Our main observation is that operator spaces can offer a very convenient starting point from which to begin an examination of Frobenius reciprocity and related matters, because the theorem assumes a particularly elementary form there. To continue, recall that an operator space is a complex vector space X equipped with a family of Banach space norms on the spaces Mn (X) of n × n matrices over X that satisfy the following two conditions: (a) If x ∈ Mn (X) and a, b ∈ Mn (C), then axb ≤ axb. (b) The norm of a block-diagonal matrix is the maximum of the norms of the diagonal 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 sup Mn (T )operator < ∞. n

The supremum is the completely bounded norm. We shall also use the related notions of completely contractive and completely isometric map. Example 2.1. Every Hilbert space H carries a number of operator space structures. In this paper we shall consider only the column Hilbert space structure, in which H is identified with the concrete operator space B(C, H) of bounded operators from C to H. Every bounded operator between Hilbert spaces is completely bounded as an operator between column Hilbert spaces, and the completely bounded 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 rise to bilinear maps Mn (Φ) : Mn (X) × Mn (Y ) −→ Mn (Z) through the formula

  n  Mn (Φ) : [xij ], [yij ] −→ k=1 Φ(xik , ykj ) .

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The Haagerup tensor product X ⊗h Y is a Banach space completion of the algebraic tensor product over C and an operator space, characterized by the property that every completely contractive Φ as above factors uniquely through a completely contractive map X ⊗h Y → Z. See [BLM04, 1.5.4]. Definition 2.2. An operator algebra is an operator space A which is also a Banach algebra with a bounded approximate unit, such that the product in A induces a completely contractive map A ⊗h A → A. Definition 2.3. A (left) operator module over an operator algebra A is an operator space X and a nondegenerate1 left A-module for which the module action extends to a completely contractive map A ⊗h X → X. One similarly defines right operator modules, and operator bimodules. We shall denote by OpModA the category 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 category OpModA contains the category HilbA of nondegenerate Hilbert space representations of A as a full subcategory (each Hilbert space being given its column operator space structure). Our first observation is that if A is of type I, then OpModA and HilbA have the same irreducible objects (that is, the same modules having no nontrivial closed submodules): Proposition 2.4. Let A be a type I C ∗ -algebra. Every irreducible operator Amodule is completely isometrically isomorphic in OpModA to an irreducible Hilbert space representation of A. Proof. Let X be an irreducible operator A-module. By [BLM04, Theorem 3.3.1] there is a nondegenerate representation of A on a Hilbert space H, a second Hilbert space K, and a completely isometric isomorphism from X to a closed Asubmodule of the space B(K, H) of bounded operators from K to H. We shall realize X as a subspace of B(K, H) in this way, and we may assume that X · K is dense in H. We are going to argue that the representation of A on H is a multiple of a single irreducible representation of A. To begin, the representation of A on H extends to the multiplier algebra M (A), and the restriction to the center Z(M (A)) is a multiple of a single irreducible representation of the center. For otherwise there would exist elements z1 , z2 ∈ Z(M (A)) with z1 z2 = 0 yet z1 H = 0

and z2 H = 0,

z1 X = 0

and z2 X = 0.

so that The subspace z1 X ⊆ 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  is a Hausdorff irreducible representation) and that furthermore the spectrum A topological space. The Dauns-Hofmann theorem (see [DH68] or [Ped79, Section 4.4]) identifies Z(M (A)) with the algebra of bounded, continuous, complex-valued  The identification is as follows: functions on A. 1 Nondegenerate

means that A · X is dense in X.

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 (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 evaluation at a point at infinity, then IA = A. In our present situation, we see that the action of Z(M (A)) on H must factor  as in item through the quotient by a maximal ideal corresponding to a point of A, (a), since the action of A on H is certainly nonzero. Therefore the action of A must factor through a quotient A/IA. Since the quotient is isomorphic to the compact operators, the action of A is a multiple of a single irreducible representation, as required. Next assume that A is a general liminal C ∗ -algebra. Let {Jα } be a composition series for A for which each quotient Jα+1 /Jα has Hausdorff spectrum. Take the least α for which Jα X = 0; by irreducibility we must then have Jα X = X. Consider X as an irreducible operator module over B = Jα /Jα−1 , where as usual Jα−1 denotes the closure of the union of all ideals in the composition series smaller than Jα . The argument above shows that H is a multiple of a single irreducible representation of B, 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 argument to a composition series for A with liminal quotients to show that H is a multiple of a single irreducible representation of A in this case too. We have now shown in general that H is a multiple of a single irreducible representation of A, say H∼ =M ⊗L where A acts trivially on M and irreducibly on L. We shall now show that there is a bounded operator S : K → M such that X consists precisely of all operators in B(K, M ⊗ L) of the form k −→ S(k) ⊗ ,

(2.5)

as  ranges over L. The map sending the operator (2.5) to S ·  will then be a completely isometric A-linear isomorphism from X to L. For each k ∈ K and m ∈ M consider the completely bounded, A-linear map from 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 for which the operator (2.6) is nonzero. Then this operator is invertible: its kernel is a proper closed submodule of the irreducible module X, while its image is a nonzero A-invariant subspace of L, which must equal L by Kadison’s transitivity theorem [Kad57] (or by a direct argument in the present rather elementary type I situation). Applying Schur’s lemma to the irreducible representation L, we find that there are 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.



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Remark 2.7. We do not know if the assumption in the previous proposition that A be of type I is actually necessary. 2.2. Functors Between Operator Module Categories. If X is a right operator B-module, and if Y is a left operator B-module, then we can of course form the algebraic tensor product of X and Y over B. The balanced Haagerup tensor product X ⊗hB Y is a completion of the algebraic tensor product and an operator space, characterized by the fact that any completely contractive bilinear map Φ : X × Y −→ Z with Φ(xb, y) = Φ(x, by) extends to a completely contractive map from X ⊗hB Y to Z. The Haagerup tensor product is associative, and functorial with respect to c.b. bimodule maps. If X and Y carry left operator A-module and right operator C-module structures, respectively, then X ⊗hB Y 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. Note that composition of functors corresponds, up to natural isomorphism, to Haagerup tensor product of bimodules. If X is an operator A-module, then the module structure induces a completely isometric 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 the identity functor up to natural isomorphism. 2.3. Adjunctions. Our aim is to study adjunction relations, in the usual sense of category theory, between the functors introduced above. So let A and B be operator algebras, and let E be an operator A-B-bimodule. In addition, let F be an operator B-A-bimodule. Following standard terminology, we say that F is left 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 automatically a uniformly (over X and Y ) completely bounded natural isomorphism of operator spaces, as the following simple lemmas make clear (the lemmas simply place the unit/counit characterization of adjoint functors within the operator module context: compare [ML98, Chapter IV]).

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Lemma 2.11. Associated to each natural isomorphism (2.10) there is a completely bounded A-bimodule map η : A −→ E ⊗hB F (the unit of the adjunction) with the property that the composition ∼ =

(2.10)

CBB (F ⊗hA X, Y ) −→ CBA (X, E ⊗hB Y ) −→ CBA (A ⊗hA X, E ⊗hB Y ) sends a morphism T : F ⊗hA X → Y to the composition η⊗id

id⊗T

A ⊗hA X −→ E ⊗hB F ⊗hA X −→ E ⊗hB Y. Proof. The definition of η is very simple (and standard). Take X = A and Y = 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 element F ⊗hA A → F on the left given by the module action. The map η defined in this way is a priori just a left A-module map, but the naturality of (2.10) implies it is a right A-module map too. The proof that the isomorphism (2.10) is given by the formula in the lemma is a straightforward consequence of naturality of the isomorphism once again, together with the following claim: the isomorphism (2.10)

CBB (F ⊗hA X, F ⊗hA X) −→ 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 commuting diagram CBB (F ⊗hA X, F ⊗hA X)

/ CBA (X, E ⊗hB F ⊗hA X)

 CBB (F ⊗hA A, F ⊗hA X) O

 / CBA (A, E ⊗hB F ⊗hA X) O

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-module map 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 completely bounded B-bimodule map ε : F ⊗hA E −→ B

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(the counit of the adjunction) with the property that the inverse of the composition ∼ =

(2.10)

CBA (X, E ⊗hB Y ) ←− CBB (F ⊗hA X, Y ) ←− CBB (F ⊗hA X, B ⊗hB Y ) sends a morphism S : X → E ⊗hB Y to the composition id⊗S

ε⊗id

F ⊗hA X −→ F ⊗hA E ⊗hA Y −→ 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 η⊗id

id⊗ε

id⊗η

ε⊗id

A ⊗hA E −−−→ E ⊗hB F ⊗hA E −−−→ E ⊗hB B −→ E and F ⊗hA A −−−→ F ⊗hA E ⊗hB F −−−→ B ⊗hB F −→ F are the the canonical isomorphisms induced from the left and right A- and B-module actions on E and F , respectively. Conversely this data determines an adjunction isomorphism.  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 tensor product functor OpModB −→ OpModA simply 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 associated tensor 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(a1 a2 ) = a1 ⊗ a2 and ε(b1 ⊗ b2 ) = b1 b2 are the unit and counit 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], as well as [Lan95] for an introduction to Hilbert C ∗ -modules. Let E be a right Hilbert C ∗ -module over a C ∗ -algebra B. The matrix space Mn (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 operator B-module. A bounded, adjointable operator between Hilbert C ∗ -B-modules is automatically completely bounded with the same norm (in fact this is true for any bounded B-module map, whether or not it is adjointable).

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We are especially interested in the situation where a Hilbert C ∗ -B-module E is equipped with a left action of a second C ∗ -algebra A by bounded and adjointable operators. One sometimes calls E a C ∗ -correspondence from A to B, and every such correspondence is an operator A-B-bimodule. Now if E is any operator space, then its adjoint E ∗ is the complex conjugate vector space, equipped with the norms     [eji ] [eij ] , ∗ = Mn (E )

Mn (E)

∗

which endow E with the structure of an operator space. See [BLM04, Section 1.2.25]. If E is an operator A-B-bimodule, where A and B are C ∗ -algebras, then E ∗ 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 ∗ -Bmodule, as follows. Denote by KB (E) the C ∗ -algebra of B-compact operators on E, that is, the closed linear span of all bounded adjointable operators on E of the form e1 ⊗ e∗2 : e −→ e1 e2 , e. The tensor product notation is particularly apt in view of the following very elegant and useful calculation of Blecher. Lemma 2.15. [BLM04, Corollary 8.2.15]. The above formula defines a completely 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 second way: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 -correspondence 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 ∗ -correspondence from A to B. If the action of A on E is through B-compact operators, then the operator 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 ∗ . 2 A third very elegant connection, which like the first is due to Blecher, will be indicated in Lemma 3.5.

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On the other hand by Lemma 2.16 the inner product on E gives us a c.b. B-bimodule map ε : E ∗ ⊗hA E −→ B. We claim that these are the unit and counit, respectively of an adjunction. According to Lemma 2.13 to prove this it suffices to show that the compositions (2.18)

η⊗id

A ⊗hA E −−−→ E ⊗hB E ∗ ⊗hA E −−−→ E ⊗hB B −→ E id⊗ε

and id⊗η

E ∗ ⊗hA A −−−→ E ∗ ⊗hA E ⊗hB E ∗ −−−→ B ⊗hB E ∗ −→ E ∗

(2.19)

ε⊗id

are the the canonical isomorphisms induced from the left and right A- and B-module actions on E and E ∗ , respectively. The composition ∼ =

E ⊗hB E ∗ ⊗hA E −→ E ⊗hB B −→ E id⊗ε

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 other hand, the map η⊗id

A ⊗hA E −→ 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 ∗ −−−→ B ⊗hB E ∗ −→ E ∗ ε⊗id

is given by the formula e∗1 ⊗ e2 ⊗ e∗3 → e1 , e2 e∗3 = (e3 e2 , e1 )∗ = (κ(e2 ⊗ e∗3 )∗ e1 )∗ , while the map id⊗η

E ∗ ⊗hA A −−−→ E ∗ ⊗hA E ⊗hB E ∗ is given by the formula 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.



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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 points of a connected reductive group defined over R, as we did in [CCH16a], although what we have to say would certainly apply to a broader class of examples. On the other hand the special linear and general linear groups will suffice to illustrate the results 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 representation, and so correspond to nondegenerate representations of the reduced C ∗ -algebra of 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 conjugacy P is a subgroup of block-upper-triangular matrices, L is the subgroup of blockdiagonal matrices, and N is the subgroup of block-upper-triangular matrices with identity diagonal blocks. See [Kna96, Section VII.7] for the general definitions. The functor of (normalized ) parabolic induction, IndG P : HilbC ∗ (L) −→ HilbC ∗ (G) , associates to a unitary representation π : L → U (H) (or equivalently, a nondegenerate representation of the full group C ∗ -algebra) the Hilbert space completion of the space of smooth functions  1 f : G → H : f (gn) = π()−1 δ()− 2 f (g) in the inner product

! f1 , f2  =

f1 (k), f2 (k)H dk, K

where K is a maximal compact subgroup of G. Here δ : L → (0, ∞) is the smooth homomorphism defined by   δ() = det Ad : n → n , where n is the Lie algebra of N and Ad denotes the adjoint action. 1 The presence of the normalizing factor δ − 2 ensures that the Hilbert space so obtained is a unitary representation of G under the left translation action. If the original representation is weakly contained in the regular representation, then so is the parabolically induced representation. For all this see for example [Kna86, Chapter VII]. 3.1. Parabolic Induction and Hilbert C*-Modules. Pierre Clare began the study of parabolic induction from the point of view of modules and bimodules over operator algebras in [Cla13]. Clare realized the functor of normalized parabolic induction as the tensor product with an explicit C ∗ -correspondence Cr∗ (G/N ), from Cr∗ (G) to Cr∗ (L), which is obtained as a completion of the space of continuous, compactly supported functions on the homogeneous space G/N in a natural (normalized, using δ) inner product valued in Cr∗ (L). Thus he exhibited a natural isomorphism ∼ C ∗ (G/N ) ⊗C ∗ (L) H IndG H = P

r

r

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of functors from HilbCr∗ (L) to HilbCr∗ (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 that we shall present below holds in either context, but for later purposes it is more appropriate for us to work with the reduced C ∗ -algebra. The Hilbert module picture of parabolic induction as a tensor product allows us to define parabolic induction of operator modules, IndG P : OpModCr∗ (L) −→ OpModCr∗ (G) using the Haagerup tensor product: ∗ IndG P X = Cr (G/N ) ⊗hCr∗ (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 Proposition 2.4, concerning irreducible objects in the categories OpModC ∗ (G) and OpModCr∗ (G) applies. Within the context of operator modules it is natural and simple to consider, in addition to Cr∗ (G/N ), the adjoint operator space Cr∗ (G/N )∗ , which is an operator Cr∗ (L)-Cr∗ (G)-bimodule. We obtain from the tensor product formula ∗ ∗ ResG P X = Cr (G/N ) ⊗hCr∗ (G) X

a functor ResG P : OpModCr∗ (G) −→ OpModCr∗ (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, as functors on operator modules. Thus there is a natural isomorphism CBC ∗ (L) (ResG X, Y ) ∼ = CBC ∗ (G) (X, IndG Y ) r

P

r

P

for all operator Cr∗ (G)-modules X and all operator Cr∗ (L)-modules Y . Proof. In [CCH16a, Proposition 4.5] we showed that the action of Cr∗ (G) on the C ∗ -correspondence Cr∗ (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 over the full group C ∗ -algebras. 3.2. Local Adjunction. Let us contrast the Frobenius reciprocity theorem proved in the previous section with the situation for categories of Hilbert C ∗ modules. In [CCH16a] we were able to show, using considerable input from representation theory, that the operator bimodule Cr∗ (G/N )∗ in fact carries the structure of a C ∗ -correspondence. In other words its operator space structure is induced from a Cr∗ (G)-valued inner product.

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It needs to be stressed that this circumstance depends in a delicate way on issues in representation theory; in fact our explicit formula for the inner product is derived from Harish-Chandra’s Plancherel formula. There is for example no similar inner 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*ModCr∗ (L) −→ C*ModCr∗ (G) and Res : C*ModCr∗ (G) −→ C*ModCr∗ (L) between categories of (right) Hilbert C ∗ -modules and adjointable operators between Hilbert C ∗ -modules. Kasparov’s interior tensor product is related to the Haagerup tensor product in a very simple way: Lemma 3.5. [Ble97, Theorem 4.3]. Let E be a C ∗ -A-B-correspondence and let 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 completely isometric isomorphism.  See also [BLM04, Theorem 8.2.11]. But despite the lemma, and despite the theorem proved in the previous section, it is not true that the two functors above are adjoint to one another. Instead, the best result available is that there are natural isomorphisms (3.6)

KCr∗ (L) (Res X, Y ) ∼ = KCr∗ (G) (X, Ind Y )

between the spaces of compact adjointable operators. See [CCH16b, Theorem 5.1]. In contrast to all this, our operator module result, Theorem 3.3, is stronger and relies only on the fact that Cr∗ (G) acts through compact operators on Cr∗ (G/N ). This is in turn an easy consequence of the geometry of G, involving no representation 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 bit further, 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 is summarized 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 in G = SL(2, R), namely the group P of upper triangular matrices, with Levi factor L the diagonal matrices in SL(2, R). The (necessarily one-dimensional) irreducible unitary   rep0 resentations of L divide into two classes—the even representations where −1 0 −1 3 There 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.

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acts as 1, and the odd representations where it acts as −1. There is accordingly a direct sum decomposition Cr∗ (L) ∼ = Cr∗ (L)even ⊕ Cr∗ (L)odd in which the even representations factor through the projection onto the even summand, and the odd representations factor through the projection onto the odd summand. 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 cuspidal representations. By definition, these are the irreducible unitary representations of G that are weakly contained in the regular representation but do not embed in any principal series representation; according to a basic calculation they are precisely the irreducible square-integrable representations of G, also called the discrete series of G. Associated to this division of the representations of Cr∗ (G) into three types (cuspidal, plus even and odd principal series) there is a three-fold direct sum decomposition Cr∗ (G) ∼ = Cr∗ (G)cuspidal ⊕ Cr∗ (G)even ⊕ Cr∗ (G)odd . Finally, there is a compatible direct sum decomposition Cr∗ (G/N ) ∼ = Cr∗ (G/N )even ⊕ Cr∗ (G/N )odd under which the reduced C ∗ -algebras of both G and L act on the even and odd parts through the projections onto their respective even and odd summands. In what follows we shall concentrate on the even summands. The odd summands are similar, but a bit harder to describe in the case of Cr∗ (G). However the situation as regards adjunctions is actually simpler and less interesting for the odd summands, and this is the reason that we shall concentrate on the even parts. The cuspidal part of Cr∗ (G) plays no role at all, since it acts trivially on Cr∗ (G/N ). There is a C ∗ -algebra isomorphism  Z/2Z Cr∗ (G)even ∼ = C0 R, K(H) where H is a separable infinite-dimensional Hilbert space, and the two-element group Z/2Z acts on R by multiplication by −1, while it acts on K(H) trivially. There is an isomorphism of Hilbert modules Cr∗ (G/N )even ∼ = C0 (R, H) under which (a) The left action of Cr∗ (G)even becomes the obvious pointwise action under the isomorphisms given above. (b) The right action of Cr∗ (L)even is by pointwise multiplication under the identification of Cr∗ (L)even with C0 (R), and the inner product is the pointwise inner product. (c) The Cr∗ (G)-valued inner product on Cr∗ (G/N )∗even takes values in the ideal Cr∗ (G)even , and is given by f1 , f2 Cr∗ (G) = 12 f1 ⊗ f2∗ + 12 w(f1 ) ⊗ w(f2 )∗ ,

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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 one adjointable operators on C0 (R, H). From all of this, and keeping in mind the obvious Morita equivalence  Z/2Z C0 R, K(H) ∼ C0 (R)Z/2Z , Morita

we find that the problem of formulating an adjunction theorem for the C ∗ -correspondence Cr∗ (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 structure on E ∗ coincides with one induced by an A-valued inner product, namely the inner product f1 , f2 A = 12 f1∗ f2 + 12 w(f1∗ f2 ). The failure of the local adjunction isomorphism to extend to an isomorphism on all adjointable operators is a consequence of the fact that the counit ε defined above is a completely bounded map of B-bimodules, but not an adjointable map of Hilbert modules when the Haagerup tensor product is identified with Kasparov’s internal tensor product using Lemma 3.5. 4. The Second Adjoint Theorem For smooth representations of reductive p-adic groups, Bernstein made the remarkable discovery that parabolic induction has not only a left adjoint, but also a right adjoint. The right adjoint, like the left adjoint, is given by parabolic restriction, 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 the representation theory of p-adic groups, leading to a direct product decomposition of the category of smooth representations into component categories. See for example [Ren10, Chapter VI] again. Similar structure can be seen in the tempered representation theory of both real and p-adic reductive groups, and one of the main motivations for the work presented in [CCH16b, CCH16a] was to obtain something similar to Bernstein’s theorem in categories of representations related to the reduced group C ∗ -algebra. The local adjunction isomorphism of [CCH16b] that we described in Section 3.2 is a partial solution. But it is not altogether satisfactory, since in the

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p-adic context Bernstein’s theorem is a geometric foundation from which representation theory may be built up,4 whereas our local adjunction theorem required an extensive acquaintance with tempered representation theory to formulate and prove. So the question remains whether or not a suitable counterpart of Bernstein’s second adjoint theorem can be developed in an operator-algebraic context. We shall investigate this issue in detail elsewhere; our purpose here is to present two computations in the simple case of the group SL(2, R) that together indicate a possibly interesting role here for operator algebras and operator modules. 4.1. Harish-Chandra’s Schwartz space. If G is a real reductive group, as before, then its Harish-Chandra algebra is a Fr´echet convolution algebra HC(G) of smooth, complex valued functions on G that is perhaps easiest to present here as a distinguished subalgebra of Cr∗ (G) that is closed under the holomorphic functional calculus. The definition of HC(G) is a bit involved. Moreover it is not by any means obvious, even after one has mastered the definitions, that HC(G) is closed under convolution 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 of Fr´echet spaces, in this section and the next, the symbol ⊗ will denote the completed projective tensor product of Fr´echet spaces. Definition 4.1. Let A be a Fr´echet algebra (that is, a Fr´echet space equipped with a (jointly) continuous and associative multiplication operation). A smooth Fr´echet module over A is a Fr´echet space V which is equipped with a continuous A-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 the quotient of the completed projective tensor product A ⊗ V by the closed subspace generated by the balancing relators a 1 a2 ⊗ v − a1 ⊗ a2 v with a1 , a2 ∈ A and v ∈ V . Definition 4.3. We denote by SFModA the category of smooth Fr´echet modules over A, with continuous A-linear maps as morphisms. If E is a smooth A-B-Fr´echet bimodule, then the tensor product construction in 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 next section. We should remark that if A is a Fr´echet algebra, then it is not necessarily true that the multiplication map A ⊗A A −→ A 4 In this context see the recent article [BK13] for a striking geometric conceptualization of Bernstein’s original theorem.

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is an isomorphism, but this is true for the Harish-Chandra algebras that we shall be studying. 4.2. The Harish-Chandra algebra of SL(2,R). We shall now specialize to 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. Both the even and odd summands are isomorphic as Fr´echet 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 Z/2Z HC(G)even ∼ = S (R, K(H)) 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 are weight vectors). (b) The right-hand algebra consists of continuous functions f from R into the compact 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 orthonormal basis 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 functions fj are of rapid decay with respect to any Schwartz space seminorm, as in (c) above. Since there is again a Morita equivalence Z/2Z

S (R, K(H))

∼

Morita

S(R)Z/2Z

(that is, an equivalence of SFMod categories) we are finally reduced to studying adjunction theorems in the following Fr´echet 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 that we made in Section 3.3. And the situation with regard to Frobenius reciprocity is similarly predictable: if we define F = S(R),

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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 −→ B

defined by η(a1 a2 ) = a1 ⊗ a2

and

ε(f ⊗ e) = f e 

are the unit and counit of an adjunction.

But the situation with regard to “Bernstein reciprocity,” or the assertion that E also has a right adjoint, is much more interesting. Surprisingly, in view of the fact that in most respects the Fr´echet algebra HC(G) behaves much like Cr∗ (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 E as follows: b1 b2 → b1 x ⊗ b2 + b1 ⊗ xb2

(4.5)

for b1 , b2 ∈ B (we are writing x for the function x → x). It is not immediately obvious that the formula is well-defined. But the following calculation shows that this is so: Lemma 4.6. The quantity in F ⊗A E described in (4.5) depends only on the product b1 b2 ∈ B, and the formula defines a continuous B-bimodule homomorphism. Proof. Let us first show that if b ∈ B, then b1 bx ⊗ b2 + b1 b ⊗ xb2 = b1 x ⊗ bb2 + b1 ⊗ xbb2 . We can write b = a1 + a2 x, where a1 , a2 ∈ A, and it suffices to consider separately the cases where a1 = 0 and a2 = 0. The latter is easy, since the tensor products are over A. As for the former, we calculate that b1 (a2 x)x ⊗ b2 + b1 (a2 x) ⊗ xb2 = b1 ⊗ a2 x2 b2 + b1 x ⊗ a2 xb2 = b1 x ⊗ a2 xb2 + b1 ⊗ xa2 xb2 , as required (we used the fact that x a2 ∈ A). So the formula defines a continuous map B ⊗B B −→ F ⊗A E, 2

and the lemma follows from the easily verified fact that the multiplication map B ⊗B B −→ B 

is an isomorphism.

Remark 4.7. Bernstein constructed the unit map for his second adjunction using the geometry of the homogeneous space G/N , and in particular the fact that if P = LN is the opposite parabolic subgroup, then the product map N × L × N −→ G

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embeds the left hand side as an open subset of G. See for example [Ber92, Section 3.1]. It is not immediately apparent, but the unit described here is essentially the same, and differs only in that we have used the function x → x in place of (the reciprocal 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 map for 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 1 (ef )− x in which the superscript “−” on the right means that we take the odd part of the function ef ∈ B (a superscript “+” will likewise denote the even part of a function). The composition (4.9)

e ⊗ f →

E ⊗B B −→ E ⊗B F ⊗A E −→ A ⊗A E −→ E is given by the formula e ⊗ b1 b2 → e ⊗ b1 x ⊗ b2 + e ⊗ b1 ⊗ xb2 1 1 → (eb1 x)− ⊗ b2 + (eb1 )− ⊗ xb2 x x 1 1 − → (eb1 x) b2 + (eb1 )− xb2 x x = (eb1 )+ b2 + (eb1 )− b2 = eb1 b2 , and this is the standard multiplication map, as required. In addition the composition B ⊗B F −→ F ⊗A E ⊗B F −→ F ⊗A A −→ F is given by the formula b1 b2 ⊗ f → b1 x ⊗ b2 ⊗ f + b1 ⊗ xb2 ⊗ f 1 1 → b1 x ⊗ (b2 f )− + b1 ⊗ (f b2 x)− x x 1 − − → b1 (b2 f ) + b1 (f b2 x) x = b1 (b2 f )− + b1 (b2 f )+ , 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 possible to detect the use of the opposite parabolic subgroup, except indirectly through the geometric role it plays in giving the formula for the unit map, as indicated in Remark 4.7.

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4.3. Bernstein’s Theorem and Operator Spaces. In this final section we shall adapt the Schwartz algebra computations of the previous section to the context of operator algebras. The formula (4.9) for the Bernstein counit does not make sense for arbitrary continuous functions, and so does not make sense at the level of (reduced) group C ∗ -algebras. We will show however that the Bernstein reciprocity theorem of the previous section can be recovered after replacing C ∗ -algebras with non-self-adjoint operator 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) with the 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 the ultimate 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/c is a smooth function, bounded at infinity (in the natural construction of the unit map, involving the actual c-function from representation theory, the boundedness condition does not hold). But this is a relatively minor issue; see Remark 4.17 below. Definition 4.11. We shall denote by B ⊆ C0 (R) the space of those functions f ∈ C0 (R) for which the product c · f − extends to a continuous (and necessarily even) function on R. Equivalently B consists of those functions in C0 (R) whose odd 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(b1 b2 ) = δ(b1 )b2 + w(b1 )δ(b2 ). for all b1 , b2 ∈ B. As a result the formula " # b 0 (4.13) b −→ δ(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 a norm-closed subalgebra. 

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We shall equip the algebra B with the operator algebra structure it receives from the embedding (4.13). In addition, let A be the C ∗ -algebra of even functions in 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 operator B-A-bimodule. Frobenius reciprocity, or the assertion that the tensor product 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 following version of Bernstein reciprocity: Theorem 4.15. The tensor product functor E is left-adjoint to the tensor product 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 b1 b2 ⊗ b2 + b1 ⊗ . c c The formula gives a well-defined map by the argument of Lemma 4.6, which applies here because every element of B is of the form a1 + a2 /c. The map is completely bounded because the function 1c is a bounded multiplier of C0 (R). Clearly η is a B-bimodule map. In addition, define a counit map (4.16)

b1 b2 −→

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

/B

δ

/A

in which the first map is just the multiplication map on B, which is completely bounded. As for δ, it is the restriction to B of the completely bounded map " #   1 T −→ 0 1 T 0 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 is exactly as in the proof of Theorem 4.8. 

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Remark 4.17. If the function 1/c was unbounded (as it would be if we were to use the natural, representation-theoretic c-function), then our formula (4.16) for the unit map η would give an unbounded, densely-defined B-bimodule map. Its domain, an ideal in B, would be an operator algebra in its own right, and we could repeat the above argument with this algebra in place of the original B. References 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. Kr¨ otz, Smooth Fr´ echet 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 operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon 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 via C*-algebras and Hilbert C ∗ -modules. Composito Math., FirstView (2016), DOI 10.1112/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. Operator 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 the American 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 on examples. MR855239 (87j:22022) [Kna96] A. W. Knapp, Lie groups beyond an introduction, Progress in Mathematics, vol. 140, Birkh¨ auser 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 algebraists. MR1325694 (96k:46100) [ML98] S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR1712872 (2001j:18001) [Ped79] 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) [Ren10] D. Renard, Repr´ esentations des groupes r´ eductifs p-adiques (French), Cours Sp´ ecialis´es [Specialized Courses], vol. 17, Soci´ et´ e Math´ ematique de France, Paris, 2010. MR2567785 (2011d:22019) [Art75]

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[Rie74] [Var89]

[Wal88] [Wal92]

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M. A. Rieffel, Induced representations of C ∗ -algebras, Advances in Math. 13 (1974), 176–257. MR0353003 (50 #5489) 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) N. R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR929683 (89i:22029) 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 E-mail address: [email protected] Department of Mathematics, Penn State University, University Park, Pennsylvania 16802 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13505

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 operators representing an element of K 1 (M ). Consider the first two components of the 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 these classes, both of which are in the spirit of holonomy. These are then studied for families parametrized by a closed 3-manifold. A connection between the multiplicity 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, Fredholm operators with compact resolvent, filtered by the maximum dimension of eigenspaces. One goal was to relate the vanishing of the components of the Chern character of their K-theory class to bounds on the multiplicity of the spectrum. In the present note we will give a simple uniform description of the first two terms in the odd Chern character and study them in the case of certain families over a closed 3-manifold. This short paper is preliminary to carrying out a similar analysis for the component of the Chern character in degree 5. The authors would like to thank Alan Carey for valuable discussions and hospitality while visiting Australian National University. 2. Invariants of families of operators We will consider families of operators {Dx } where x ∈ M , M a closed smooth manifold. The operators will be unbounded, self-adjoint, Fredholm operators with compact resolvent. We follow the notation and ideas of the paper, [7], and refer there 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 family determines the trivial element in K 1 (M ). In [7], the approach taken was to adapt topological obstruction theory to determine when one can deform the family to one with a disconnected graph. The obstructions met along the way were related to the components of the Chern character of the K-theory class defined by the family. Under the assumption that the multiplicity of the spectrum of the family was 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. c 2016 American Mathematical Society

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spectral flow and the second to the index gerbe. In the next two subsections we will give elementary definitions of these classes. To this end, we note that there is a 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 will assume that the sets in the cover and their finite intersections are contractible. We will build the invariants relative to this data. ˇ 2.1. Spectral flow. We will define a Cech cocycle in Z 1 ({Ui }, Z), where Z denotes the presheaf of Z valued functions on M . Fix an open set Ui in the cover and, for x ∈ Ui , let PUi (x) be the orthogonal projection onto the subspace of H spanned 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 and that 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 refer to [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 Z 1 ({Ul }, Z) whose image, sf ({Dx }) = [c] ∈ H 1 (M, Z) ⊆ H 1 (M, Q) is equal to the spectral flow class of the family {Dx }, which is a rational multiple of the first component of the Chern character of the K-theory class represented by the family. Proof. Using the definition of spectral flow described in [7], the result follows by a direct modification.  2.2. Index gerbe. We will construct a 3-dimensional integral cohomology class on M and show that it agrees with the Dixmier-Douady invariant of the index gerbe. 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 open cover, {Ui }. Fix a point x ∈ Ui . The projection PUi (x) determines a quasi-free representation 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 different representations, αUi (x) and αUj (x), which are equivalent. This follows since the projections yielding them differ by a finite rank projection and one can construct a canonical intertwining unitary operator, (see Appendix), SPUj (x),PUi (x) : FPUj (x) (H) → FPUi (x) (H), satisfying (1)

SPUj (x),PUi (x) αUi (x)SP∗ U

j

(x),PUi (x)

= αUj (x).

Note that SPUj (x),PUi (x) is defined only up to a choice of a scalar z ∈ S 1 . This ambiguity disappears in the following, (2)

SˆPUj (x),PUi (x) = AdSPU

j

(x),PU (x) i

: B(FPUj (x) (H)) → B(FPUi (x) (H)).

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Next, suppose that x ∈ Ui ∩ Uj ∩ Uk , for sets Ui , Uj , Uk in the cover. Then there exists a function g(Ui , Uj , Uk ) : Ui ∩ Uj ∩ Uk → S 1 such that one has (3)

SˆPUj (x),PUi (x) ◦ SˆPUi (x),PUk (x) ◦ SˆPUk (x),PUj (x) = g(Ui , Uj , Uk )(x) ∈ S 1 ,

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 the projections 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 orthonormal basis for Hi,j (x0 ), a volume element can be determined in a continuous manner for each Hi,j (x). This is what is needed to define the intertwining unitaries in a continuous way. See the appendix for more on this matter. We will be using sheaf cohomology with respect to the presheaf of S 1 -valued functions on M , [3]. Proposition 2.2. The family, {g(Ui , Uj , Uk )} defines a cocycle in C 2 ({Ui }, S 1 ) and, hence, a cohomology class [g] ∈ H 2 ({Ui }, S 1 ). Proof. This is a direct computation.



ˇ 3 (M, Z), which is an isomorphism if There is a natural map H 2 ({Ui }, S 1 ) → H ˇ 3 (M, Z). the cover is well chosen. We will denote the image of [g] by G({Dx }) ∈ H One can check that it depends on the family {Dx } only up to homotopy. Every element in this group is the Dixmier-Douady element of an appropriate equivalence class of gerbes. We next identify this class. Theorem 2.3. The class G({Dx }) is equal to the Dixmier-Douady invariant of the index gerbe of the family as defined by Lott, [9]. Proof. Following Lott, we use the definition of a gerbe, as described by Hitchin, [10]. Thus, there are line bundles LUi ,Uj over Ui ∩ Uj possessing the necessary properties. It is easily seen that tensoring the bundle of Fock spaces over Ui ∩ Uj with LUi ,Uj is the same as applying the family of unitary operators  SPUj (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 3 of 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 → S 1 , satisfying certain conditions, determines a Deligne 3 (X, Z(3)), as in [3] p. 216. It is shown there that equivalence cohomology class in HD classes of gerbes with curving and connection are in 1-1 correspondence with the 3 (X, Z(3)). Thus, from the exact sequence, group HD i π 3 ˇ 3 (X, Z) −−−−−−→ 0 (X, Z(3)) −−−−−−→ H 0 −−−−−−→ Ω2 (X)/Ω2Z (X) −−−−−→ HD

one sees that there are many Deligne classes with the same Dixmier-Douady invariant. In the present context there is a natural choice of connection obtained by pulling back the Chern connection from the Hilbert space bundle over the Grassmannian. If our family is obtained geometrically as a family of Dirac operators on odddimensional closed manifolds, then it follows from [9] that there is a natural choice for curving yielding a specific Deligne class.

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3. A low dimensional example In this section we will analyze certain families over a closed 3-manifold. First recall that when the spectral flow of the family is zero one is able to decompose the graph into a union of levels indexed by the integers. Among the families that have this property we single out those satisfying: Condition (A): Level G0 can have a non-empty intersection with only G1 or G−1 , and G0 ∩ G−1 ∩ G1 = ∅. For such families we will provide a complete classification using the methods of [7]. It is possible, under more general circumstances, to reduce a family to one satisfying Condition (A), but it is more complicated and we leave that to a later paper. We make some preliminary observations. Recall the addition operation for families. If {Dx } and {Dx } are families on the Hilbert spaces H and H respectively, then the sum of the families is the direct sum on the Hilbert space H ⊕ H . The graph of the sum is the union of the graphs of each summand, and the multiplicity of eigenvalues is the sum of the multiplicities on the intersection of the graphs and the multiplicity for the individual families on the symmetric difference of the graphs. 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 manifold, M . Consider sf ({Dx }) ∈ H 1 (M, Z) = [M, S 1 ]. Let α : M → S 1 represent this class. Let {Dz } be a family on S 1 with spectral flow equal to 1. Then {Dx } − α∗ ({Dz }) has spectral flow zero. Note that, in general, one can’t do an analogous construction to eliminate the index gerbe class in H 3 (M, Z) because of the possibility of torsion. However, if M is a closed 3-manifold then this reduction can be done and we will make use of this in what follows. Now, assume we are given a family, {Dx }, parametrized by a smooth closed 3-dimensional manifold M . Let Γ = Γ({Dx }) ⊆ M × R with projection π : Γ → M be the graph. We will assume that sf ({Dx }) = 0, so that we have a spectral decomposition, $ Gk . (4) Γ= k∈Z

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 arbitrary families using the “moves” developed in [7]. Although we are able to apply them to 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 addressed later. Thus, we will assume now that (A) holds and describe next how we can use the methods of [7] to ensure that the set π(G0 ∩ G1 ), is contained in a ball whose complement contains π(G0 ∩ G−1 ). Proposition 3.1. Given a family satisfying (A), one can apply the procedure of [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.

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Proof. We proceed as in [7, p. 325]. Since the multiplicity is at most 2 on G0 , we can find open sets with disjoint closures, V and W contained in M , with π(G0 ∩ G1 ) ⊆ W and π(G0 ∩ G−1 ) ⊆ V . Triangulate M finely enough so that any simplex 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 family so that G0 is separated from G1 and G−1 , over the 0, 1 and 2 skeletons. At this stage, G0 ∩ G1 and G0 ∩ G−1 for the deformed families are contained in the interiors of 3-simplices, and hence in sets homeomorphic to 3-balls. Now, if we remove from M the 3-balls containing G0 ∩ G−1 , the result is path connected. Thus we may find paths between the 3-balls containing G0 ∩ G1 and avoiding the 3-balls containing G0 ∩ G−1 , and then thicken them to be tubes. This can be done in such a way that the result is a single closed 3-ball containing G0 ∩ G1 with G0 ∩ G−1 in its exterior.  We continue our construction. Let gi : M → Gi be the cross section map which sends x to pr2 (({x} × R) ∩ Gi ), i.e. it sends x to the eigenvalue of Dx on Gi . There is a 2-plane bundle, E, over the 2-sphere boundary of B, whose fiber at a point x is the space of eigenvectors of g0 (x) and g1 (x) in H. This bundle extends over B and, hence, is trivial. There is a splitting of E|∂B as the direct sum of two line bundles, L0 and L1 . The splitting is determined by the orthogonal eigenspaces for g0 (x) and g1 (x), for x in some neighborhood, W , of B, with W ⊆ M \ G0 ∩ G−1 . Note that on W we have g0 (x) < g1 (x). Identifying ∂B with S 2 the line bundle L0 is determined by the homotopy class of its clutching map κ : S 1 → U1 , and hence by an integer k. Moreover, the integer associated to L1 is −k. Unless k = 0, the splitting will not extend over the ball, B. We fix this integer k associated to the family{Dx } and the ball B. There is a tautologous family of operators over S 3 , [7, 8, 11]. It is obtained by ˆ ∈ SU (2). One then associates to x the Dirac identifying x ∈ S 3 with an element x operator on S 1 twisted by the 2-plane bundle on S 1 determined by x ˆ. This family represents the element 1 ∈ K 1 (S 3 ) ∼ = Z. For an integer n, the family determined (n) by using x ˆn will be denoted by {∂x }. It is a family corresponding to the integer n ∈ K 1 (S 3 ). Next, observe that there is a degree one map c : M → S 3 which sends the interior of the ball B, int(B), to S 3 \ {(0, 0, −1)} and M \ int(B) to (0, 0, −1). (−k) Consider the family {Dˆx } = {Dx } + c∗ ({∂x }) over M where k is the integer obtained above. We will show that this family is equivalent to a trivial family and, (k) hence, [{Dx }] = c∗ ([{∂x }]) ∈ K 1 (M ). This will provide canonical representatives for classes in K 1 (M ). Proposition 3.2. The family {Dˆx } can be deformed to a family {D˜x } which (−k) satisfies G˜0 ∩ G˜1 = ∅. Moreover, {D˜x } is equal to {Dx } + c∗ ({∂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 to B  ⊆ B. We may arrange, by an initial homotopy of our families, that g0 (x) = 0 for x ∈ B  for {Dx }, and g0 (x) = 0 for x ∈ c(B  ) for c∗ ({∂z−k }). Now we (−k) apply the “flattening” operation, [7, p. 322], to both {Dx } and c∗ ({∂z }) on  B relative to B. This yields two new families, homotopic to the original ones, but with the eigenvalues g0 (x), g1 (x), g0 (x), and g1 (x), constantly 0 on B  . The

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span of their eigenspaces defines a 4-dimensional vector bundle E over B  . More specifically, the graph of the sum of the flattened families has the following structure on 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 direct sum of four line bundles, L0 , L1 , L0 , and L1 , where L0 , and L1 are associated with (−k) c∗ ({∂z }) and L0 , L1 with {Dx }. Now, L0 and L0 are determined by integers which are negatives of each other. Thus, the sum L0 ⊕ L0 is trivial, as is L1 ⊕ L1 . This implies that the splitting E ∼ = (L0 ⊕ L0 ) ⊕ (L1 ⊕ L1 ) extends across π −1 (B  ). We may then use the “scaling operation” from [7, p. 324] to deform the family by increasing the eigenvalue on the subspaces spanned by (L1 ⊕ L1 ) 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 G˜0 ∩ G˜1 = ∅ as required.  Note that a family {D˜x } for which G˜0 ∩ G˜1 = ∅ satisfies that g˜0 (x) < g˜1 (x) for all 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∗ ({∂zk }),

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 satisfies condition (A) is equivalent to one of the form α∗ (sf ) + c∗ ({∂zk }), where sf is the family over the circle with spectral flow 1, and α : M → S 1 is the map representing sf ({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 pass from the given family to the one in standard form imply homotopy of the families, ˆ 1 (M ) to be the group generated the converse may not hold. Thus, if we define K by families of operators modulo flattening and scaling, there is a surjection ˆ 1 (M ) → K 1 (M ) → 0. (6) K It would be interesting to know if it is not always injective. Remark 3.4.2. The method of studying K 1 (M ) we use, actually starts with a subset of M × R with an eigenspace associated to each point–that is, one has an 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 could then try to define twisted cohomology groups. A theory along these lines was developed in [8]. It would also be useful to consider a notion of concordance, apriori weaker than homotopy, for enhanced graphs and obtain a result stating that 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 3manifold which satisfies condition (A). We can deform the family as in Proposition 3.1 and proceed further so that G0 ∩ G1 consists of a single point, {x}. Then k{x} defines a 0-dimensional homology class which is Poincar´e dual to the 3-dimensional class G({Dx }). This is analogous to a result of Cibotaru, [6].

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4. Appendix For the sake of exposition we will review the construction, c.f.[12], of the intertwining operators in the case at hand– that is, we assume we are given two projections P, Q with P − Q of dimension N. Given a Hilbert space H, we will ¯ the conjugate Hilbert space. Decompose H using P and Q, denote by H H = P H ⊕ (1 − P )H = QH ⊕ (1 − Q)H.

(7)

(8)

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).

One defines an isomorphism SQ,P : FP (H) → FQ (H) via (9) SQ,P = IF (QH) ⊗ S˜Q,P ⊗ IF ((I−P )H) , where S˜P,Q : F((P − Q)H) → F((P − Q)H) is defined as follows. Choose a volume element ω ∈ F((P − Q)H), that is, a non-zero element of the top exterior power, ΛN ((P − Q)H). Then S˜P,Q , on Λk ((P − Q)H), is the composition, → ΛN −k ((P − Q)H)∗ → Λk ((P − Q)H) − Θ

(10)

→ ΛN −k (((P − Q)H)∗ ) → ΛN −k ((P − Q)H),

where the first map is Θ(x)(y) =< x ∧ y, ω > and the latter two are canonical isomorphisms. Thus, the composite isomorphism depends only on the choice of ω. One may view (10) as defining an isomorphism (11)

N N % %  k  Λ ((P − Q)H) ⊗ ΛN ((P − Q)H) → ΛN −k ((P − Q)H). k=0

k=0

If the projections, P and Q, depend continuously on x ∈ M , then (11) extends to an isomorphism of bundles, (12)

F((P − Q)H) ⊗ LP,Q → F((P − Q)H).

The map S˜Q,P is obtained by fixing a non-zero cross-section of LP,Q , and it depends on that choice. In the cases considered in the paper one has that LP,Q is trivial. References [1] H. Araki, Bogoliubov automorphisms and Fock representations of canonical anticommutation relations, Operator algebras and mathematical physics (Iowa City, Iowa, 1985), Contemp. Math., vol. 62, Amer. Math. Soc., Providence, RI, 1987, pp. 23–141, DOI 10.1090/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, Modern Birkh¨ auser Classics, Birkh¨ auser Boston, Inc., Boston, MA, 2008. Reprint of the 1993 edition. MR2362847 (2008h:53155) [4] A. Carey, Private communication. [5] A. Carey and J. Mickelsson, A gerbe obstruction to quantization of fermions on odddimensional manifolds with boundary, Lett. Math. Phys. 51 (2000), no. 2, 145–160, DOI 10.1023/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

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[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 Monographs, 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 Symmetry, 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 Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR900587 (88i:22049) Department of Mathematics, Texas A&M University, College Station, Texas 778433368 E-mail address: [email protected] Department of Mathematics, IUPUI, Indianapolis, Indiana Department of Mathematics, University of California Davis, Davis, California 95616 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13506

On the classification of simple amenable C*-algebras with finite 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. Then A 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 and Sato showed that A ⊗ UHF can be tracially approximated by finite dimensional C*-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 trace simplex 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 decomposition rank—see [2]—and (so far) of restricting the K0 -group to have torsion-free rank equal to one. Theorem 1.1. Let A be a simple unital separable C*-algebra satisfying the UCT. 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-Su algebra ([12]). This theorem can also be regarded as an abstract version (still in a special case) of the classification result of [10] and [7], where any simple unital locally approximately subhomogeneous C*-algebra is shown to be rationally tracially approximated by Elliott-Thomsen algebras (1-dimensional noncommutative CW complexes) ([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 let us 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 Research Council of Canada (NSERC) Discovery Grant, and the research of the second author was supported by a Simons Foundation Collaboration Grant. ©2016 American Mathematical Society

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Definition 2.1 (N. Brown, [3]). Let A be a unital C*-algebra, and denote by Tqd (A) the tracial states with the following property: For any (F, ε), there is a unital 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 (Definition 3.3.1 of [3]), the UHF algebra Q was replaced by a matrix algebra. It is easy to see that 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 ⊆ A and 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 contractions φ, 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 unitarily equivalent projections, there exists a unitary u ∈ Un+1 (B) such that        ∗ φ(a) ψ(a)  < ε, a ∈ F. u u −  n · ξ(a) n · ξ(a)  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 sum of n copies of ξ(a) (or 1). This notation is also used in the proof of Corollary 2.6 below. Remark 2.5. In the theorem above (and also Corollary 2.6 below), one assumes by convention that the finite subset G is sufficiently large and δ is sufficiently small that [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 with coefficients. More precisely, one has Corollary 2.6. Let A be a simple, unital, exact, separable C*-algebra satisfying the UCT. For any finite subset F ⊆ A and any ε > 0, there exist n ∈ N and a K-triple (P, G, δ), with P ⊆ Proj∞ (A), with the following property: For any three completely positive contractions φ, ψ, ξ : A → Q which are δ-multiplicative 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, 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  G, δ) with the property of Theorem 2.3. Set K-triple (P, n0 + 1 = n and

 ∩ Proj∞ (A) = P. 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-multiplicative 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.

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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, if the 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) such that       ∗ φ(a)  ψ(a) w  < ε, a ∈ F, w −    n0 · ξ (a)  n0 · ξ (a) 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 := 1Q w1Q is a unitary of Q. Clearly, if the approximation of ξ by ξ  ⊕· · ·⊕ξ  on F is sufficiently good, 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 any finite 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.

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Proof. By Theorem 5.2 of [14] and its corollary, there is an increasing sequence of finite-dimensional subspaces of Aff(Δ) with dense union, containing the canonical order unit 1 ∈ Aff(Δ), and such that each map Rmk → Rmk+1 and Rmk → Aff(Δ) is positive, with respect to the canonical (pointwise) order relations:  / Aff(Δ). / Rm 2   / · · ·  Rm 1  (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 positive linear functional ρ˜i : Aff(Δ) → R. Then the map  : Aff(Δ)  f → (˜ ρ1 (f ), ρ˜2 (f ), . . . , ρ˜m1 (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 Banach space with order unit u, and let γ : Rm → Aff(T(C)) be a unital positive linear map. Then, for any finite set F ⊆ Rm and any ε > 0, there are n and a unital positive 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 = c1 e1 + · · · + cm em , where c1 , . . . , cm > 0. Since C is simple, each affine function γ(ei ) is strictly positive on 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 e1 , e2 , . . . , em−1 ∈ Aff(T(Cn )) such that ιn,∞ (ei ) − γ(ei )∞ < min{ε,

cm δm cm ε δi , , }, 2 2(c1 + · · · + cm−1 ) 2(c1 + · · · + cm−1 )

1 ≤ i ≤ m − 1.

In particular, by (2.2), ιn,∞ (ei )(τ ) ≥ δi /2,

τ ∈ T(C), 1 ≤ i ≤ m − 1.

Setting em :=

1 (1 − c1 e1 − · · · − cm−1 em−1 ) ∈ Aff(T(Cn )), cm

one has ιn,∞ (em ) − γ(em )∞

1 (1 − c1 ιn,∞ (e1 ) − · · · − cm−1 ιn,∞ (em−1 )) cm 1 − (1 − c1 γ(e1 ) − · · · − cm−1 γ(em−1 ))∞ cm ≤ min{δm /2, ε}. = 

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In particular, by (2.2), ιn,∞ (em )(τ ) ≥ δm /2,

τ ∈ T(C).

Then, considering instead the images of e1 , e2 , . . . , em in a building block further out (replacing n by the later index), one may assume that ei (τ ) > δi /4,

τ ∈ T(Cn ), 1 ≤ i ≤ m.

In particular, all the affine functions ei ∈ Aff(T(Cn )) are positive. Define γn : Rm → Aff(T(Cn0 )) by γn (ei ) = ei , 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 satisfying the UCT. Assume that T(A) = Tqd (A) and that K0 (A) is of rank one. Then, for any finite set F ⊆ A ⊗ Q and any ε > 0, there are unital completely positive linear maps 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 ∼ = Q (as order= A ⊗ Q, we have K0 (A) ∼ 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 positive maps,  / Rm θ / Aff(T(A)), Aff(T(A)) such that θ((fˆ)) − fˆ∞ < 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 (ι1,∞ )∗ (γ(fˆ)) − Ξ(fˆ)∞ < ε/8,

(2.3)

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],

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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 the proof.) Since [0, 1] is compact, there is a partition 0 = t0 < t1 < · · · < tl−1 < tl = 1 such 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(fˆ)(τt ) − γ(fˆ)(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] such that 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 ) ⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm ) : A → Q. & '( ) & '( ) qj,1

qj,m

Note that it follows from (2.4), (2.5), and (2.8) that   tr(ϕj (f )) − γ ∗ (τtj )(f ) < ε/4, (2.9)

f ∈ F.

By (2.8), (2.6), one has that (2.10)

|qj,k − qj+1,k | 1 < , p mn

k = 1, . . . , m, j = 0, . . . , l − 1.

For each j = 0, . . . , l − 1, compare the direct sum maps ϕj = (φ1 ⊕ · · · ⊕ φ1 ) ⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm ) & '( ) & '( ) qj,1

qj,m

and ϕj+1 = (φ1 ⊕ · · · ⊕ φ1 ) ⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm ), & '( ) & '( ) qj+1,1

qj+1,m

and consider the common direct summand of these two maps, ψj := (φ1 ⊕ · · · ⊕ φ1 ) ⊕ · · · ⊕ (φm ⊕ · · · ⊕ φm ). & '( ) & '( ) min{qj,1 ,qj+1,1 }

min{qj,m ,qj+1,m }

By (2.10), one has 1 1 |qj,k − qj+1,k | < . p n m

|tr(1 − ψj (1))| =

k=1

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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+1 ϕj+1 (f )uj+1  < ε/4, f ∈ F · F, 0 ≤ j ≤ l − 1. Define v0 = 1, and set uj uj−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+1 ϕj+1 (f )uj+1  < ε/4. Replacing each homomorphism ϕj by Ad(vj ) ◦ ϕj for j = 1, . . . , l, and still denoting it 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 t − tj tj+1 − t ϕj (f ) + ϕj+1 (f ), if t ∈ [tj , tj+1 ]. φ(f )(t) := tj+1 − tj tj+1 − tj Then, by (2.11), the map φ is F-ε-multiplicative. By (2.9) and (2.7), one has (2.12)

φ∗ (fˆ) − γ(fˆ)∞ < ε/2,

f ∈ F.

+ Note that A and C have cancellation for projections, and also K+ 0 (A) = K0 (C) = ∼ Q (unital identification) and Aff(T(A)) = Aff(T(C)). By Theorem 4.4 and Corollary 6.8 of [9] (see also Theorem 2.6 of [5] and Theorem 5.5 of [4], expressed in terms of W instead of Cu), it follows that the Cuntz semigroup of A and the Cuntz semigroup of C are isomorphic. Applied to the canonical unital map Cu(C1 ) → Cu(C) ∼ = Cu(A), Theorem 1 of [22] implies that there is a unital homomorphism 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 an embedding. By (2.12), (2.13), and (2.3), one then has φ∗ ◦ ψ∗ (fˆ) − fˆ∞ < ε, as desired.

f ∈ F, 

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 one writes A ∈ TAS, if the following condition holds: For any finite set F ⊆ A, any ε > 0, and any non-zero a ∈ A+ , there is a non-zero sub-C*-algebra S ⊆ A such that S ∈ S, and if p = 1S , then (1) pf − f p < ε, f ∈ F, (2) pf p ∈ε S, f ∈ F, and

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(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 by interval 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 amenable simple 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 if Ell(A) ∼ Ell(B). = The following is the main result of this note, which asserts that certain abstract C*-algebras are covered by the classification theorem above. Theorem 2.13. Let A be a separable simple unital C*-algebra satisfying the UCT. Assume that A ⊗ Q has finite nuclear dimension, T(A) = Tqd (A), and K0 (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] directly.  Proof of Theorem 1.1. By Proposition 8.5 of [2], as A⊗Q has finite decomposition rank, T(A ⊗ Q) = Tqd (A ⊗ Q). Furthermore, by [13], A ⊗ Q is stably finite and nuclear and so by [1] and [11], T(A) = ∅. Then K0 (A ⊗ Q) = Q (as order-unit groups), and the statement follows from Theorem 2.13. (The classifiability of A⊗Z holds by Theorem 2.12.)  References [1] B. Blackadar and M. Rørdam, Extending states on preordered semigroups and the existence of quasitraces on C ∗ -algebras, J. Algebra 152 (1992), no. 1, 240–247, DOI 10.1016/00218693(92)90098-7. MR1190414 (93k:46049) [2] J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White, and W. Winter. Covering dimension of C*-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, and dimension functions on C ∗ -algebras, J. Reine Angew. Math. 621 (2008), 191–211, DOI 10.1515/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. London Math. 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 locally ASH-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 on a C ∗ -algebra, Amer. J. Math. 133 (2011), no. 4, 969–1005, DOI 10.1353/ajm.2011.0027. MR2823868 (2012f:46120)

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[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 French summaries), 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 representing 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 (electronic), 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. Reine Angew. Math. 692 (2014), 233–243. MR3274553 [19] H. Lin and Z. Niu, Lifting KK-elements, asymptotic unitary equivalence and classification of simple C ∗ -algebras, Adv. Math. 219 (2008), no. 5, 1729–1769, DOI 10.1016/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 probl`eme 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 E-mail address: [email protected] Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13507

Topology of natural numbers and entropy of arithmetic functions Liming Ge Dedicated to Professor Richard V. Kadison on the occasion of his ninetieth birthday. Abstract. Compactification of natural numbers is studied in association with their arithmetics. Stone-Gelfand-Naimark’s theory on function algebras and abelian C*-algebras is applied to characterize the associated compact Hausdorff spaces. Entropy for arithmetic functions is introduced. It is shown that zero entropy functions form a C*-algebra. Sarnak’s M¨ obius Disjointness Conjecture 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 nurtured by von Neumann, Kadison and several others, has matured and become an important branch of Mathematics. S. S. Chern once commented that Number Theory is an applied science—any good mathematics should have deep connections and applications in it. Although Operator Algebras has its roots in Quantum Physics, its connections with Number Theory are evident from its basic examples. Most of the constructions of operator algebras come from groups, group algebras and group actions on manifolds or measure spaces. The set of natural numbers N = {0, 1, 2, . . .} and its compactifications provide a vast class of spaces both topologically and measure-theoretically. Then operations on natural numbers give rise to actions of the semigroup, either additive or multiplicative, of natural numbers on the 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 Ndynamics given by compact Hausdorff spaces obtained through compactifications of N in association with its arithmetic structures. Suppose X is a compact Hausdorff space and T a map from N to X with a dense range. Denote by l∞ (N) the algebra of all uniform bounded, complexvalued functions on N. It is an abelian C*-algebra (as well as a maximal abelian von Neumann algebra) acting on l2 (N), the Hilbert space of all square summable functions on N. Then T induces an injective homomorphism, denoted by T again, Key words and phrases. Compactification, arithmetic functions, C*-algebras, M¨ obius function. Supported in part by NNSF (No: 11321101) of China, Morningside Center of Mathematics and Mathematical Sciences Center (Yau Center of Mathematics) at Tsinghua University . c 2016 American Mathematical Society

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from C(X) (all bounded continuous functions on X) into l∞ (N), i.e., for f ∈ C(X) and n ∈ N, (T f )(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), there is a compact Hausdorff space X such that A ∼ = C(X) by Stone-Gelfand-Naimark theory. The space X is also known as the maximal ideal space of A, or, equivalently the pure (or, multiplicative) state space of A. For l∞ (N), we shall use βN to denote ˇ its maximal ideal space, also known as Stone-Cech compactification of N. It is easy to see that any multiplicative state (i.e., a homomorphism into C given by the point evaluation at an element in βN) on l∞ (N) is also a multiplicative state on A which corresponds to an element in X. This correspondence gives rise to a canonical map from βN to X. Since βN is the maximal compactification of N, the above map from βN to X is always continuous. From the density of N in βN and that of T (N) in X, one easily checks that this map is also surjective. Thus any compactification X of N (given by a map T : N → X with a dense range) is given uniquely by a unital C*-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 take into consideration some additional structures of N. There are two elementary algebraic operations on N: addition and multiplication. Naturally N is equipped with a discrete topology. Throughout the paper, we shall use A to denote the map on N given by n → n + 1, for all n ∈ N. Then Am : n → n + m is well defined for each m ∈ N. Similarly, for each m ∈ N, we define Mm : n → mn, for all n ∈ N. With the discrete topology, any map T from N to any Hausdorff space Y is continuous including those to N itself. Moreover if the space Y is second countable, such a map may have a dense range. The maps A and Mm on N may or may not induce maps on T (N) (in Y ). When they do, then A : T n → T (n+1) and Mm : T n → T (nm) are well-defined on T (N). In general, these maps may not be extendable to continuous maps on Y . We shall be interested in the case when such a map or maps can be extended continuously onto Y . From discussions in the previous paragraph, instead of Y , it is natural to consider the algebra C(Y ) of all bounded continuous functions on 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 a compact Hausdorff space. In this paper, we shall assume that all C*-algebras are unital. We again use T to denote the map from N to X, i.e., T (n) : f → f (n) is the multiplicative state of point evaluation at n. In general, T (n) → T (n + 1) may not 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 into itself, then we call the C*-algebra A an anqie1 of N. Without ambiguity, we may also call the N-dynamical system (X, A) an anqie of N (with a specified transitive point T (0) in X). One can easily see that a C*-subalgebra A of l∞ (N) is an anqie if (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 map Mm 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. 1 The

word “anqie”, pronounced like “an-chy”, is the Chinese translation of the word “angel.”

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Topologically, suppose X is a compact Hausdorff space and A a continuous map on X. If A has a transitive point x0 in X (i.e., {An x0 : n ∈ N} is dense in X), then the map T : n → An x0 gives rise to a map from N to X with a dense range. The map A on X coincides with the map n → n + 1 on N and the density of 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 widely used one is defined in terms of open sets: (X, A) is transitive if, for any two open subsets 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) is called transitive if there is an x0 ∈ X such that {An x0 : n ∈ N} is dense in X. Transitive dynamics have been well studied by many mathematicians. Our focus is not the transitivity of an N-dynamics, but the arithmetics associated with it. More specifically, we are interested in number theoretic implications or properties of anqies or N-dynamics generated by specific arithmetic functions. Arithmetic functions we encounter in this paper are well known ones in number theory or other areas of mathematics. The anqies they generate often give rise to nice geometrical objects with smooth structures. A major tool we shall use in this paper to study anqies is the notion of “entropy”. Roughly speaking, the entropy we shall associate with an anqie A is the topological entropy of the additive map A. Surprisingly, this dynamical entropy resembles more of Shannon’s entropy for random variables. This might hint a connection between the two philosophically different notions of entropies and reveal certain underlying complexity in a system with or without an action. Our results, even though primitive and preliminary, will show certain connections between geometry and number theory. The natural bridge between the two is analysis and algebras. We hope to bring more geometrical and analytical tools into the study of number theory, and strongly believe that geometry and number theory are the two faces of one angel (see also [3]). Our paper is organized as follows. Section 2 contains two basic constructions of anqies. Many of our examples are well known in classical dynamics and operator algebras. Topological characterizations are established in terms of the generating arithmetic functions of anqies. An additive (or arithmetic) entropy, which we call “anqie entropy”, of arithmetic functions is introduced in Section 3. Examples, computations and basic properties of the entropy are also discussed. Further properties of anqie entropy are explored in Section 4. Among them, we show that “orthogonality” (or certain independence) of arithmetic functions implies the additivity of anqie entropy. Also our entropy has a lower semi-continuity property with respect to uniform convergence of functions. These properties resemble those in both Shannon’s and Kolmogorov’s entropies. As a consequence, we shall see that all zero entropy functions form a C*-algebra. In Section 5, GNS constructions are performed with respect to invariant mean states on anqies. With respect to vector norms given by the states, different notions of periodicity are introduced. Asymptotically periodic functions can be weakly approximated by zero entropy functions. Zero entropy functions can be further weakly approximated by zero entropy functions with finite ranges. Some number theoretically connections with anqies and topological spaces are discussed in Section 6. Arithmetic compactification of N is studied. As an application, we show that the M¨obius function μ is disjoint from

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the C*-algebra generated by “essentially periodic” functions and also with finite products of translations of μ2 . Detailed proofs of most of our results and some further studies will appear in [4] and [5]. Most of the content in this article resulted from a course taught at Tsinghua University in Spring 2014 and seminar talks at the Chinese Academy of Sciences, Morningside Center of Mathematics and UNH. The author wishes to thank Yunping Jiang, Jianya Liu, Zeev Rudnick, Dong Wang, Fei Wei, Boqing Xue, Xiangdong Ye, Wei Yuan, Xin Zhang, Yitang Zhang and many others for collaboration and fruitful discussions. Special thanks go to Professors Yuan Wang, Le Yang and S. T. Yau for 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, we see that there are two standard methods to obtain anqies (or hyper-anqies) of N: one is to construct A-invariant (or, A- and Mm -invariant, respectively) C*-subalgebras of 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). In this 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 of mathematics. 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 S 1 = {z ∈ C : |z| = 1} in the complex plane C. It is easy to see that T has a dense range in S 1 and A : e2πinθ → e2πi(n+1)θ = e2πiθ e2πinθ , known as an irrational rotation map, induces a continuous map z → e2πiθ z on S 1 . Thus (S 1 , A) (or, C(S 1 ) ⊂ l∞ (N) determined by T ) is an anqie of N. Also it is not hard to show that multiplication maps Mm (m ∈ N∗ ) induce continuous maps z → z m on S 1 . So the irrational rotation action by θ on S 1 gives rise to a hyper-anqie of N. The above map A is metric preserving when S 1 is equipped with its usual metric. 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 when 0 ≤ x ≤ 12 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). The choices of different transitive points give rise to many embeddings of C[0, 1] into l∞ (N) and thus many anqies. By using dyadic rational  approximation, Dr. Zhang showed us many transitive points given by sums like n 2rnn , where rn are rationals in (0, 1). Topologically, the tent map has winding number equal to zero on S 1 (when 0 and 1 are identified in [0, 1]). Measure theoretically, it is unitarily equivalent to the map A : z → z 2 on S 1 which has a winding number equal to 2. For any θ n irrational, the embedding T : n → e2πi2 θ of N into S 1 gives rise to an anqie C(S 1 ) with the additive map A also given by z → z 2 (extending T (n) → T (n + 1)). As an n arithmetic function, T (n) = e2πi2 θ and similar exponentials have been extensively

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studied in additive number theory. Obviously, T (n) is closely related to the tent map in dynamics. At topological level, there is no simple way to recognize hyper-anqies from an N-dynamics. Geometrically (see [3]), the unit circle S 1 and the unit disk D in C are 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 to obtain a hyper-anqie from C[0, 1]. The above tent map and more general piece-wise monotone (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 us as well. Now, we return to the C*-algebra construction of anqies. Our starting point is a C*-subalgebra of l∞ (N). First of all, we know that l∞ (N) is not a separable C*-algebra. Thus it is not countably generated as a C*-algebra (it is indeed singly generated as a von Neumann algebra). Correspondingly, the maximal ideal space βN of l∞ (N) is not metrizable. C*-subalgebras of l∞ (N) we will be interested in are often countably generated. When a C*-subalgebra A of l∞ (N) is countably generated, the maximal ideal space X of A is metrizable. Moreover if A is singly generated by a bounded arithmetic function f , its maximal ideal space X is homeomorphic to the closure of f (N) in C. Now, we assume that A is an anqie of N. Then f ∈ A implies that Af ∈ A (again (Af )(n) = f (n + 1), for n ∈ N). One can easily check that the smallest anqie containing a bounded arithmetic function f is the unital C*subalgebra of l∞ (N) generated by f, Af, A2 f, . . .. We shall call this C*-algebra the anqie 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 two spaces may not be the same. The following theorem gives a description of Xf in term of f (N) and a representation 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). Denote by DN the Cartesian product of D indexed by N. Assume that B is the Bernoulli shift on DN defined by B : (a0 , a1 , a2 , . . .) → (a1 , a2 , a3 , . . .). Note that DN is a compact Hausdorff space endowed with the product topology. Let Df be the closure of {(f (n), f (n+1), . . .) : n ∈ N} in DN , and Xf the maximal ideal space of the anqie Af generated by f . Then Df ∼ = Xf and the restriction of B on Df is identified with A on Xf . The proof of the theorem is straight forward. A generalization of the above theorem and its detailed proof are given in [5]. Since D is a metric space (induced by the usual metric on C), the product space DN can be equipped with a metric equivalent to the product topology. The following metric on DN is one of such  |aj −bj | . Then Df (⊆ DN ) becomes a and will be often used: d({aj }, {bj }) = ∞ j=0 2j compact metric space and we also know that C(Df ) ∼ = Af . For different f ’s, we get many interesting examples of anqies (Xf , A). Sometimes, Xf ’s might be the same as topological spaces, but have very different anqie structures. For example, n let f1 (n) = e2πinθ and f2 (n) = e2πi2 θ with θ irrational (see Example 1 and the paragraph after Example 2). Then Xf1 ∼ = X f2 ∼ = S 1 . But the map A is metric preserving on Xf1 but not on Xf2 .

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Now we introduce some basic definitions related to anqies (similar definitions can also be introduced for hyper-anqies). Let A ⊆ l∞ (N) be an anqie of N. An A-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 compact Hausdorff spaces X, Y with respective continuous maps A1 , A2 and their respective transitive points x0 ∈ X and y0 ∈ Y , we can define anqie homomorphisms and isomorphisms between (X, A1 ) and (Y, A2 ) accordingly. From the transitivity of A2 , 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 of ∞ l (N) containing S, then we say anqie A is generated by S. In fact, in this case, A is generated by {An f : n ∈ N, f ∈ S} as a unital C*-algebra. Similarly, A is a hyper-anqie generated by S if A is the smallest unital C*-algebra containing An f and Mm f for all m ∈ N∗ , n ∈ N and f ∈ S. Since all anqies are A-invariant unital C*-subalgebras of l∞ (N), such inclusions induce 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 continuous, 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 from introducing exterior operations between anqies in any natural way, but meantime also gives us advantages to study certain relationships among them. The following proposition helps us understand the connection between anqie inclusions and their corresponding topological dynamics. Proposition 2.1. Suppose A is a subanqie of B, or equivalently there is an injective, dynamics preserving *-homomorphism from A into B. Let X and Y be the maximal ideal spaces of A and B, respectively. Then there is an induced continuous surjective map from Y onto X (which is also dynamics preserving). The proof of the above proposition follows from the fact that each algebraic homomorphism from B to C is again an algebraic homomorphism from A to C and that every maximal ideal in A extends to a maximal ideal (may not be unique) in B. To summarize, all anqies are A-invariant C*-subalgebras of l∞ (N), or equivalently, 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 transitive N-dynamics that can be used freely to study anqies. Our hope is to build a bridge between number theory and geometry through arithmetic functions so that analytical, geometrical and other tools can be used to study problems in number theory. Or, number theoretical results can help in understanding analysis, geometry and dynamics 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 space such as S 1 , there might be different anqie structures given by different transitive Nactions. Even with the same action, different transitive points may also determine different anqies. To understand the dynamics on a given space, it is natural to study the complexity of the system. A good invariant to describe such complexity is the notion of (topological) entropy. In this section, we shall introduce an entropy for anqies.

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The notion of “entropy” was first introduced and studied by physicists in the 19th century. It measures some uncertainty or disorderness of a thermodynamical system. A mathematical notion of entropy was first introduced by Shannon. After Shannon’s introduction of a measurement of certain information contained in random variables (or signals), von Neumann made the following comment: You should call it entropy, for two reasons. In the first place, your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage. Von Neumann’s comment may be viewed as an indication that Shannon’s entropy may not have much to do with the one in physics. Shannon’s entropy agrees with Fisher’s information measure, can be described by certain simple axioms and is the foundation for information theory. Not long after Shannon’s entropy, another mathematical notion of entropy was introduced by Kolmogorov for measure preserving transformations (or automorphisms) on probability spaces. This entropy measures the complexity of the dynamics determined by the transformations. In comparison, Kolmogorov’s entropy captures the essence of the same notion in physics in many ways and is an excellent invariant for measure-preserving ergodic Z-actions (or automorphisms) on probability spaces. There are many generalizations of these two seemingly distinct notions of entropy. One direction is often spatial and the other dynamical, e.g., topological entropy is one of the generalizations on the dynamical side. Is there a deep connection between the two? The “entropy” for anqies defined in this section is based formally on the topological entropy of a dynamical system associated with an anqie (X, A), i.e., h(A)—the topological entropy of A on X. From its definition, one obtains many properties of a dynamical entropy. When we use anqie generators, i.e., certain arithmetic functions, 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 map on X. Suppose U and V are two open covers for X. Denote by U ∨ V the open cover 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 will cover X. Define   1 h(T, U) = lim {log min(U ∨ T −1 (U) ∨ · · · ∨ T −n+1 (U)) }, n n h(T ) = sup {h(T, U) : U is an open cover of X} . U

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 defined equivalently by the following: Proposition 3.1. Suppose (X, d) is a compact metric space and T a continuous map on X. For each n ∈ N, define a metric dn on X by dn (x, y) = max d(T j x, T j y). For each δ > 0, let Ωn (δ) be the δ-covering number of X with

0≤j≤n−1

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respect to metric dn , i.e., the minimal number of (open) δ-balls needed to cover X. log(Ωn (δ)) Then h(T ) = sup lim sup . n n δ 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 anqie entropy of A to be the topological entropy h(A) of the additive map A extending the map n → n + 1 on N to the maximal ideal space of A. We use Æ(A) to denote it, i.e., Æ(A) = h(A). If A is generated by arithmetic functions {fi }i (as an anqie), we often use Æ({fi }i ) to denote Æ(A), which is called the anqie entropy of 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. Their proofs 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 defined on the maximal ideal space of the√anqie generated by f1 , . .√. , fn . For example, if f ≥ 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 us with 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 = S 1 and A : z → e2πiθ z, z ∈ S 1 ⊂ C. Then A is metric preserving. Thus h(A) = 0 and so Æ(f ) = 0. One-dimensional transitive dynamics has been extensively studied by many people [1]. We know that there are transitive N-dynamics on [0, 1] or S 1 with any positive 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 choice of a transitive point, anqie entropy Æ(C(X)) does not. For example, the tent map on [0, 1] has topological entropy log 2 (see Section 2, Example 2). We know that C([0, 1]), as an anqie whose dynamics is given by the tent map and with any transitive point, has anqie entropy log 2. Suppose f ∈ C([0, 1]) that generates C([0, 1]) as a unital C*-algebra. Then, for any given transitive point of the tent map, the restriction of f on its dense orbit gives rise to a bounded arithmetic function. By our definition, we have Æ(f ) = log 2, here f may represent different arithmetic 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.

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Theorem 3.5. Suppose f ∈ l∞ (N) has a finite range, i.e., f (N) is a finite set. Let Df denote the image of Xf in f (N)N , and f (N)n the Cartesian product of f (N) with itself n times. Define Φn : f (N)N → f (N)n to be the projection map from f (N)N onto its first n coordinates. Then Φn (Df ) is a finite set. Let |Φn (Df )| log |Φn (Df )| be the cardinality of Φn (Df ). Then we have Æ(f ) = limn . Moreover, n 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. The equality holds when Φn (Df ) = f (N)n . In general, it is very hard to compute the exact value of Æ(f ) even when f takes finitely many values. From [10], we know that both Æ(μ) and Æ(μ2 ) are positive, where μ is the M¨obius function (we may define μ(0) = 0 when needed). Arithmetic functions with finite ranges are very important in number theory. We will show in [5] that Æ(χS ) = 0 for many characteristic functions defined on subsets S of N, e.g., when S is the set of primes or the set of all prime powers. The result in the following example follows easily from a result in topological dynamics: If T : X → X is a continuous map with X compact and X1 ⊂ X is a T -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 S 1 . Let Af be the anqie generated by f and Xf the maximal ideal space of Af . In fact, it is not hard to show that Xf is homeomorphic to the following subset of C: X = {e− n f (n) : n ∈ N} ∪ S 1 , 1

and the induced action of the additive map A on X (still denote by A) is given 1 1 by A : e− n f (n) → e− n+1 f (n + 1) and A is the identity map on S 1 . Since S 1 is 1 A-invariant and X \ S is countable, it follows that Æ(f ) = 0. to X in the above When we examine the map M2 : n → 2n√and its extension √ 1 1 example, we see formally that M2 : e− n e2πi n → e− 2n e2πi 2n , which is not, or cannot be extended to, a continuous map on X (or S 1 as a subset √ of X). Thus this anqie 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 Kolmogorov’s entropies. More properties of anqie entropy will be obtained and more examples 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 topological entropy of an N-dynamics with a transitive map corresponding to the addition in N. Naturally it possesses all properties of a dynamical entropy. Notation-wise, an anqie entropy is also defined based on anqie generators which are functions defined on 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’s entropy. 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 to maps (on a given space).

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Independence has been generalized in several different ways, even in a noncommutative setting (see, e.g., [2]). Topologically, a natural way to describe “independence” 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*-subalgebras of l∞ (N)). We call them anqie independent (or simply “independent”) if the C*-algebra generated by A and B (in l∞ (N)) is canonically isomorphic to A ⊗ B as a C*-algebra tensor product. Two families of arithmetic functions are called anqie independent if the anqies they generate, respectively, are independent. C*-algebra tensor products are an important and subtle topic. As far as we are concerned here, most C*-algebras involved are abelian ones (and thus amenable). Based on this, we see that all anqies are always mutually commuting. Their relations are completely determined by their inclusions in l∞ (N). Now we state one of the main results in this section. Theorem 4.2. For any arithmetic functions f1 , . . . , fn , g1 , . . . , gm ∈ l∞ (N), we always 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 independent families. The proof of the above theorem follows easily from Proposition 2.1 and some basic facts in dynamics. The following simple example shows some aspect of our independence and how it is related to certain arithmetic structures related to natural numbers. For simplicity of notation, we sometimes denote e2πix by e(x). Example 4.3. For any k ∈ N∗ , define fk (j) = e( kj ), for j ∈ N. Then fk is a periodic function of period k. It is not hard to show that fn and fm are anqie independent if and only if (n, m) = 1. Before we continue with more examples, let us recall a result of Weyl’s which is not 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, is a sequence of real vectors in Rk . Then the following are equivalent: (a) {αn } is uniformly distributed modulo 1 in the following sense: Let αn = ({xn1 }, . . . , {xnk }), where {x} denotes the fractional part of a real number x. For any [aj , bj ] ⊆ [0, 1], j = 1, . . . , k, then we have     k k  1   lim {m : αm ∈ [aj , bj ], 1 ≤ m ≤ n} = (bj − aj ). n n   j j (b) For any continuous function f : Rk /Zk → R, ! n 1   lim f (αm )= f (x1 , . . . , xk )dx1 · · · dxk . n n Rk /Z k m=1

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(c) For any (l1 , . . . , lk ) ∈ Zk \ {0}, n 1  2πi(l1 xm1 +···+lk xmk ) e = 0. lim n n m=1

Example 4.4. Let f (n) = e(n2 θ), for n ≥ 0 and θ irrational. Denote by Af the anqie generated by f . It is well known that f (N) = S 1 and Af (N) = S 1 . Now A2 f (n) = e((n + 2)2 θ) = e(((n + 1)2 + 2(n + 1) + 1)θ) = e(3θ)Af (n)e(2nθ). Since z → z¯ is a continuous function on S 1 and the C*-algebra generated by f contains all continuous functions on S 1 , we have that g(n) = e(2nθ) ∈ Af . Inductively one can show that Af is generated by f and g, and thus also by f and Af as a C*-algebra. A simple application of Weyl’s criterion shows that Af ∼ = C(S 1 × S 1 ). We can also write g(n) = e(((n + 1)2 − n2 − 1)θ) = (Af )(n)f (n)e(−θ). Thus A2 f (n) = e(2θ)(Af (n))2f (n). If we use (z1 , z2 ) to denote the coordinates A  is given by f and Af in S 1 × S 1 , then A : (z1 , z2 ) → (z2 , e(2θ)z1 z22 ). This  0 1 the composition of a toral automorphism with a matrix representation −1 2 followed by an irrational rotation (by 2θ) on the second coordinate. Then we know that Æ(f ) = h(A) = 0. Moreover, one can show that Af is a hyper-anqie. Note that, for f (n) = e(2n θ), the situation is much easier than the above example. 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 any f ∈ l∞ (N) and λ = 0, we know that Æ(λf ) = Æ(f ). Letting λ tend to 0, we have that λf tends to 0 in l∞ (N). Thus we cannot expect a general continuity property for Æ with respect to uniform convergence of arithmetic functions in l∞ (N). For topological entropy on a given compact (metric) space, we do have a lower semi-continuity property for the entropy, where the limit is taken over a family of continuous maps. Here we shall discuss the limit of anqie entropy over a family of arithmetic functions. Given fk in l∞ (N), assume that fk converges uniformly to f . Maximal ideal spaces 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 arbi1 ), then the new sequence of functions trarily “small” function gk with gk l∞ ≤ 1+k will have the same limit as the original. But their resulting anqie entropies may be very 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., for any f ∈ l∞ (N) with Æ(f ) < ∞ and > 0, there is a δ > 0, such that, whenever g ∈ l∞ (N) with f − gl∞ < δ, we have Æ(g) > Æ(f ) − . 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). We may 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 to that for the lower semi-continuity for topological entropy. The following corollary is equivalent to the theorem. Corollary 4.1. If fk ∈ l∞ (N) converges uniformly to f , then lim inf k Æ(fk ) ≥ Æ(f ).

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From Lemma 3.1, we see that zero entropy functions are closed under algebraic operations. Now they are also closed under uniform limit. Thus we have the following: Theorem 4.6. Let E0 (N) = {f ∈ l∞ (N) : Æ(f ) = 0} be all bounded arithmetic functions with vanishing anqie entropy. Then E0 (N) is a unital C*-subalgebra of l∞ (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. Then E0 (N) = C(E0 (N)). A similar construction can be carried out for Z and we obtain a compactification of Z. For simplicity, we sometimes use E0 to denote E0 (N) in this 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*-algebra containing both f and g, we have Æ(f + ig) = Æ(f ) + Æ(g). We do not expect the converse to be true due to the above theorem. But it is interesting to ask: to what extent, will the additivity of anqie entropy determine the relations between two (or two 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 preparations. In number theory, we are often concerned with certain estimates or limits of expressions of the form x1 n≤x f (n). For this purpose, we shall consider states N −1 on l∞ (N) given by certain limits along “ultrafilters” from a sum like N1 n=0 f (n). Then the inner product of two functions f and g given by the states involves exactly N −1 (certain limits of) sums like N1 n=0 f (n)g(n). Recall that elements in βN \ N are called free ultrafilters. Free ultrafilters are closely related to invariant means on locally compact groups introduced by von Neumann. 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 by A : n → n + 1 on N, i.e., Af (n) = f (n + 1). A state ϕ on l∞ (N) is called Ainvariant, 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. n n n be a subset of Example 5.2. Let S = ∪∞ n=1 {2 − n, 2 − n + 1, . . . , 2 − 1} N. Suppose Sn = {j ∈ S : 0 ≤ j ≤ n}. Define Fn (f ) = |S1n | j∈Sn f (j), for f ∈ l∞ (N). Choose ω ∈ βN \ N and define Fω (f ) = limn→ω Fn (f ). Then Fω is an 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 n1 n−1 j=0 f (j).

On the other hand, there are invariant states depending on average values of functions.

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Definition 5.3. Suppose ω ∈ βN \ N is a given free ultrafilter. For any n ∈ N n−1 and any f in l∞ (N), we define En (f ) = n1 j=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 an A-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 on subsets 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 inf n→∞

n−1 n−1 1 1 f (j) ≤ E(f ) ≤ lim sup f (j). n j=0 n→∞ n j=0

When En (f ) has a limit, then E(f ) = limn→∞ En (f ). For example, when a subset S 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 of ∞ l (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 1 ˜ ˜ ˜ 2 the inner product on A and f E = f , f  for the (Hilbert space) vector norm on A. The completion of A under this norm is denoted by HE . The action of A on HE 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 can be borrowed freely to our setting. Two anqies A, B are called orthogonal (or perpendicular) (with respect to E) if E(f g) = E(f )E(g), for all f ∈ A, g ∈ B. Two arithmetic functions f and g are orthogonal if E(f g¯) = 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 (depending on the choice of a free ultrafilter), e.g., orthogonality. But when we refer to the notions, we rarely keep track on which mean state E we refer to. Therefore, when we refer to a notion in future defined by a mean state E, we usually mean that the properties used to define the notion are independent of the choices of E (or unless it is clearly stated otherwise). For example, if f and g are orthogonal, E(f g¯) = 0 holds 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” arithmetic functions. We have seen that anqie entropy is stable under perturbations by zero entropy 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 minimal anqie entropy if Æ(f ) ≤ Æ(f + g), for any g ∈ K. Problem 1. Is every bounded arithmetic function the sum of a minimal anqie entropy function and an element in K? Is the M¨ obius function μ a minimal anqie entropy function?

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For zero entropy functions, the above question has a positive answer. In general, we have the following theorem that describes the relation between zero entropy functions 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 a C*-subalgebra of l∞ (N). Proof. Let K0 = K ∩ E0 . Then K0 is a closed two-sided ideal in E0 . Thus E0 /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 ψ has a trivial kernel and thus it is a *-isomorphism. It is also easy to see that ψ has a dense range in B. Therefore ψ is onto. This proves that f + K0  in C*-algebra E0 /K0 is the same as f + K in B, which shows that all elements in B are given  by 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 that the simplest zero (anqie) entropy functions are periodic ones. For the purpose of approximation, we introduce some generalized notions of periodicity. An arithmetic function f in l∞ (N) is said to be essentially periodic (or “en0 f (or equivalently f = An0 f in  periodic”) if there is an n0 ≥ 1 such that f˜ = A is called an e-period of f . HE ) and the smallest such n0 (≥ 1) √ 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 shall construct e-periodic 1 functions taking values only 0 and 1 but with an arbitrary vector norm between 0 and 1. First we construct, for each 0 ≤ √ t ≤ 1, e-periodic functions ft of e-period 1 such that Æ(ft ) = 0 and E(ft ) = t. This is easy when t is rational. Suppose n t is irrational and choose rational numbers mjj (for j ≥ 1 and nj , mj ∈ N non n decreasing, respectively, with respect to j) so that limj mjj = t. We construct ft successively: ft takes value 1 at the first n1 natural numbers in N, then followed by 0’s at the next m1 − n1 numbers in N; repeat this process for n2 and m2 − n2 , n3 and m3 − n3 and so on. and mj (repeating themselves if necessary) so  Choose nj  that 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 eperiodic. 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 if there is a sequence of positive numbers nj ∈ N such that f − Anj f has limit zero in HE (for any mean state E). Asymptotically periodic functions exist naturally in many settings. Here we list some preliminary results, some of which might be well known. The detailed discussions and some of the proofs can be found in [4]. Again μ(n) is the M¨ obius function.

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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 periodic functions are again asymptotically periodic. Asymptotically periodic functions themselves may not have zero entropy, but they are weak (or l2 -) limits of zero entropy functions (by certain averaging process). The orthogonality of arithmetic functions may be viewed as disjointness in number theory. Theorem 5.9. All asymptotically periodic functions in l∞ (N) are contained in H0 , the closure (in HE ) of all zero entropy functions in l∞ (N). For any irrational θ, the function f (n) = e(n2 θ) (in E0 ) is perpendicular to all asymptotically periodic functions in l∞ (N). Thus measure-theoretically, asymptotically periodic functions are weak limits of zero entropy functions. The proof of the above theorem is in [4]. A similar argument will show that the linear span of projections in E0 (or characteristic functions on subsets 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 arithmetic functions. Anqie independence of functions is a much stronger version of orthogonality (or disjointness). Interesting arithmetic functions span a wide range in number theory. Some of them satisfy certain additive or multiplicative properties. For example, e(nθ) can be viewed as a homomorphism from N (with respect to addition) to S 1 and thus is additive. On the other hand, the M¨obius 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 limN N1 N n=1 f (n)g(n) = 0. When f, g are complex-valued functions, we usually 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 M¨obius function μ 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 to E0 (N) (or H0 ) in HE , the completion of all zero anqie entropy functions, for all mean states E (see Section 5 for notations). The comments after Theorem 5.3 combined with Theorem 3.1 show that Sarnak’s M¨ obius disjointness conjecture can be further reduced to the disjointness of μ from zero-dimensional, zero entropy dynamical systems. From Theorem 5.3, we see that a positive answer to Sarnak’s disjointness conjecture implies that the M¨obius function is disjoint from many functions with a nonzero anqie entropy (or equivalently functions given by dynamics of nonzero entropy). So one may ask: Is zero entropy essential for functions disjoint from the M¨ obius function? Or is it the dynamics (or the topology of the system) that is more important for the disjointness of the functions from μ? In [5], we have an example where μ is embedded into C(S 1 ) given by the dynamics (S 1 , T ), where T : z → z 4 and x0 ∈ S 1 is constructed depending on μ(n).

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Now, let us explore a little more the connections between topologies and arithˇ metics of N. Our “largest” possible transitive N-dynamics is the Stone-Cech βcompactification βN, also known as the maximal compactification of N. Similar property 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 diagram commutes: (βN, A) −→ (E0 (N), A) ϕ ↓ψ (X, T ). It is probably known to specialists that any compact N-dynamics has a maximal 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 easy to show that βN is not homeomorphic to E0 (N). But E0 (N) seems much harder to understand. 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), but not on E0 (N)). Many questions may have easier answers for E0 (Z) than that for E0 (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 countable) compact spaces. From our examples, we also see that there are continuous maps from E0 (N) onto S 1 or S 1 × S 1 . From the fact that the entropy of any transitive 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 that there is no surjective continuous map from E0 (N) onto [0, 1]. Whether S 2 has a transitive N-dynamics of zero entropy is still an open question. Thus far, we have seen some obstructions in topological side in relations to arithmetic functions (or anqies) having zero entropy. The following definition seems natural. Definition 6.3. A compact Hausdorff space X is called an E0 -space if X admits a transitive N dynamics of zero entropy. For f ∈ l∞ (N), we say that f is totally disjoint from a compact Hausdorff space X if for any anqie embedding of C(X) into l∞ (N), f is disjoint from C(X) (with respect to any mean state). One can easily show that the M¨ obius function μ is totally disjoint from any finite set. Problem 2. Is μ disjoint from minimal systems arising from [0, 1] or S 1 ? In the above, the entropy zero condition is removed. More topological restrictions are imposed. Similar questions can be asked for S 2 , S 1 × S 1 etc.

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Now let us return to zero entropy dynamics. Arithmetic functions arising from any zero entropy N-dynamics belong to C(E0 (N)) which is the same as E0 (N) (the C*-algebra of all zero entropy bounded arithmetic functions). Thus to understand zero entropy functions, it is important to know E0 (N). It is easy to see that there are many infinite subsets of N having uncountable closures in E0 (N). To see this, we can choose an irrational θ and let A be the anqie generated by f (n) = e(nθ). Then there is an onto continuous map ψ from E0 (N) to S 1 , the maximal ideal space of A. For any infinite subset S of N, if the closure of f (S) in S 1 is uncountable, then the closure of S in E0 (N) is also uncountable. For example, this is the case when S = {n2 : n ∈ N}. A measure-theoretical argument can be applied to give the following theorem. Theorem 6.4. The closure of any infinite subset S of N in E0 (N) is always uncountable. Thus E0 (N) is not metrizable. Finally we return to Sarnak’s disjointness conjecture. The following result on the M¨ obius function μ generalizes a recent result in [9]. Its proof is similar to that in [9]. Theorem 6.5. For any given k ∈ N∗ , we have N h 1   lim lim ( μ(n + jk))2 = 0. h→∞ N →∞ h2 N n=1 j=1

The following corollary is an immediate consequence of the above theorem. Corollary 6.1. The M¨ obius function μ is disjoint from the following functions: 1. Arithmetic functions in the C*-algebra generated by all e-periodic functions in l∞ (N); 2. For any n1 , . . . , nk in N, functions of the form An1 (μ2 ) · · · Ank (μ2 ), i.e., for any n0 , n1 , . . . , nk in N, N 1  μ(n + n0 )μ2 (n + n1 ) · · · μ2 (n + nk ) = 0. N →∞ N n=1

lim

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 to Sarnak’s disjointness conjecture. So after all, Sarnak’s disjointness conjecture may reveal 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, Proceedings 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 M¨ obius 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.

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[7] R. Kadison and J. Ringrose, “Fundamentals of the Operator Algebras,” vols. I and II, Academic Press, Orlando, 1983 and 1986. [8] Jianya Liu and Peter Sarnak, The M¨ obius function and distal flows, Duke Math. J. 164 (2015), no. 7, 1353–1399, DOI 10.1215/00127094-2916213. MR3347317 [9] K. Matom¨ aki and M. Radziwill, Multiplicative functions in short intervals, preprint, arXiv 1501.04585. [10] P. Sarnak, “Three Lectures on the M¨ obius Function, Randomness and Dynamics”, IAS Lecture 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, Beijing 100190, China – and – Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13508

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 groups on C ∗ -algebras are studied, as well as their dual analogues for coactions. To motivate the properness conditions for actions, the commutative cases (actions on spaces) are surveyed; here the conditions are known: proper, locally proper, and pointwise proper, although the latter property has not been so well studied in the literature. The basic theory of these properness conditions is 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 some of the results are believed to be new.

1. Introduction ∗

In our recent study of C -covariant systems (A, G, α) and crossed product algebras between the full crossed product A α G and the regular crossed product A α,r G, it turns out that various generalizations of the concept of proper actions of G play an important role. We therefore start by taking a closer look at this concept, and it turns out that even for a classical action of G on a space X we made what we believe to be new discoveries. Classically (going back to Bourbaki [Bou60]), a G-space X is called proper if the 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 strongly properly discontinuous instead of pointwise proper, locally proper, and proper, respectively. 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. c 2016 American Mathematical Society

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terms P2 , P1 , and P , respectively. A characteristic property of properness (see Lemma 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 G is isomorphic to the reduced crossed product C0 (X) α,r G (see [Phi89] for the second countable case). In Proposition 6.12 (perhaps also folklore) we show that this carries over to locally proper actions. We will show in Theorem 6.2 (believed to be new) that this is true also if X is first countable, but the action is only assumed to be pointwise proper. We propose the following as natural generalizations of properness to a general C ∗ -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 representation of A, and s → Us a continuous map into the unitaries (but not necessarily a homomorphism) such that π(αs (a)) = Us π(a)Us∗ . Then for all ξ, η in the Hilbert space the coefficient function s → Us ξ, η is in C0 (G). We treat the classical situation of a G-space X in Sections 2 and 3, and discuss general 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 Section 5 that they imply s- or w-properness. The main purpose of these integrability properties is to define a suitable fixed point algebra in M (A), so our properness definitions are too general for this purpose. The natural dual concept of a C ∗ -covariant system is that of a coaction. As we briefly describe in Section 7, it turns out that s- and w-properness can be defined in a similar way for coactions, and we describe some of the relevant results. In Section 8 we describe a general construction of crossed product algebras between A α G and A α,r G. We claim that the interesting ones are obtained by 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 λ of C ∗ (G)). We showed in [KLQ13] that I is a small ideal of C ∗ (G) if and only if the 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). There are various interesting examples (see [BG] and [KLQ13]). Now to a C ∗ -covariant system (B, G, α) and E as above one can define an Ecrossed 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 theory describing these crossed products.

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Finally we mention the work by Baum, Guentner, and Willet [BGW] on the Baum-Connes conjecture. They have shown that there is a unique minimal exact and Morita compatible functor that assigns to a C ∗ -covariant system (A, G, α) a C ∗ -algebra between A α G and A α,r G. At least one of the authors doubts that this minimal functor is an E-crossed product for some large ideal E, although this remains an open problem. In Sections 2–6 we give a fairly detailed exposition, in particular proofs of results we believe to be new. Sections 7–8 will be more descriptive, referring to the literature for details and proofs. 2. Actions on spaces Throughout, G will be a locally compact group, A will be a C ∗ -algebra, and X will be a locally compact Hausdorff space. We will be concerned with actions α of G on A, and we just say (A, α) is an action since the group G will typically be fixed. If G acts on X then we sometimes call X a G-space, and the associated action (C0 (X), α) is defined by αs (f )(x) = f (s−1 x)

for s ∈ G, f ∈ C0 (X), x ∈ X.

Recall that, since the map (s, x) → sx from G×X to X is continuous, the associated action 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, but at 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, but lack of such a reference should not be taken as any claim of originality. In much of the literature on proper actions the spaces are only required to be Hausdorff, or completely regular; in the proofs we will take full advantage of our assumption that our spaces are locally compact Hausdorff. Definition 2.2. A G-space X is proper if the map φ : X × G → X × X defined by 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 to as “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 the set ((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 that the action of G on the homogeneous space G/H by translation is proper if and only if H is compact. The following result is contained in [Pal61, Theorem 1.2.9].

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Proposition 2.5. A G-space X is proper if and only if for all x, y ∈ X there are 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 to choose the neighborhoods U and V to be compact. Conversely, assume the condition involving pairs of points x, y, and let K ⊂ X be 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 that si xi ∈ K. Passing to subnets and relabeling, we can assume that xi → x and si xi → y for some x, y ∈ K. By assumption we can choose compact neighborhoods U of x and V of y such that ((U, V )) is compact. Without loss of generality, for all i we have xi ∈ U and si xi ∈ V , and hence si ∈ ((U, V )). Thus {si } has a convergent subnet by compactness.  Definition 2.6. A G-space X is locally proper if it is a union of open Ginvariant sets on which G acts properly. Palais uses the term Cartan instead of locally proper. The forward direction of 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 a neighborhood U such that ((U, U )) is compact. Proof. First assume that the action is locally proper, and let x ∈ X. Choose an open G-invariant set V containing x on which G acts properly. Then choose a compact neighborhood U of x contained in V . Then ((U, U )) is compact by properness. Conversely, assume the condition involving compact sets ((U, U )). Choose an open 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 ∈ G such 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 )), and also 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 a neighborhood V of x such that ((V, V )) ⊂ U . Proof. First assume that the action is locally proper. We argue by contradiction. Suppose we have x ∈ X and a neighborhood U of Gx such that for every neighborhood V of x there exists s ∈ ((V, V )) such that s ∈ / U . Fix a neighborhood R of x such that ((R, R)) is compact. Restricting to neighborhoods V of x with V ⊂ R, we see that we can find nets {si } in the complement U c and {yi } in R such that • si yi ∈ R for all i, • yi → x, and • si yi → x.

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Then si ∈ ((R, R)) for all i, so passing to subnets and relabeling we can assume that si → s for some s ∈ G. Then si yi → sx, so sx = x. Thus s ∈ Gx . But then eventually si ∈ U , which is a contradiction. Conversely, assume the condition regarding isotropy groups and neighborhoods thereof, and let x ∈ X. Since Gx is compact, we can choose a compact neighborhood U 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 locally proper.  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 and G\X is Hausdorff. Proof. First assume that the action is proper. Then it is locally proper, and to show that G\X is Hausdorff, we will prove that if a net {Gxi } in G\X converges to both Gx and Gy then Gx = Gy. Since the quotient map X → G\X is open, we can pass to a subnet and relabel so that without loss of generality xi → x. Then again passing to a subnet and relabeling we can find si ∈ G such that si xi → y. Choose compact neighborhoods U of x and V of y, so that ((U, V )) is compact by properness. Without loss of generality xi ∈ U and si xi ∈ V for all i. Then si ∈ ((U, V )) for all i, so by compactness we can pass to subnets and relabel so that {si } converges to some s ∈ G. Then si xi → sx, so sx = y, and hence Gx = Gy. Conversely, assume that the action is locally proper and G\X is Hausdorff. Let x, 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 that the action is proper, we will prove that ((U, V )) is compact. Let {si } be any net in ((U, V )). For each i choose xi ∈ U such that si xi ∈ V . By compactness we can pass to subnets and relabel so that xi → z and si xi → w for some z ∈ U and w ∈ V . Then by Hausdorffness we can write Gz = lim Gxi = lim Gsi xi = Gw, so we can choose s ∈ G such that w = sz. Then si xi → sz, so s−1 si xi → z. Without loss of generality, for all i we can assume that s−1 si xi ∈ U , so that s−1 si ∈ ((U, U )). By compactness we can pass to subnets and relabel so that s−1 si → t for some t ∈ G. Thus si → st, and we have found a convergent subnet  of {si }. Thus ((U, V )) is compact. Example 2.10. It is a well-known fact in topological dynamics that there are actions 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 neighborhood of {(1, 0), (0, 1)} meets itself infinitely often. This action is locally proper because 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 R act on the same space by s(x, y) = (es x, e−s y). Definition 2.11. A G-space X is pointwise proper if for all x ∈ X and compact K ⊂ X, the set ((x, K)) is compact.

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The above properness condition does not seem to be very often studied in the dynamics literature, and the term we use is not standard, as far as we have been able 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 ∈ X the 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 {si x} in the orbit Gx converging to y ∈ X. Choose a compact neighborhood U of y. Without loss of generality, for all i we have si x ∈ U , and hence si ∈ ((x, U )). This set is compact by pointwise properness, so passing to a subnet and relabeling we can assume that si → s for  some s ∈ G. Then si x → 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 ∈ X the isotropy subgroup Gx is compact and the map s → sx from G to Gx is relatively open, equivalently, the action of G on the orbit Gx is conjugate to the action on the homogeneous space G/Gx . Proof. First assume that the action is pointwise proper, and let x ∈ X. Then Gx is trivially compact. By homogeneity it suffices to show that the map s → sx from G to Gx is relatively open at e. Let W be a neighborhood of e. Suppose that W x is not a relative neighborhood of x in the orbit Gx. Then we can choose a net / W x and si x → x. Let U be a compact neighborhood of {si } in G such that si x ∈ x. Then ((x, U )) is compact. Without loss of generality, for all i we have si x ∈ U , and so si ∈ ((x, U )). By compactness we can pass to a subnet and relabel so that si → s for some s ∈ G. Then si x → sx. Thus sx = x, and so s ∈ Gx . But then eventually si ∈ W Gx , which is a contradiction because W Gx x = W x. The converse is obvious, since if Gx is compact the action of G on G/Gx is proper.  We will show that pointwise properness is weaker than local properness, but for this we need a version of Proposition 2.13 for local properness. The following result 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 si x → y. Choose an open G-invariant subset U containing y on which G acts properly. Then the action of G on U is pointwise proper, so the orbit Gx is closed in U , and hence y ∈ 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 such that 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 on U. 

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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 X by identifying {0} × (0, ∞) with (0, ∞) × {0} via (0, y) ∼ (1/y, 0). Then the action descends to the identification space, and the quotient action is pointwise 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 are second countable, and that every isotropy subgroup is compact. Then the following are 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 countable case, (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). In this section we examine the ramifications for the action α of the various properness conditions covered in Section 2. For the state of the art in the case of proper actions, see [EE11]. Notation 3.1. If ψ : X → Y is a continuous map between locally compact Hausdorff 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 Hausdorff 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 map s → 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).

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Since the map φ : G × X → X × X given by φ(s, x) = (sx, x) is proper, we have φ∗ (f × g) ∈ Cc (G × X), so there exist compact sets K ⊂ G and L ⊂ X such that for 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 supported in the compact set K −1 . Conversely, assume the condition regarding αs (f )g. To show that the action is proper, we will show that the map φ is proper, and by Lemma 3.2 it suffices to show that if h ∈ Cc (X × X) then φ∗ (h) ∈ Cc (G × X). The support of h is contained in a product 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 show that φ∗ (f × g) has compact support. By assumption the support K of s → αs (f )g is compact, and letting L be the support of g we see that for all (s, x) not in the compact 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 all f ∈ C0 (X) and μ ∈ M (X) = C0 (X)∗ the map ! f (sx) dμ(x) g(s) = X

is in C0 (G). Proof. First assume that the action is pointwise proper. Let f ∈ C0 (X) and μ ∈ M (X), and define g as above. Note that g is continuous since the associated action (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 {sn V } are pairwise disjoint. Then for each x ∈ X we have limn→∞ f (sn x) = 0, because for fixed x and any δ > 0 it is an easy exercise to see that the compact set {s ∈ G : |f (sx)| ≥ δ} can only intersect finitely many of the sets {sn V }. Thus by the Dominated Convergence theorem limn→∞ g(xn ) = 0, contradicting sn ∈ S for all n. The converse follows immediately by taking μ to be a Dirac measure and applying Lemma 2.12.  Proposition 3.5 below is the first time we need vector-valued integration. There are numerous references dealing with this topic. We are interested in integrating functions f : Ω → B, where Ω is a locally compact Hausdorff space equipped with a Radon measure μ (sometimes complex, but other times positive, and then frequently infinite), and B is a Banach space. Rieffel [Rie04, Section 1] handles continuous bounded functions to a C ∗ -algebra using C ∗ -valued weights. Exel [Exe99, Section 2] develops a theory of unconditionally integrable functions with values in a Banach space, involving convergence of the integrals over relatively compact subsets of G. Williams [Wil07, Appendix B.1] gives an exposition of the general theory of L1 (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 a positive measure throughout, and we occasionally need complex measures; this

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poses no problem, since the theory of [Wil07] can be applied to the positive and negative variations of the real and imaginary parts of a complex measure. We prefer to use [Wil07] as our reference for vector-valued integration, mainly because it entails absolute integrability rather than unconditional integrability (see the first item in the following list). Here are the main properties of L1 (Ω, B) that we need:  1 • The map  f → Ω f dμ from L (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 f (x) dμ(x) = ω(f (x)) dμ(x). Ω

Ω

• If f ∈ L (Ω) and b ∈ B then !  ! (f ⊗ b) dμ = f dμ b, 1

Ω

Ω

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 integrable with respect to any complex measure. Of course, we refer to the elements of L1 (Ω, B) as the integrable functions from Ω to B. If X is a G-space, then C0 (X) gets a Banach-module structure over M (G) = C0 (G)∗ by ! μ ∗ f (x) = f (sx) dμ(s) for μ ∈ M (G), f ∈ C0 (X), x ∈ X. G

Here we are integrating the continuous bounded function s → αs−1 (f ) with respect to 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 each f 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 by s → aαs (b). Definition 4.2. An action (A, α) is w-proper if for all a ∈ A and all ω ∈ A∗ the map   g(s) = ω αs (a) is in C0 (G). We use the admittedly nondescriptive terminology s-proper and w-proper to avoid confusion with the myriad other uses of the word “proper” for actions on C ∗ -algebras.

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Remark 4.3. It is almost obvious that a G-space X is locally proper if and only if there is a family of α-invariant closed ideals of C0 (X) that densely span C0 (X) and on each of which α has the property in Proposition 3.3. In fact, we will use this in the proof of Proposition 6.12. This could be generalized in various ways to actions on arbitrary C ∗ -algebras, but since we have no applications of this we will 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 the Cohen-Hewitt factorization theorem every functional in A∗ can be expressed in the form 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= αs (a) dμ(s) for μ ∈ M (G), a ∈ A. G

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 ∈ A the 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* in M (G), and let ω ∈ A∗ . Then !  ! ω(μi ∗ a) = ω αs (a) dμi (s) = ω(αs (a)) dμi (s) → 0, G

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 ! ω(αs (a)) dμi (s) = ω(μi ∗ a) → 0 G

by continuity. By the well-known Lemma 4.7 below, the element s → ω(αs (a)) of  Cb (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 morphisms: Proposition 4.8. Let φ : A → M (B) be a nondegenerate homomorphism that is equivariant for actions α and β, respectively. If α is s-proper or w-proper, then β has the same property.

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Proof. First assume that α is s-proper. Let c, d ∈ B. By the Cohen-Hewitt Factorization 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 that the 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 with a ∈ A and c ∈ B. By the Cauchy-Schwarz inequality for positive functionals on C ∗ -algebras, we have     2 ω ◦ βs (b)2 = ω φ(αs (a∗ ))βs (c)  ≤ 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, because it 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 a pair of nondegenerate homomorphisms A

jA

/ M (A δ G) o

jG

C0 (G)

such that (jA , jG ) is a universal covariant homomorphism. The dual action δ of G on 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 homomorphism jG : C0 (G) → M (A δ G) is rt − δ equivariant. Thus δ is s-proper since rt is.  [BG12, Corollary 5.9] says that if an action of a discrete group G on a compact Hausdorff space X is a-T-menable in the sense of [BG12, Definition 5.5], then every covariant representation of the associated action (C(X), α) is weakly contained in a representation (π, U ), on a Hilbert space H, such that for all ξ, η in a dense subspace of H the function s →< Us ξ, η is in c0 (G). The following proposition shows 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 representation of A on a Hilbert space H, and for each s ∈ G suppose we have a unitary operator Us on H such that Ad Us ◦ π = π ◦ αs . Then for all ξ, η ∈ H the function s → Us ξ, η

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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 |Us π(a)ξ  , η| = |Us ξ  , π(αs (a∗ ))η| ≤ ξ  π(αs (aa∗ )η, η1/2 , 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 aT-menable actions in the sense of [BG12] and pointwise proper actions. As it stands, the connection would be subtle, because an infinite discrete group cannot act pointwise properly on a compact space. Action on the compacts. The following gives a strengthening of a special case of Proposition 4.10: Proposition 4.13. Let H be a Hilbert space, and let α be an action of G on K(H). For each s ∈ G choose a unitary operator Us such that αs = Ad Us . The following are equivalent: (1) α is s-proper; (2) α is w-proper; (3) s → Us ξ, ξ vanishes at infinity for all ξ ∈ H. (4) s → Us ξ, 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, ηξ. For ξ, η, γ, κ ∈ H, a routine computation shows E(ξ, η)αs (E(γ, κ)) = Us γ, ηE(ξ, κ)Us∗ , so

    E(ξ, η)αs (E(γ, κ)) ≤ Us γ, ηE(ξ, κ), which vanishes at infinity. Thus s → aαs (b) is in C0 (G, K(H)) whenever a and b are 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 have the following: Corollary 4.14. Let U be a representation of G on a Hilbert space H, and let α = Ad U be the associated action of G on K(H). Suppose that ξ is a cyclic vector for the representation U . If s → Us ξ, 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 the dense subspace of H spanned by {Us ξ : s ∈ G} the function s → Us η, κ vanishes at 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 → Us ηn , κn  vanishes at infinity. Then a routine estimation shows that the functions s → Us ηn , κn  converge uniformly to the function s → Us η, κ, and hence this latter function vanishes at infinity. The result now follows from Proposition 4.13. 

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5. Rieffel properness We will show that if an action (A, α) is proper in Rieffel’s sense [Rie90, Definition 1.2] (see also [Rie04, Definition 4.5] then it is s-proper. Rieffel’s definitions of proper action in both of the above papers involve integration of A-valued functions on G, and we have recorded our conventions regarding vector-valued integration in the 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 is integrable for all a, b in some dense subalgebra, plus other conditions that we will not 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 all a, 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, it follows immediately from the elementary lemma Lemma 5.2 below that s → αs (a)b is 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 scalarvalued.  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 continuous and integrable. Then f vanishes at infinity. Proof. Since  the composition of f with the norm on B is uniformly continuous, 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 routine adaptation of a classical result about scalar-valued functions on R, sometimes referred 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) is s-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 X is proper, and then it follows from [Rie04, Theorem 4.7 and and its proof] that α is Rieffel proper. Conversely, if α is Rieffel proper, then in particular it satisfies the hypothesis of Corollary 5.1 (1), so α is s-proper. 

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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 the following: for f ∈ Cc (X) the integral !  f (Gx) := f (sx) ds G

defines f ∈ Cc (G\X). If the action on X is just pointwise proper, the integral  f (sx) ds still makes sense for f ∈ Cc (X). It would be interesting to know what G properties persist in this case. Example 5.5. Proposition 5.3 is not true for arbitrary actions (A, α). For example, 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 positive definite. For k ∈ N define hk (s) = e−l(s)/k , and let Uk be the associated cyclic representation 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 = Ad Uk of G on K(Hk ) is s-proper, by Corollary 4.14. We claim that not all these actions αk can be Rieffel proper. Rieffel shows in [Rie04, Theorem 7.9] that the action α is proper in the sense of [Rie04, Definition 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 that it uses concepts from the theory of left Hilbert algebras. Also, Rieffel’s definition of proper action in [Rie04] is somewhat complicated in that it involves C ∗ -valued weights. In this paper we prefer to deal with the more accessible definition of Rieffel-proper action in [Rie90, Definition 1.2], which Rieffel shows implies the properness condition [Rie04, Definition 4.5]. Actually, we need not concern ourselves here with Rieffel’s definition of square-integrable representations, rather all we need is his reassurance (see [Rie04, Corollary 7.12 and Theorem 7.14]) that a cyclic representation of G is square-integrable in his sense if and only if it is contained in the regular representation of G — so his notion of square integrability is 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], and so the representation Uk is contained in the regular presentation λ. Now we argue exactly as in [BG12, proof of Proposition 4.4]: since the functions hk converge to 1 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 representation. 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.

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6. Full equals reduced Definition 6.1. Let (A, α) be an action. We say the full and reduced crossed products 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-space then 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 the proof of Proposition 6.12 and Remark 6.14. We extend this to pointwise proper actions and weaken the countability hypothesis: Theorem 6.2. If X is a first countable pointwise proper G-space, then the full and 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) be an injective α − β equivariant homomorphism. Suppose that the crossed-product homomorphism 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 and reduced crossed products of α are equal. Proof. We have a commutative diagram A α G Λα

φG

/ M (B β G) Λβ



A α,r G

φr G

 / 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 products equal, 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 the quotient map. We have a commutative diagram J G

φG

ΛJ

 J r G

/ AG

ψG

ΛA/J

ΛA

φr G

 / A r G

/ A/J  G

ψr G

 / A/J r G.

The argument is a routine diagram-chase. The vertical maps are the regular representations, which are surjective, and moreover ΛJ and ΛA/J are injective by hypothesis. Since J is an ideal, the map φ  G is an isomorphism onto the kernel of

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ψ  G [Gre78, Proposition 12]. Further, since J is an invariant subalgebra, φ r G is 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 full and reduced crossed products are equal for every αi . Then the direct sum action 

% % Ai , αi i∈I

i∈I

also has full and reduced crossed products equal. Proof. By Lemma 6.4, the conclusion holds if I has cardinality 2,* and by induction 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 a commutative diagram *     / * A G A  G i∈F

ΛF

* i∈F

i

i∈I



   Ai r G 

i

ΛI

/

  A i∈I i r G,

*

where the vertical arrows are the regular representations. Thus ΛI must be an isomorphism, by properties of inductive limits.  Corollary 6.6. Let (A, α) be an action, let {(Ai , αi )}i∈I be a family of actions for 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 Ai i∈I

be the associated equivariant homomorphism. Suppose that that the crossed-product homomorphism 

%  Ai αi G A α G → M

+ i∈I

ker φi = {0}, and

i∈I

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 recall the notion of quasi-regularity, and we only need this in the special case of closed orbits:

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Definition 6.7 (special case of [Gre78, Page 221]). Let G act on X, and assume that all orbits are closed. Then the associated action of G on C0 (X) is quasi-regular if for every irreducible covariant representation (π, U ) of (C0 (X), G) there is an orbit G · x such that ker π = {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 orbit space G\X is second countable or almost Hausdorff in the sense that every closed subset contains a dense relative open Hausdorff subset. Here we will prove a variant of this result: Proposition 6.8. If a G-space X is pointwise proper and first countable, then the associated action of G on C0 (X) is quasi-regular. We first need a topological property of pointwise proper actions on first countable spaces: Lemma 6.9. If a G-space X is pointwise proper and first countable, then each orbit is a countable decreasing intersection of open G-invariant sets. Proof. Since orbits are closed, the quotient space G\X is T1 . Since the quotient map is continuous and open, G\X is first countable. In particular, every point is a countable decreasing intersection of open sets, and the result follows.  Remark 6.10. In Lemma 6.9 the first countability assumption could be weakened 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 a special case: Lemma 6.11. If a G-space X is pointwise proper and first countable, and if there is an irreducible covariant representation (π, U ) of (C0 (X), G) such that π is faithful, then X consists of a single orbit. Proof. We can extend π to a representation of the algebra of bounded Borel functions 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-countable case; Murphy states the theorem for compact Hausdorff spaces, but it applies equally well to locally compact spaces by passing to the one-point compactification). 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 at most one orbit with measure 1. Claim: every nonempty G-invariant open subset O of X has spectral measure 1. It suffices to show that P (O) = 0. Since O = ∅, we can choose a nonzero f ∈ C0 (X) supported in O. Then 0 = π(f ) = π(f χO ) = 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 the properties of spectral measures, we have P (G · x) = lim P (On ) = 1. n

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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 representation of (C0 (X), G) on a Hilbert space H. Then ker π is a G-invariant ideal of C0 (X), so there is a closed G-invariant subset Y of X such that ker π = {f ∈ C0 (X) : f |Y = 0}. We will show that Y consists of a single orbit. The restriction map f → f |Y is a G-equivariant homomorphism of C0 (X) to C0 (Y ), and ker π = C0 (X \ Y ), so π factors through a faithful representation ρ of C0 (Y ) such that (ρ, U ) is an irreducible covariant representation of (C0 (Y ), G). Then Y is a single orbit, by Lemma 6.11.  Proof of Theorem 6.2. For each x ∈ X, the orbit G·x is closed, the isotropy subgroup Gx is compact, and the canonical bijection G/Gx → G·x is an equivariant homeomorphism. Thus Gx is in particular amenable, so it follows from the above and [QS92, Corollary 4.3] (see also [Kas88, Theorem 3.15]) the associated action of G on C0 (G · x) has full and reduced crossed products equal. The restriction map φx : C0 (X) → C0 (G · x) is equivariant, and we get an equivariant injective homomorphism 

% φ : C0 (X) → M C0 (G · x) . x∈X

By Proposition 6.8 the action of G on C0 (X) is quasi-regular, so every irreducible 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 homomorphism 

 % φ  G : C0 (X)  G → M C0 (G · x)  G x∈X

is faithful. Therefore the theorem follows from Corollary 6.6.



The above strategy can also be used to prove the following folklore result, which is a mild extension of Phillips’ full-equals-reduced theorem. Actually, we could not find the following result explicitly recorded in the literature, but it seems to us that it 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 Ginvariant ideals that densely span A. If for every i the restriction of the action to Ji has full and reduced crossed products equal, then the action on A has the same property. Proof. For each i let αi = α|Ji , let φi : A → M (Ji ) be the α − αi equivariant homomorphism induced by the A-bimodule structure on Ji , and let φ : A →

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163

* M ( i∈I+ Ji ) be the associated equivariant homomorphism, since A = spani∈I Ji , we have i∈I ker φi = {0}. Thus, by Corollary 6.6 we only need to show that 

%  φ  G : A α G → M Ji αi G i∈I

is faithful. Suppose that ker(φ  G) = {0}. The ideals Ji αi G densely span A α G, since the Ji ’s densely span A. Thus we can find i ∈ J such that {0} = ker(φ  G) ∩ (Ji αi G) = ker(φ|Ji  G). But φ|Ji  G is faithful since φ|Ji is faithful and Ji is a G-invariant ideal, so we have a contradiction.  Proof of Proposition 6.12. First, if the G-space X is actually proper, then G\X is Hausdorff, so by [Gre78, Corollary 19] the action of G on C0 (X) is quasiregular, so the conclusion follows as in the proof of Proposition 6.2. In the general case, X is a union of open G-invariant subsets Ui , on each of which G acts properly. Then C0 (X) is densely spanned by the ideals C0 (Ui ), so by properness the associated actions αi have full and reduced crossed products equal, and hence the conclusion follows from Lemma 6.13.  Remark 6.14. In the above proof we appealed to [Gre78, Corollary 19], whose proof involved dense points in irreducible closed sets. In the spirit of the techniques of the current paper, we offer an alternative argument: assume that X is a proper G-space. To see that the action is quasi-regular, as in the proof of Proposition 6.8 we can assume without loss of generality that there is an irreducible covariant representation (π, U ) of (C0 (X), G) such that π is faithful. We must show that X 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 open neighborhoods of G · x and G · y in G\X, and hence nonempty disjoint open Ginvariant sets U and V in X. But, as in the proof of Lemma 6.11, letting P denote the spectral measure associated to the representation π of C0 (X), every nonempty G-invariant open subset O of X has P (O) = 1. Since we cannot have two disjoint open 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], if every irreducible representation is by compacts. In the second countable case, the following result is contained in [Wil07, Proposition 7.31]. Proposition 6.15. Let X be a G-space. In either of the following two situations, 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. To illustrate how the above methods apply, we give the following argument. We have seen above that the action is quasi-regular, and hence every irreducible covariant representation (π, U ) of (C0 (X), G) factors through an irreducible representation of the restriction of the action to (C0 (G · x), G) for some x ∈ X. The G-spaces G · 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

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the isotropy subgroup Gx is compact, C ∗ (Gx ) is CCR, and hence the image of the integrated form ρ × U , which equals the image of π × U , is the algebra of compact operators. In the general case, X is a union of open G-invariant proper G-spaces Ui , so C0 (X)  G is the closed span of the CCR ideals C0 (Ui )  G. Since every C ∗ -algebra has 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 action is amenable (a condition involving approximation by positive-definite functions). By [AD02, Theorem 5.3], if a G-space X is amenable then the associated action α on C0 (X) has full and reduced crossed products equal. This raises a question: is every pointwise proper action amenable? It seems that amenability of the G-space is closely related to equality of full and reduced crossed products: by [Mat14, Theorem 3.3], for an action of a discrete exact group G on a compact space X, if α has full and reduced crossed products equal then the action is amenable. Unfortunately, this is of no help for our question, because a noncompact group cannot act pointwise properly on a compact space. 7. Properness conditions for coactions We will now dualize the properness properties of Definitions 4.1 and 4.2. To motivate how this will go, we pause to recall some basic facts regarding C ∗ -tensor products, 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, M β (A)), where M β (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). (A ⊗ C0 (G)), where for any In fact, the image of α  lies in the C ∗ -subalgebra M ∗ C -algebras A and D (A ⊗ D) := {m ∈ M (A ⊗ D) : m(1 ⊗ D) ∪ (1 ⊗ D)m ⊂ A ⊗ D}. M Using the above facts, Corollary 4.4 can be restated as follows:

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165

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 that the commutative C ∗ -algebra C0 (G) is replaced by C ∗ (G). Here we will use the standard conventions for tensor products and coactions (see, e.g., [EKQR06, Appendix A], in particular, the coaction is a homomorphism (A ⊗ C ∗ (G)). δ:A→M 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 properness conditions, but in that paper we used the term proper coaction for the above sproper coaction, and slice proper coaction for the above w-proper coaction (because it involves the slice map ω ⊗ id). After we submitted [KLQ], we learned that Ellwood had defined properness more generally for coactions of Hopf C ∗ algebras [Ell00, Definition 2.4]. Indeed, Proposition 3.3 is essentially [Ell00, Theorem 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 the algebraic tensor product. Remark 7.4. An action on C0 (X) can be w-proper without being s-proper, and a 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, every coaction 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) given by the comultiplication is s-proper, because it is symmetric in the sense that δG = Σ ◦ δ G , 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 FourierStieltjes 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 map f → f · a is weak*-to-weakly continuous. Both s-properness and w-properness are preserved by morphisms. For wproperness this is proved in [KLQ, Proposition 5.3], and here it is for s-properness:

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Proposition 7.7. Let φ : A → M (B) be a nondegenerate homomorphism that is equivariant for coactions δ and ε, respectively. If δ is s-proper then ε has the same 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 homomorphism iG : C ∗ (G) → M (A α G) is δG − α  equivariant, where δG is the canonical  is s-proper since δG is.  coaction on C ∗ (G) given by the comultiplication. Thus α Recall that if (A, δ) is a coaction then the spectral subspaces {As }s∈G are given by 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 let ae ∈ 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* continuous  linear 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. Also note that when G is nondiscrete a coaction (A, δ) can be s-proper and still have nonzero spectral subspaces As (and hence nontrivial fixed-point algebra Aδ , but these 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 δp on 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 in M (A). If (A, δ) is s-proper, then so is the corner coaction (pAp, δp ) defined above.

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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 , δ n ) is s-proper; (3) The maximalization (Am , δ m ) 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 coaction δ is automatically s-proper  because s-properness is not and the maximalization δ m is Morita equivalent to δ, preserved by Morita equivalence — otherwise every coaction of an amenable group would be s-proper! Recall from [KMQW10, Proposition 3.1] that if A → G is a Fell bundle then there is a coaction δA of G on the (full) bundle algebra C ∗ (A). (That result was stated 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 the form f · b for f ∈ A(G) ∩ Cc (G): !   δ(a)(f · b ⊗ 1) = a(t)f · b ⊗ t dt !G   = a(t)b ⊗ tf dt (justified below) G

∈ C ∗ (A) ⊗ C ∗ (G), because the integrand t → 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,

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where we must now justify the equality f · b ⊗ 1 = b ⊗ f : both sides can be regarded as compactly supported strictly continuous functions from G to M (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 r δA = (id ⊗ λ) ◦ δA : C ∗ (A) → M (C ∗ (A) ⊗ Cr∗ (G)) r is be the reduction of the coaction δA . [Bus10, Theorem 3.10] shows that δA r integrable in the sense that the set of positive elements a in A for which δA (a) is in the domain of the operator-valued weight id ⊗ ϕ is dense in A+ , where ϕ is the Plancherel weight on Cr∗ (G).

Corollary 7.16 below is a dual analogue of Corollary 5.1 (1). To explain the terminology, we recall a few things from Buss’ thesis [Bus07]. Buss worked with reduced coactions, but as he points out in [Bus07, Remark 2.6.1 (4)], the theory 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 = span Mϕ , so that Mϕ is a hereditary cone in C ∗ (G), and coincides with both Mϕ ∩ C ∗ (G)+ and span Nϕ∗ Nϕ , 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 that   θ(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 and c ∈ Mϕ . Let Λ : Nϕ → L2 (G) be the canonical embedding associated to the GNS construction 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 associated to 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 all a ∈ 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 Mϕ , N ϕ , and Mϕ . Similarly for the canonical extension of id ⊗ ϕ to +

M (A ⊗ C ∗ (G)), Mid⊗ϕ , etc. Let + Asi = {a ∈ A : δ(aa∗ ) ∈ Mid⊗ϕ }.

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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, Section 6.8 and Proposition 6.9.4] gives a Fell bundle A over G and a δA −δ equivariant surjective homomorphism κ : C ∗ (A) → A. By Proposition 4.8, every quotient of an 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 Bα G and the reduced crossed product B α,r G. But there are frequently many “exotic” crossed products in between, i.e., quotients (B α G)/J where J is a nonzero ideal properly contained in the kernel of the regular representation Λ. In [KLQ13], inspired by work 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 Cr∗ (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 ⊥ in B(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) = Cr∗ (G)∗ [KLQ13, Lemma 3.14], and the map E → ⊥ E gives a bijection between the large ideals of B(G) and the small ideals I of C ∗ (G). For a large ideal E the quotient map ∗ qE : C ∗ (G) → CE (G) := C ∗ (G)/⊥ E E . is equivariant for δG and a coaction δG

Example 8.1. E = B(G) ∩ C0 (G) is a large ideal, and if G is discrete then G has the Haagerup property if and only if E = B(G) [BG12, Corollary 3.5]. Example 8.2. For 1 ≤ p ≤ ∞, E p := B(G) ∩ Lp (G) is a large ideal. Of course E ∞ = B(G). For p ≤ 2 we have E p = Br (G) [KLQ13, Proposition 4.2] (and [BG12, Proposition 2.11] for discrete G). If G = Fn for n > 1, it has been attributed to Okayasu [Oka] and (independently) to Higson and Ozawa (see [BG12, Remark 4.5]) that for 2 ≤ p < ∞ the ideals E p are all distinct.

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Given an action (B, α), we use large ideals to produce exotic crossed products by involving the dual coaction α  on B α G. As in [KLQ], the process is most cleanly expressed in terms of an abstract coaction (A, δ). An ideal J of A is called δ-invariant if δ descends to a coaction on the quotient A/J. We call an ideal J small if it is invariant and contained in the kernel of jA , where (jA , jG ) is the canonical covariant homomorphism of (A, C0 (G)) into the multiplier algebra of the crossed product A δ G. For the coaction (C ∗ (G), δG ), this is consistent with the above 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} ), we is a small ideal of A [KLQ, Observation 3.10]. For a dual coaction (B α G, α call 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 form a 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 J of A. Since every dual coaction is s-proper, and hence w-proper, Lemma 8.3 is ap) for any action (B, α). In [KLQ, Theorem 6.10] we used plicable to (B α G, α this Galois correspondence to exhibit examples of small ideals J that are not of the 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 the ). Consequently, there are exotic crossed products that are not form (B α G, α E-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 crossedproduct as a functor (B, α) → B α,τ G, from the category of actions to the category of C ∗ -algebras, equipped with natural transformations / B α,τ G B α G q qqq q q q  xqqq B α,r G, where the vertical arrow is the regular representation, such that the horizontal arrow is surjective. For a large ideal E of B(G), the E-crossed product (B, α) → B α,E G

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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 untwisting isomorphism (B ⊗ KG )  G " (A  G) ⊗ KG , * 2 where KG denotes the compact operators on ∞ n=1 L (G), descends to an isomorphism (B ⊗ KG ) τ G " (A τ G) ⊗ KG of τ -crossed products. [BGW, Theorem 3.8] (with an assist from Kirchberg) shows that 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 Morita compatible [BGW, Lemma A.5]. It is an open problem whether the minimal functor of [BGW] is an E-crossed product for some large ideal E. The counterexamples of [BE13] do not necessarily give a negative answer, because it is unknown whether they fit into a crossedproduct functor. The state of the art regarding E-crossed products is depressingly meager 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 group G (where Br (G) = Cr∗ (G)∗ denotes the reduced Fourier-Stieltjes algebra). But nonexact groups are quite mysterious. References [BGW] [Bou60]

[Bou63]

[BG12]

[Bus07] [Bus10]

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School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287 E-mail address: [email protected] Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway E-mail address: [email protected] School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13509

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 operators affiliated with a finite von Neumann algebra R. Such an algebra contains both bounded and unbounded operators on a Hilbert space. In this article, we study reflexivity of Murray-von Neumann algebras. We discuss the stability of closed subspaces of a Hilbert space H under closed, densely defined operators on H,  of a set S of closed, densely defined operators based on which we define LatS  of a set P of closed subspaces of H. We show that Murray-von on H, and AlgP  LatA  f (R). We Neumann algebras Af (R) are reflexive, that is, Af (R) = Alg  also define Ref a Af (R), and show that Murray-von Neumann algebras Af (R)  a Af (R). are algebraically reflexive, that is, Af (R) = Ref

1. Introduction and Preliminaries One of the productive avenues through the study of Hilbert spaces and the linear transformations (“operators”) acting on them is indicated by the vague (but easily understood) question: Which operators can put which vectors in which places in the Hilbert space? A sample of such a question might be: What is the common null space of a certain given set of operators or the common eigenvectors for that set? To amplify on the nature of such questions, we may be asking the question about a single operator or families (perhaps, algebras) of operators, about a single vector or families (perhaps, subspaces) of vectors, and a variety of places. Another sample of such a question might be: What are the stable (invariant) subspaces under the operators in a given von Neumann algebra? (The answer would be: The ranges of projections in the commutant of that von Neumann algebra.) The answers, in general, will often be given in terms of conditions on the operators. In the illustration just given, the condition is that the set of operators form a von Neumann algebra. The “variety of places” may be described in terms of conditions on the vectors. For example: Which non-zero vectors does a given operator T place in 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 may be interested in finding the vectors they place at 0, that is, in finding the common null space. Of course, this is the main step in solving linear ordinary and partial differential equations. Such “action-location” questions are ubiquitous throughout analysis, and by contact and extension, throughout geometry and algebra. Reflexivity is a topic that has become an important theme in the study of operator algebra in recent decades. (See [H][Ha][L][RR] for general references.) c 2016 American Mathematical Society

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It also illustrates some aspect of action-location questions where operators and placement of vectors in subspaces are concerned. A main thrust in this article is the development of the basics of reflexivity in the framework of unbounded operators. As we shall see, the dominance of domain considerations adds a significant difficulty to the study of reflexivity when unbounded operators are involved. Suppose H is a separable Hilbert space and B(H) denotes the algebra of all bounded linear operators on H. Let P be a set of (orthogonal) projections in B(H). Define AlgP = {T ∈ B(H) : T E = ET E, 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 of B(H). Define LatS = {Projections E : T E = ET E, 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 of B(H) is said to be reflexive if A = AlgLatA. We know that every von Neumann algebra containing the identity is reflexive. Similarly, a lattice L of projections in B(H) is said to be reflexive if L = LatAlgL. For a subset S of B(H), let Sx = {Ax : A ∈ S}. Define Refa S = {T ∈ B(H) : T x ∈ Sx for each x ∈ H}. So, T ∈ Refa S if and only if for each x ∈ H, there is an Ax ∈ S, depending on x, such that T x = Ax x. A subspace A of B(H) is said to be algebraically reflexive if A = Refa A. In this article, we study reflexivity and algebraic reflexivity of Murray-von Neumann algebras. A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. Definition 1. We say that a closed densely defined operator T on a Hilbert space H is affiliated with a von Neumann algebra R when U  T = T U  for each unitary operator U  in R , the commutant of R. (We use D(T ) to denote the domain of T . Note that the equality U  T = T U  means that D(U  T )(= D(T )) = D(T U  ) and U  T x = T U  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 of operators affiliated with a factor of type II1 (or, more generally, affiliated with a finite von Neumann algebra, those in which the identity operator is finite) admits surprising operations of addition and multiplication that suit the formal algebraic manipulations used by the founders of quantum mechanics in their mathematical model. This is the case because of very special domain properties that are valid for finite families of operators affiliated with a finite von Neumann algebra. (Unbounded operators, even those that are closed and densely defined, can often neither be 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, and let 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]). More precisely, if A and B are in Af (R), then A + B and AB are preclosed (closable) and

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 B and A · B, respectively, are in Af (R). We refer their closures, denoted by A + to Af (R) as the Murray-von Neumann algebra (associated with R). In [Z2], basic structures of Af (R) (the center of Af (R), maximal abelian self-adjoint subalgebras of 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 “Ref a ,” respectively. Due to subtleties involving domains of the unbounded operators, we cannot adopt the standard definitions of the bounded case for these sets. These domain difficulties have been dealt with by Richard V. Kadison and Simon A. Levin in an article [KL] in which they introduce a concept they call “proper stability.” Using this concept and their “Operator-Projection Commutativity Principle” (appearing as our Theorem 19), we are able to press on and prove the basic facts about reflexivity in the unbounded-operator setting. The author wishes to thank Professors Kadison and Levin for the advance view of their work and for their permission to include their Operator-Projection Commutativity Principle. Our thanks are also due to our colleague Professor Deguang Han for valuable conversations while this research was in progress. For a set P of projections in B(H), define , = {Closed, densely defined T : T leaves E(H) properly stable for every E ∈ P}. AlgP And for a set S of closed, densely defined operators on H, define , = {Projections E : E(H) is properly stable under every T ∈ S}. LatS 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 , a Af (R) = {Closed, densely defined T : T x ∈ Af (R)x for every x ∈ D(T ), Ref and U  (D(T )) ⊆ D(T ) for every unitary U  ∈ R }, , a Af (R) if and only if for every x ∈ D(T ), there is an Ax ∈ Af (R) So, T ∈ Ref such that x ∈ D(Ax ), T x = Ax x and U  (D(T )) ⊆ D(T ) for every unitary U  ∈ R . , a Af (R), we cannot use “ T x ∈ Af (R)x for every Note that, for the definition of Ref x ∈ H ” since T is, in general, not everywhere-defined and x may not be in the domain of every element in Af (R). We shall prove that the Murray-von Neumann algebra Af (R) is reflexive (The, LatA , f (R), and that it is algebraically reflexive orem 23), that is, Af (R) = Alg , a Af (R). (Theorem 24), that is, Af (R) = Ref 2. Commutativity and Stability Suppose that E is the projection from the Hilbert space H onto a closed subspace Y . With T in B(H), clearly, T (Y ) ⊆ Y if and only if T Ex = ET Ex for each x 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) = (T E − ET E)∗ . It follows that Y is invariant under T if and only if Y ⊥ (the orthogonal complement of Y ) is invariant under T ∗ . Moreover, we have that T and E commute if and only if Y is invariant under both T and T ∗ (that is, Y and Y ⊥ are both invariant under T ). To see this, if Y is invariant under both T and T ∗ , then ET = (T ∗ E)∗ =

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(ET ∗ E)∗ = ET E = T E. Conversely, if T E = ET , then ET ∗ = T ∗ E, so that ET E = ET = T E and ET ∗ E = ET ∗ = T ∗ E. We ask the question of whether or not the above result (Operator-Projection Commutativity Principle for bounded operators) holds when replacing a bounded operator T with a closed, densely defined operator C on the Hilbert space H. Since such 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 bounded operator) on H” needs to be carefully defined. Definition 2. If A and C are closed, densely defined operators on a Hilbert space 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 simply the equality of AB and BA. In the case of operators on a Hilbert space (or, more generally, 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 are unbounded, even closed, this simple equality no longer serves as the expression of an adequate concept of commutativity. A hint of what can cause difficulties can be 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 on D(A), the domain of A. Of course, we want to think of 0 as commuting with each closed operator. We do have that 0A ⊆ A0. For a closed operator A and a 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 space H and B is a bounded, everywhere-defined operator on H, we say that A and B commute when BA ⊆ AB. We state some basic facts about unbounded operators (from a Hilbert space H into H) here without proofs. We use [KR] (in particular, Section 2.7) as our basic reference for results in the theory of unbounded operators. Remark 4. If T is closed and B is bounded, then T B is closed (while BT , in general, 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 ∗ ⊆ T0∗ . 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-adjoint subset F of B(H), then T C ⊆ CT for each T in the von Neumann algebra generated by F. ([KRI]; Lemma 5.6.13)

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Proposition 11. If A and C are densely defined preclosed operators and B is a bounded operator such that A = BC, then (BC)∗ = C ∗ B ∗ . ([KRII]; Lemma 6.1.10) Remark 12. Commutativity of a closed, densely defined operator C and a projection E, now means that EC ⊆ CE. From this commutativity, CE must be 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 same time, E(D(C)) is dense in E(H), as D(C) is dense in H and E is continuous on H. Also, if EC ⊆ CE, then C maps D(C) ∩ E(H) into E(H). To see this, if x ∈ 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 (compare with the “bounded” case: for T in B(H), T (E(H)) ⊆ E(H) if and only if T E = ET E and that T E = ET if and only if E(H) is invariant under both T and T ∗ ). In Theorem 19, we demonstrate that the “operator-projection commutativity principle” holds not only when the operator is bounded, but also when it is closed and 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 that E(H) is properly stable under C if and only if E(H)⊥ , the orthogonal complement of 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 results established in this paper and elsewhere, we prove our reflexivity theorems in Section 3. Definition 13. The range E(H) of a projection E on a Hilbert space H is said to be properly stable under a closed, densely defined operator C on H when C maps 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 same time, 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 Proposition 11, (BC)∗ = C ∗ B (note that BC is densely defined since D(BC) = D(C) and that BC is preclosed since BC ⊆ CB and CB is closed). From Proposition 9, and the 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 on a 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 ECE is 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 this holds for each x in D(CE), CE ⊆ ECE. Therefore, CE = ECE.

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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 on a 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 ∗ are densely 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 ∗ E are 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)∗ (Proposition 9) . 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. Thus

CE = 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 on a Hilbert space H such that C and E commute (that is, EC ⊆ CE), then CE = EC and C ∗ E = EC ∗ . Proof. Since C and E commute, from Remark 12, C maps D(C) ∩ E(H) into E(H) and CE is densely defined. From Proposition 15, C ∗ and E commute; hence C ∗ maps D(C ∗ ) ∩ E(H) into E(H) and C ∗ E is densely defined. Thus Lemma 17 applies 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, we obtain C ∗ E = EC ∗ .  Theorem 19. (Operator-Projection Commutativity Principle for unbounded operators) A closed, densely defined operator C and a projection E commute (that is, 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 under C. From Proposition 15, EC ⊆ CE implies that EC ∗ ⊆ C ∗ E. Again, the proper stability 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), ∗

∗

C (D(C ) ∩ E(H)) ⊆ E(H),

E(D(C)) ⊆ D(C), E(D(C ∗ )) ⊆ D(C ∗ ).

Since E(D(C)) ⊆ D(C), D(C) ⊆ D(CE); hence CE and, similarly, C ∗ E are densely 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.

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Our present goal is to show that EC ⊆ CE. If x ∈ D(EC), then x ∈ D(C) and Ex ∈ 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) and Ex ∈ D(C)). Of course, (I − E)x ∈ D(CE) since E(I − E)x = 0 ∈ D(C). From Lemma 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 H is properly stable under a closed, densely defined operator C on H if and only if E(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 in E(H). From Remark 14, under the assumption that E(D(C)) ⊆ D(C), we have E(D(C)) = D(C) ∩ E(H). Our goal is to show that C ∗ maps D(C ∗ ) ∩ E(H)⊥ into E(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 unique extension from D(C) ∩ E(H) to E(H) (since E(D(C)) = D(C) ∩ E(H) is dense in E(H)). It follows that the mapping y → C ∗ x, y is 0 on E(H) and hence C ∗ x is in 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) to H and Riesz’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, consider those 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, the mapping w → Cw, Ex is bounded and therefore the mapping w → Cw, (I −E)x is bounded on w ∈ D(C) ∩ E(H)⊥ .

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If E(H)⊥ is properly stable under C ∗ , applying what we have just proved, we obtain that (E(H)⊥ )⊥ is properly stable under C ∗∗ . Since (E(H)⊥ )⊥ = E(H) and C ∗∗ = 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, and let T be a closed, densely defined operator on H. If U T ⊆ T U for every unitary operator U in R, then U T = T U . Proof. Since U T ⊆ T U , multiplying both sides on the right and on the left by U ∗ , we obtain T U ∗ ⊆ U ∗ T . Since this holds for the adjoint of every unitary in R, it still holds if U ∗ is replaced by U , that is, T U ⊆ U T for every unitary U in R. It follows, from the assumption U T ⊆ T U , that U T = T U . 

Proposition 22. If R is a finite von Neumann algebra acting on a Hilbert space 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 if Eλ Fλ = 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 , LatA , f (R). Af (R) is reflexive. That is, Af (R) = Alg Proof. By definition, , f (R) = {Projections E : E(H) is properly stable under every T in Af (R)}. LatA Since Af (R) is a self-adjoint algebra (that is, when it contains T , it contains T ∗ ), from Theorem 19, , f (R) = {Projections E : ET ⊆ T E for every T in Af (R)}. LatA From Lemma 18, T E = ET if T and E commute. Recall that, by definition, E · T = ET in Murray-von Neumann algebras. It follows that E · T = ET = T E = T E = T · E (note that T E is closed) and , f (R) = {Projections E : E · T = T · E for every T in Af (R)} LatA = {Projections E : E ∈ R } (Proposition 22)

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Then , LatA , f (R) Alg = {Closed, densely defined T : T leaves E(H) properly stable , f (R)} for every E ∈ LatA = {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 ⊆ T E for every projection E ∈ R } (Theorem 19) = {Closed, densely defined T : U T ⊆ T U for every unitary U ∈ R } (Proposition 10) = {Closed, densely defined T : U T = T U for every unitary U ∈ R } (Lemma 21) = Af (R).



Theorem 24. Let R be a finite von Neumann algebra. The Murray-von Neu, a Af (R). mann algebra Af (R) is algebraically reflexive, that is, Af (R) = Ref , a Af (R). To show that Ref , a Af (R) ⊆ Af (R), Proof. Clearly, Af (R) ⊆ Ref , , , LatA , , f (R). Supsince Af (R) = AlgLatAf (R), we shall show that Ref a Af (R) ⊆ Alg , a Af (R). From the proof of the preceding theorem, to show that T pose T is in Ref , , is in AlgLatAf (R), it suffices to show that for every projection E in R , E(H) is properly 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 T E = ET E and E(D(T )) ⊆ D(T ). , a Af (R), U  (D(T )) ⊆ D(T ) for every unitary U  ∈ R and hence Since T ∈ Ref E(D(T )) ⊆ D(T ) for every projection E ∈ R . For every x ∈ D(T E)(= D(ET E)), Ex ∈ D(T ), and T Ex = AEx Ex = EAEx Ex = ET Ex, , LatA , f (R). for some AEx ∈ Af (R). Therefore, T ∈ Alg  References 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 Operators 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

[H]

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[Z2]

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H. Radjavi and P. Rosenthal, On invariant subspaces and reflexive algebras, Amer. J. Math. 91 (1969), 683–692. MR0251569 (40 #4796) Zhe Liu, On some mathematical aspects of the Heisenberg relation, Sci. China Math. 54 (2011), no. 11, 2427–2452, DOI 10.1007/s11425-011-4266-x. MR2859703 (2012m:46066) 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 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13510

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 Hochschild cohomology for von Neumann algebras, concentrating on the most recent theorems concerning tensor products. We describe methods rather than giving proofs, but we include detailed references to the literature on this topic.

1. Introduction The algebraic theory of cohomology was initiated by Hochschild [19–21], based on multilinear maps into a bimodule and coboundary operators. Subsequently Johnson, Kadison and Ringrose [22, 23, 27, 28] developed an appropriate theory for Banach algebras and operator algebras by requiring the relevant cocycles and coboundaries to be bounded. The cohomology group H n (A, V ) for an operator algebra A and an A-bimodule V is the quotient of the space of bounded n-cocycles by the subspace of bounded n-coboudaries. There are many choices for V , but we will focus almost exclusively on the case where A is a von Neumann algebra M and V 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 by the same pair of authors [50]. For this reason we will give only a brief exposition of the main results up to the publication of [50] and concentrate on more recent theorems on tensor products. The starting point for cohomology in the functional analytic setting is the theorem of Kadison and Sakai [26, 46] which states that every derivation δ : M → M is inner. In cohomological terms this means that H 1 (M, M ) = 0. A similar question for the bimodule B(H) is still open, and is known from [30] to be equivalent to Kadison’s similarity problem [25]. It should also be noted that vanishing of first cohomology 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 groups H n (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, complete 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 Antonio A.M.S. meeting, January 2015. RRS was partially supported by NSF grant DMS-1101403. c 2016 American Mathematical Society

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all cases where these groups have been determined. We note, however, that beyond the self-adjoint setting cohomology groups of arbitrarily specified dimension are possible [17], although we will not pursue this direction. In Section 2 we give the definition of cocycles and coboundaries and also of H n (A, V ). We also give a brief review of the theory of completely bounded linear and multilinear maps on operator algebras that has become indispensible in the theory of cohomology. There are several good detailed sources for complete boundedness, see [15, 34, 38]. Since two cocycles that differ by a coboundary define the same element of H n (M, M ), when studying a particular cocycle we have the freedom to perturb it by a suitably chosen coboundary in order to improve its properties. This is the topic of Section 3 and Theorem 3.1 summarizes what can be achieved in this direction. The subsequent section is devoted to completely bounded cohomology. Theorem 4.1 is the crucial one for studying this type of cohomology and is derived from structural results for completely bounded multilinear maps using the techniques of [46]. It leads to vanishing of the completely bounded cohomology groups in Theorem 4.2 and the vanishing of H n (M, M ) for various classes of von Neumann algebras in Theorem 4.3. The final section is devoted to a discussion of the results of [41] on the vanishing of H 2 (M ⊗ N, M ⊗ N ) for tensor products of type II1 factors, where once again complete 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 H and also V = A. For n ≥ 1, we denote by Ln (A, V ) the space of n-linear bounded maps φ : An → V where An is the n-fold Cartesian product A×A×· · ·×A of copies of 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 ≥ 1 and φ ∈ Ln (A, V ) we let ∂φ(x1 , . . . , xn+1 ) = x1 φ(x2 , . . . , xn ) +

n−1 

(−1)i φ(x1 , . . . , xi−1 , xi xi+1 , xi+2 , . . . , xn )

i=1

(2.1)

+ (−1)n φ(x1 , . . . , xn )xn+1 ,

for xi ∈ A, 1 ≤ i ≤ n + 1. A routine calculation shows that ∂∂ = 0. The kernel of ∂ : Ln (A, V ) → Ln+1 (A, V ) is the space of cocycles, and contains the space of coboundaries which is the image of ∂ : Ln−1 (A, V ) → Ln (A, V ). The quotient of these spaces is the nth Hochschild cohomology group H n (A, V ). For the case n = 1, the cocycles are the derivations and the coboundaries are the inner derivations, so all derivations are inner precisely when H 1 (A, V ) = 0. Having given the definition for a general module V , we will eventually concentrate on the situation where A is a von Neumann algebra M and V = M . An important role in the theory is played by the completely bounded multilinear maps. If φ : A → B(H) is a bounded map, then φ lifts to a bounded map φn : Mn (A) → Mn (B(H)) by (aij ) → (φ(aij )). Then φ is completely bounded if supn≥1 φn  < ∞ and this supremum defines φcb . There is an extensive theory

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of such maps (see [8, 9, 15, 34, 38]) and there is a very useful representation result: if φ : A → B(H) is completely bounded, then there exists a representation π : A → B(K) and maps V : K → H, W : H → K so that (2.2)

a ∈ A.

φ(a) = V π(a)W,

Moreover, V and W can be chosen to satisfy the optimal estimate V  = W  = 1/2 φcb . 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 give the definition for bilinear maps and the extension to n-linear ones should be clear since we are mimicing matrix multiplication. (A) × Mn (B) → If φ : A × B → B(H), then φ lifts to a bilinear map φn : Mn 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 supremum defining φcb . The representation theorem in this case is as follows. For simplicity we state it for two variables but the extension to n variables only involves 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) = V1 π1 (x)V2 π2 (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)

1/3

V1  = V2  = V3  = φcb .

Since it is easy to check that any map with such a representation is completely bounded, it becomes a routine exercise to see that ∂ preserves complete boundedness. Thus there is a parallel theory of completely bounded cohomology groups n Hcb (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 by the requirements (2.6)

aφ(b1 , b2 ) = φ(ab1 , b2 ),

(2.7)

φ(b1 a, b2 ) = φ(b1 , ab2 ),

(2.8)

φ(b1 , b2 a) = φ(b1 , b2 )a,

for a ∈ A and b1 , b2 ∈ B, with an obvious extension to A-multimodularity of maps φ : B n → B(H). As will be seen below, reduction to cocycles that are multimodular with respect to a suitably chosen subalgebra is a crucial part of the theory. 3. Reduction of cocycles If we have a bounded cocycle φ : M n → M and a hyperfinite subalgebra R ⊆ M , then by averaging repeatedly over an amenable group of unitaries in R that generates a weakly dense C∗ -algebra A, we find that φ is equivalent to a cocycle

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ψ : M n → M which vanishes whenever one of its arguments is in A. We illustrate this 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 abuse notation and write the action of μ as an integral. For u, v ∈ G, ! ! ! ∗ ∗ u δ(uv) dμ(u) = u uδ(v) dμ(u) + u∗ δ(u)v dμ(u) (3.1) G

G

so

!

u∗ δ(uv) dμ(u) −

G

!

u∗ δ(u)v dμ(u) ! ! ∗ = v(uv) δ(uv) dμ(u) − u∗ δ(u)v dμ(u).

δ(v) =

G

(3.2)

G

G

Let t ∈ M be (3.3)

 G

G

u∗ δ(u) dμ(u). The invariance of μ gives 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 Rmultimodular. One consequence is that H n (R, R) = 0 for all hyperfinite von Neumann algebras. While it is unknown whether cocycles φ : M n → M are coboundaries into M , there is a result on extended cobounding [28] which expresses a cocycle φ as ∂ψ where ψ maps M n−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 (m1 b1 , . . . , mn bn ) = 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 that W ∗ (M, B) is hyperfinite so its cohomology is trivial. Consequently φ1 = ∂ψ1 for some 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 that M ⊆ 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 a subsequent averaging argument allows it to be separately normal. We summarize these reductions, proved in [23, 28] as Theorem 3.1. Let φ : M n → M be a bounded n-cocycle. Let R ⊆ M be a hyperfinite von Neumann subalgebra. Then φ is equivalent to an n-cocycle ψ : M n → M that is separately normal in each variable, vanishes whenever any of the arguments lies in R, and is R-multimodular. If N is a hyperfinite von Neumann algebra containing M , then there exists a separately normal R-multimodular map ψ : M n−1 → N so that φ = ∂ψ.

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4. Completely bounded cohomology n In this section, we restrict attention to the cohomology groups Hcb (M, M ) where the cocycles and coboundaries are assumed to be completely bounded. This is the part of the theory that is fully worked out as we will see below. To motivate the next result, consider a completely bounded map φ : M → M of the special form φ(x) = axb, x ∈ M , where a, b ∈ M are fixed. By the Dixmier approximation theorem [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 so that

(4.1)

z −

kn 

λi,n ui,n bu∗i,n 
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 an integer n, elements x1 , . . . , xm ∈ M and ε > 0, there exist orthogonal projections p1 , . . . pn ∈ M summing to 1 with τ (pi ) = n1 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 equivalent to ones that are completely bounded, establishing that H n (M, M ) = 0 for n ≥ 1. Subsequently property Γ was extended to the non-factor case in [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 proof of the complete boundedness of derivations δ : M → M . This will be deduced from the following proposition concerning column boundedness. A similar inequality was proved by Christensen [3], but with constant 14 instead of 4.

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Proposition 4.4. If M ⊆ B(H) is a von Neumann algebra and δ : M → B(H) is a derivation, then ||

(4.10)

n 

δ(ai )∗ δ(ai )|| ≤ 4 ||δ||2 · ||

i=1

n 

a∗i ai ||, ai ∈ M, 1 ≤ i ≤ n.

i=1

Proof. Fix an element a ∈ M with polar decomposition a = vh, where v is a partial isometry and h = (a∗ a)1/2 ≥ 0. For every unitary u ∈ M we have δ(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||2 h2 .

(4.12)

Fix ξ ∈ H, ||ξ|| ≤ 1. Then [37, Theorem 7.3] ensures the existence of an ultrafilter ω on N, a sequence of unitary operators un ∈ M, and a state f on M such that lim||δ(un a)ξ||2 ≤ ||δ||2 f (a2 ) for all self-adjoint elements a ∈ M. We obtain ω

  δ(a)∗ δ(a)ξ, ξ ≤ 2 ||δ(un h)ξ||2 + 2 ||δ||2 h2 ξ, ξ ,

(4.13)

and by taking the limit along ω we get

  δ(a)∗ δ(a)ξ, ξ ≤ 2 ||δ||2 f (h2 ) + 2 ||δ||2 h2 ξ, ξ .

(4.14)

For a1 , . . . , an ∈ M we have / / . n . n n    ∗ 2 ∗ 2 ∗ δ(ai ) δ(ai )ξ, ξ ≤ 2 ||δ|| f ( ai ai ) + 2 ||δ|| ai ai ξ, ξ i=1

i=1

≤ 4 ||δ||2 ||

(4.15)

n 

i=1

a∗i ai ||

i=1

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 a derivation, 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 columnnorms 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 (4.16)

≤ 4 ||δ|| · ||X|| · ||C|| + ε ≤ 4 ||δ|| · ||X|| + ε.

Since ε > 0 was arbitrary, the conclusion follows.



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5. Tensor products In this section we describe some recent work [41] on the second cohomology of tensor products M ⊗ N . We restrict to the case where both algebras are II1 factors with separable preduals for simplicity, but the results are true for tensor products of general II1 von Neumann algebras. From [42], this allows us to specify subalgebras A ⊆ R ⊆ M and B ⊆ S ⊆ N such that A and B are masas in M and N respectively while R and S are hyperfinite subfactors with trivial relative commutants in their containing factors. These subalgebras play an important role in proving the main result of this section: Theorem 5.1. ([41]) Let M and N be II1 factors with separable preduals. Then H 2 (M ⊗ N, M ⊗ N ) = 0.

(5.1)

We will sketch the proof of this result, setting out the required lemmas. As always in cohomology, complete boundedness is an important component. Lemma 5.2 ([10]). Let M ⊆ B(H) and S ⊆ B(K) be II1 factors with S hyperfinite, 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)-modularity implies that ψ maps into S  so φ maps M ⊗min S into C∗ (S  , S). From [14] there is an isomorphism ρ : S  ⊗min S → C ∗ (S  , S) given by ρ(s s) = s ⊗s. Then φM ⊗ S min can be realized as ρ◦(ψ ⊗idS ), and complete boundedness follows from normality of φ, 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 bilinear maps. Lemma 5.3. Let φ : (M ⊗ S) × (M ⊗ S) → M ⊗ S be a bounded separately normal (R ⊗ S)-multimodular bilinear map. Then φ is completely bounded. This result depends on an inequality [47, 5.4.5 (ii)] stemming from Grothendieck’s inequality [18, 36],   n  1/2  n 1/2 n           ∗ ∗  (5.2) φ(xi , yi ) ≤ 2φ  xi xi   yi yi  ,        i=1

i=1

i=1

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 been obtained. 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 inclusion of finite von Neumann algebras, then each derivation δ : M → N is inner and is implemented by an element of N . In considering a 2-cocycle φ : (M ⊗ N ) × (M ⊗ N ) → M ⊗ N , adding coboundaries allows us to reduce to the situation where the following four conditions are satisfied:

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(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 φ maps (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 φM ⊗ S = ∂α, and  similarly a normal (R ⊗ S)-modular map β : R ⊗ N → R ⊗ N so that φR ⊗ N = ∂β. These maps extend to α, ˜ β˜ : M ⊗ N → M ⊗ N by (5.3)

α ˜ = α ◦ EM ⊗ S ,

β˜ = β ◦ ER ⊗ N

˜ satisfies (C1)-(C4). and one checks that ψ = φ − ∂(α ˜ + β) 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  is the von Neumann algebra generated by Q and eP . The map x → x∗ on Q extends to a conjugate linear isometry J : L2 (Q) → L2 (Q) and Q, eP  = JP J. (See [24] or [51, Chapter 4] for details of the basic construction). If P is hyperfinite then so is Q, eP , while if P is a masa in Q then JP J 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)J is 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 EM ⊗ N,eR ⊗ S  to define γ : M ⊗ N → M ⊗ N, eR ⊗ S  by γ = EM ⊗ 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 (5.4)

f (m ⊗ n) = φ(m ⊗ I, I ⊗ n) + γ(m ⊗ n),

(5.5)

g(m ⊗ n) = φ(I ⊗ n, m ⊗ I) + γ(m ⊗ n), h(m ⊗ n) = g(m ⊗ n) − f (m ⊗ n)

(5.6)

= φ(I ⊗ n, m ⊗ I) − φ(m ⊗ I, I ⊗ n).

The objective is to show that these maps are bounded. This is perhaps surprising 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 helpful to look at a particular example of such a φ. Let α : M → M be a completely bounded normal R-modular map satisfying α(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) (5.7)

+ (α(m) ⊗ β(I))(I ⊗ n) = −ξ(m ⊗ n),

and we see boundedness of m ⊗ n → φ(m ⊗ I, I ⊗ n) from the boundedness of ξ.

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The next lemma lists some properties of these functions f , g, and h. The map γ 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 respective restrictions of γ to these subalgebras, and are all bounded derivations spatially implemented by elements of M ⊗N, eR⊗S . (iii) The restrictions h|M ⊗I and h|I⊗N are both 0.   Sketch. Since φM ⊗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  by applying the conditional expectation onto this algebra. The same argument works  for 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 . We include the proof from [41]. Proposition 5.8. There exists a bounded normal map ξ : M ⊗N → M ⊗N such 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 completely bounded derivation implemented by an element t ∈ M, eR ⊗N, eS . Define a derivation 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) there is 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 . This algebra is (M  ∩ M, eR )⊗N, eS , equal to (JM J ∩ (JRJ) )⊗N, eS , and in turn equal to I⊗N, eS . The latter algebra is hyperfinite, so if we take a conditional

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expectation onto it and apply this to (5.10), then we conclude that the element b of (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] (5.12)

= b(m ⊗ n) − (m ⊗ n)b,

m ∈ M, n ∈ N.

Thus δ has a unique bounded normal extension to M ⊗N , and (5.9) shows that the same is then true for f . Since ξ = f − γ on M ⊗ N from (5.4), and γ is already bounded and normal on M ⊗N , this gives a bounded normal extension of  ξ to M ⊗N . The proof of this proposition and equation (5.4) show that f has a bounded normal 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 Theorem 5.1. If Fn is the free group on n generators, n ≥ 2, then the II1 factors L(Fn ) do not have a Cartan subalgebra [52], do not have property Γ [33], and are prime so are not tensor products of II1 factors [16]. Thus none of the theorems in this paper apply to L(Fn ) and so nothing is known of the cohomology beyond the first cohomology group [26, 46] in this case. Any progress on the higher cohomology of free group factors would be very interesting. References [1] J. M. Cameron, Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras, 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, DOI 10.1016/0022-1236(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 groups of certain finite von Neumann algebras, Math. Ann. 307 (1997), no. 1, 71–92, DOI 10.1007/s002080050023. MR1427676 (98c:46130) [6] E. Christensen, F. Pop, A. M. Sinclair, and R. R. Smith, Hochschild cohomology of factors with property Γ, Ann. of Math. (2) 158 (2003), no. 2, 635–659, DOI 10.4007/annals.2003.158.635. MR2018931 (2004h:46072) [7] E. Christensen, F. Pop, A. M. Sinclair, and R. R. Smith, Property Γ factors and the Hochschild 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 operators, 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. London Math. 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 injectivity, 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)

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[13] J. Dixmier, Quelques propri´ et´ es 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 suspensions, 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, DOI 10.1007/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. Operator Theory 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 Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 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 on Grothendieck’s constants, J. Funct. Anal. 29 (1978), no. 3, 397–415, DOI 10.1016/00221236(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)

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[38] G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, 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, DOI 10.1016/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 Property Γ. 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, London Mathematical 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 with Cartan 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 Neumann algebras, Proc. Natl. Acad. Sci. USA 95 (1998), no. 7, 3376–3379 (electronic), DOI 10.1073/pnas.95.7.3376. MR1622277 (99b:46110) [50] A. M. Sinclair and R. R. Smith, A survey of Hochschild cohomology for von Neumann algebras, Operator algebras, quantization, and noncommutative geometry, Contemp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 383–400, DOI 10.1090/conm/365/06712. MR2106829 (2005h:46089) [51] A. M. Sinclair and R. R. Smith, Finite von Neumann algebras and masas, London Mathematical 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 probability 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 Island, New York 10301 E-mail address: [email protected] Department of Mathematics, Texas A&M University, College Station, Texas 77843 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13511

On the optimal paving over MASAs in 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, there exists a partition of 1 √ with projections p1 , p2 , . . . , pn ∈ Aω (i.e. a paving) such p xp  ≤ 2 n − 1/n, and give examples where this is sharp. Some that Σn i i i=1 open problems on optimal pavings are discussed.

1. Introduction A famous problem formulated by R.V. Kadison and I.M. Singer in 1959 asked if the diagonal MASA (maximal abelian ∗ -subalgebra) D of the algebra B(2 N), of all linear bounded operators on the Hilbert space 2 N, satisfies the paving property, requiring that for any contraction x = x∗ ∈ B(2 N) with 0 on the diagonal, and any  ε > 0, there exists a partition of 1 with projections p1 , . . . , pn ∈ D, such that  i pi xpi  ≤ ε. This problem has been settled in the affirmative by A. Marcus, D. Spielman and N. Srivastava in [MSS13], with an actual estimate n ≤ 124 ε−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 requires that for any x = x∗ ∈ M and any ε > 0, there exist n ≥ 1, a net of partitions of 1 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 (Σnk=1 pk,i xpk,i − ai )qi  ≤ ε, ∀i, and qi → 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(2 N). We conjectured in [PV14] that any MASA A ⊂ M satisfies so-paving. We used the results in [MSS13] to check this conjecture for all MASAs in type I von Neumann algebras, and all Cartan MASAs in amenable von Neumann algebras and in group measure space factors arising from profinite actions, with the estimate 124 ε−4 for the so-paving size derived from [MSS13] as well. We also showed in [PV14] that if A is the range of a normal conditional expectation, 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. c 2016 American Mathematical Society

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equivalent to the usual Kadison-Singer paving for the ultrapower MASA Aω ⊂ M ω , with the norm paving size for Aω ⊂ M ω coinciding with the so-paving size for A ⊂ M . In the case A is a singular MASA in a II1 factor M , norm-paving for the ultrapower inclusion Aω ⊂ M ω has been established in [P13], with paving size 1250ε−3 . This estimate was improved to < 16ε−2 + 1 in [PV14], while also shown to be ≥ ε−2 for arbitrary MASAs in II1 factors. In this paper we prove that the paving size for singular MASAs in II1 factors is in fact < 4ε−2 + 1, and that for certain singular MASAs this is sharp. More precisely, we prove that for any contraction x ∈ M ω with 0 expectation onto Aω , ω and for any n ≥ 2, there √ exists a partition of 1 with n projections pi ∈ A such n that Σi=1 pi xpi  ≤ 2 n − 1/n. In fact, given any finite set of contractions F ⊂ M ω % Aω , we can find a partition p1 , . . . , pn ∈ Aω that satisfies this estimate for all 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 that any word with alternating letters from {uk | 1 ≤ k ≤ n − 1} and F ∪ F ∗ has trace 0. This implies that for each x ∈ F the set X = {ui−1 xu−i+1 | i = 1, 2, . . . , n} ∗ m ∗ satisfies the conditions τ (Πm k=1 (x2k−1 x2k )) = 0 = τ (Πk=1 (x2k−1 x2k )), for all m and all xk ∈ X with xk = xk+1 for all k. We call L-freeness this property of a subset of a II1 factor. We then prove the general result, of independent interest, that n any √ L-free set of contractions {x1 , . . . , xn } satisfies the norm estimate Σi=1 xi  ≤ 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 Σni=1 Ui  = 2 n − 1 from results in [AO74]. This implies the inequality for the L-free contractions as well. By applying this to {ui−1 xu1−i | i = 1, . . . , n} and taking into account that n1 Σni=1 ui−1 xu1−i = Σni=1 pi xpi , where p1 , . . . , pn are the √ minimal spectral projections of u, we get Σni=1 pi xpi  ≤ 2 n − 1/n, ∀x ∈ F . We also notice that if M is a II1 factor, A ⊂ M is a MASA and v ∈ M a self-adjoint unitary of trace 0 which is free with respect to A, then Σni=1 pi vpi  ≥ √ 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 denotes the canonical generator. This shows that the estimate 4ε−2 + 1 for the paving size is 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 Ramanujan graphs. Its occurence in this context leads us to a more refined version of a conjecture formulated in [PV14], predicting that for any MASA A ⊂ M which is range of a normal conditional expectation, any n ≥ 2 and any contraction x = x∗ ∈ M with 0 expectation onto A, the infimum ε(A ⊂ M ; n, x) over all norms of pavings of x,√Σni=1 pi xpi , with n projections p1 , . . . , pn in Aω , Σi pi = 1, is bounded above ∗ by √ 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(2 Z). 2. Preliminaries A well known result of H. Kesten in [K58] shows that if Fk denotes the free group with k generators h1 , . . . , hk , and λ is the left regular representation of Fk

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−1 k on 2 Fk , then √ the norm of the Laplacian operator L = Σi=1 (λ(hi ) + λ(hi )) is if k elements equal to 2 2k − 1. It was also shown in [K58] that, conversely, √ −1 k h1 , . . . , hk in a group Γ satisfy Σi=1 λ(hi ) + λ(hi ) = 2 2k − 1, then h1 , . . . , hk are freely independent, generating a copy of Fk inside Γ. The calculation of the norm of L in [K58] uses the formalism of random walks on groups, but it really amounts 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 on the group von Neumann algebra L(Fk ). Kesten’s result implies that whenever u1 , . . . , uk are freely independent Haar (i.e., u1 , . . . , uk generate a copy of L(Fk ) inside M ), unitaries in a type II1 factor M √ then one has Σki=1 ui + u∗i  = 2 2k − 1. In particular, if M is the free group factor k ∗ L(F √ k ) and ui = λ(hi ), where h1 , . . . , hk ∈ Fk as above, then Σi=1 αi ui + αi ui  = 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 obtained in [L73], [B74], [AO74]1 . These estimates involve elements in L(Γ) (viewed as convolvers on 2 Γ) that are supported on a subset {g1 , . . . , gn } ⊂ Γ satisfying the following weaker freeness condition, introduced in [L73]: whenever k ≥ 1 and is = js , js = is+1 for all s, we have that

· · · gik gj−1 = e . gi1 gj−1 1 k In [B74] and [AO74], this is called the Leinert property and it is proved to be equivalent with {g1−1 g2 , . . . , g1−1 gn } freely generating a copy of Fn−1 . The most general calculation of norms of elements x = Σi ci λ(gi ) ∈ L(Γ), supported on a Leinert set {gi }i , with arbitrary coefficients ci ∈ C, was obtained by Akemann and Ostrand in [AO74]. The calculation shows in particular √ that if {g1 , . . . , gn } satisfies Leinert’s freeness condition then Σni=1 λ(gi ) = 2 n − 1. Since h1 , . . . , hk ∈ Γ freely independent implies {hi , h−1 | 1 ≤ i ≤ k} is a Leinert set, the result in [AO74] i does recover Kesten’s theorem as well. Like in [K58], the norm of an element of the form L = Σni=1 ci λ(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=1 λ(gi ) = 2 n − 1, then g1 , . . . , gn is a Leinert set. On the other hand, note that if g1 , . . . , gn are n arbitrary elements in an arbitrary group Γ and we denote L = Σni=1 λ(gi ) the corresponding Laplacian, then the n’th moment τ ((L∗ L)n ) is bounded from below by the n’th moment of the Laplacian obtained √ by taking gi to be the generators of Fn . Thus, we always have Σni=1 λ(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 word vi1 vj∗1 vi2 vj∗2 . . . .vim vj∗m , ∀m ≥ 1, ∀1 ≤ ik , jk ≤ n, has trace with non-negative real √ n − 1. In particular, for any unitaries u1 , . . . , un ∈ M part, then Σni=1 vi  ≥ 2 √ one has Σni=1 ui ⊗ 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: 1 See also the more “rough” norm estimates for elements in L(F ) obtained by R. Powers in n 1967 in relation to another problem of Kadison, but published several years later in [Po75], and which motivated in part the work in [AO74].

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Proposition 2.1 ([AO74]). If v1 , v2 , . . . , vn−1 ∈ M are freely independent Haar unitaries, then √ (2.1) 1 + Σn−1 i=1 vi  = 2 n − 1. Also, if α0 , . . . , αn−1 ∈ C, Σi |αi |2 = 1, then

α0 1 + Σn−1 i=1 αi vi  ≤ 2 1 − 1/n.

(2.2)

Note that (2.1) above shows in particular that if p, q ∈ M are projections with τ (p) = 1/2 and τ (q) = √ 1/n, for some n ≥ 3, and they are freely independent, 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 the two 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 denote u = Σnj=1 λj−1 qj , where λ = 2 exp(2πi/n). It is then easy to see that the elements vk = vuk vu−k , k = 1, 2, . . . , n − 1 are freely independent Haar √ unitaries. n−1 k −k k −k u vu  = 1 + Σ vu vu  = 2 n − 1. But By (2.1) we thus have Σn−1 k=0 k=1 k −k n u vu = n(Σ q vq ), implying that Σn−1 j j=1 j k=0 √ 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 projections q, p of arbitrary trace in M (in fact, of the whole spectral distribution of qpq) was obtained by Voiculescu in [Vo86], as one of the first applications of his multiplicative free convolution (which later became a powerful tool in free probability). We recall here these norm estimates, which in particular show that the first of the above norm 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 with τ (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 von Neumann algebra M are called freely independent sets2 if the trace of any word with letters alternating from X and Y is equal to 0. Also, a subalgebra B ⊂ M is called 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” B inside a given subalgebra Q ⊂ M that is freely independent of a given countable set X. Motivated by a condition appearing in one such result, namely [P13, Theorem 4.1], and by a terminology used in [AO74], we consider in this paper the following free independence condition for arbitrary elements in tracial algebras: 2 We specifically consider this condition for subsets X, Y ⊂ M C1, not to be confused with the freeness of the von Neumann algebras generated by X and Y .

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Definition 3.1. Let (M, τ ) be a von Neumann algebra with a normal faithful tracial state. A subset X ⊂ M is called L-free3 if τ (x1 x∗2 · · · x2k−1 x∗2k ) = 0 and

τ (x∗1 x2 · · · x∗2k−1 x2k ) = 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 the set 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 to F being a Leinert set. But the key example of an L-free set that is important for us here occurs from a diffuse algebra B that is free independent from a set Y = 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−1 yk u−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∗1 x2 x∗3 x1 ) = 0 from τ (y1 y2∗ y3 y4∗ ) = 0 for all yi ∈ X with y1 = y2 , y2 = y3 , y3 = y4 . However, if X ⊂ U(M ) consists of unitaries, 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 are equivalent (a) X is an L-free set. (b) τ (ui1 u∗j1 · · · uik u∗jk ) = 0 whenever k ≥ 1 and is = js , js = is+1 for all s. (c) u∗1 u2 , . . . , u∗1 un 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 Σni=1 ui  = 2 n − 1. Moreover, if α1 , . . . , αn ∈ C with Σni=1 |αi |2 ≤ 1, then n     αi ui  ≤ 2 1 − 1/n.  i=1

Proof. Since = α1 1 + Σni=2 αi u∗1 ui , the statement follows by applying (2.2) to the freely independent Haar unitaries vj = u∗1 uj , 2 ≤ j ≤ n.  Σni=1 αi ui 

Proposition 3.4. Let M be a finite von Neumann algebra with a faithful tracial state τ . If {x1 , . . . , xn } ⊂ M is an L-free set with xi  ≤ 1 for all i, then there exists a tracial von Neumann algebra (M, τ ), a trace preserving unital embedding M ⊂ M  with M  = Mn+1 (C) ⊗ M so and an L-free set of unitaries {U1 , . . . , Un } ⊂ U(M) that, denoting by (eij )i,j=0,...,n the matrix units of Mn+1 (C), we have e00 Ui e00 = xi for all i. Proof. Define M = M ∗ L(Fn(n−1) ) and denote by ui,j , i = j, free generators of L(Fn(n−1) ). For every i ∈ {1, . . . , n}, define ci = 1 − xi x∗i and di = − 1 − x∗i xi .  = Mn+1 (C) ⊗ M and define the unitary elements Ui ∈ U(M)  given by Put M  (ejj ⊗ ui,j ) . Ui = (e00 ⊗ xi ) + (eii ⊗ x∗i ) + (e0i ⊗ ci ) + (ei0 ⊗ di ) + j=i 3 Note that this notion is not the same as (and should not be confused with) the notion of L-sets used in [Pi92].

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Note that Ui is the direct sum of the unitary   xi c i di x∗i in positions 0 and i, and the unitary

%

ui,j

j=i

in the positions j = i. By construction, we have that e00 Ui e00 = xi e00 . So, it remains 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 must prove that τ (Ui1 Uj∗1 · · · Uik Uj∗k ) = 0 .

(3.1)

Consider V := Ui1 Uj∗1 · · · Uik Uj∗k as a matrix with entries in M. Every entry of this matrix 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 xi1 x∗j1 · · · xik x∗jk , which has zero trace; • if i1 = jk = i, in the entry Vii as w = di x∗j1 xi2 x∗j2 · · · xik−1 x∗jk−1 xik d∗i . Then we have τ (w) = τ (x∗j1 xi2 · · · x∗jk−1 xik d∗i di ) = τ (x∗j1 xi2 · · · x∗jk−1 xik ) − τ (x∗j1 xi2 · · · x∗jk−1 xik x∗i xi ) = 0 − τ (xi1 x∗j1 · · · xik x∗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 (M ua,b M ) = {0}. 3◦ Words w with two or more letters of the type ua,b or u∗a,b . Consider two consecutive such letters in w, i.e. a subword of w of the form 

uεi,j w0 uεi ,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 have that τ (w0 ) = 0.

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A subword of the form ui,j w0 u∗i,j can only arise from the jj-entry of Uis Uj∗s · · · Uit Uj∗t

with is = jt = i , js = it = j

(and thus, t ≥ s + 2). In that case, w0 = c∗j xis+1 x∗js+1 · · · xit−1 x∗jt−1 cj . Thus, τ (w0 ) = τ (xis+1 x∗js+1 · · · xit−1 x∗jt−1 cj c∗j ) = τ (xis+1 x∗js+1 · · · xit−1 x∗jt−1 ) − τ (xis+1 x∗js+1 · · · xit−1 x∗jt−1 xj x∗j ) = 0 − τ (xjs x∗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 Uj∗s−1 Uis · · · Uj∗t−1 Uit

with js−1 = it = i , is = jt−1 = j

(and thus, t ≥ s + 2). In that case, w0 = dj x∗js xis+1 · · · x∗jt−2 xit−1 d∗j . As above, it follows that τ (w0 ) = 0. So, we have proved that every word that appears in a diagonal entry Vii of V has trace zero. Then also τ (V ) = 0 and it follows that {U1 , . . . , Un } is an L-free set of unitaries.  Corollary 3.5. Let (M, τ ) be a finite von Neumann algebra with a faithful normal tracial state. If {x1 , . . . , xn } ⊂ M is L-free with xi  ≤ 1 for all i, then n   √   xi  ≤ 2 n − 1 .  i=1

More generally, given any complex scalars α1 , . . . , αn with Σni=1 |αi |2 ≤ 1, we have n     αi xi  ≤ 2 1 − 1/n .  i=1

Proof. Assuming n ≥ 2, with the notations from Proposition 3.4 and by using   n  Corollary 3.3, we have  i=1 αi Ui  ≤ 2 1 − 1/n. Reducing with the projection e00 , it follows that n     αi xi  ≤ 2 1 − 1/n .  i=1

 4. Applications to paving problems Like in [P13], [PV14], if A ⊂ M is a MASA in a von Neumann algebra and x ∈ M, then we denote by n(A ⊂ M; x, ε) the smallest n for  which there exist n projections p , . . . , p ∈ A and a ∈ A such that a ≤ x, 1 n i=1 pi = 1 and     n  i=1 pi xpi − a ≤ εx (with the convention that n(A ⊂ M; x, ε) = ∞ if no such finite partition exists), and call it the paving size of x.

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Recall also from [D54] that a MASA A in a von Neumann algebra M is called singular, if the only unitary elements in M that normalize A are the unitaries in A. Theorem 4.1. Let An ⊂ Mn be a sequence of singular MASAs

in finite von Neumann algebras and ω a free ultrafilter on N. Denote M = ω Mn and A =

A . Given any countable set of contractions X ⊂ M%A and any integer n ≥ 2, n ω there exists a partition of 1 with projections p1 , . . . , pn ∈ A such that n   √   pj xpj  ≤ 2 n − 1/n, for all x ∈ X .  j=1

In particular, the paving size of A ⊂ M, n(A ⊂ M; ε) = sup{n(A ⊂ M; x, ε) | x = x∗ ∈ M % A} , def

is less than 4ε−2 + 1, for any ε > 0. Proof. By Theorem 4.1(a) in [P13], there exists a diffuse abelian von Neumann subalgebra A0 ⊂ A such that for any k ≥ 1, any word with alternating letters x = x0 Πki=1 (vi xi ) 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=1 λj−1 pj , where λ = exp(2πi/n), then for any x ∈ X the set {ui−1 xu−i+1 | i = 1, 2, . . . , n} is L-free. Since n1 Σni=1 ui−1 xu1−i = Σni=1 pi xpi , where p1 , . . . , pn are the minimal spectral projections of u, by Proposition 3.4 it follows that for all x ∈ X we have √ 1 Σni=1 pi xpi  = Σni=1 ui−1 xu−i+1  ≤ 2 n − 1/n. n To derive the last part, let ε > 0 and denote by n the integer with the property that 2n−1/2 ≤ ε < 2(n − 1)−1/2 . If x ∈ M % A, x ≤ 1, and p1 , . . . , pn ∈ A are mutually orthogonal projections of trace 1/n that satisfy the free independence relation with X = {x} as above, then n < 4ε−2 + 1 and we have √ Σni=1 pi xpi  ≤ 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 measuring the so-paving size over a MASA in a von Neumann algebra A ⊂ M admitting a normal conditional expectation, is by considering the quantity def

ε(A ⊂ M ; n) =

sup x∈(Mhω Aω )1

(inf{Σni=1 pi xpi  | pi ∈ P(Aω ), Σi pi = 1}).

With this notation, the above theorem√shows that for a singular MASA in a II1 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 ⊂ M ω ; ε) < 4ε−2 + 1. Also, the conjecture (2.8.2◦ in [PV14]) about the so-paving √size can this way be 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 particularly interesting to study this question in the classical Kadison-Singer case of the diagonal MASA D ⊂ B = B(2 N), and more generally for Cartan MASAs A ⊂ M . So

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far, the solution to the Kadison-Singer paving problem in [MSS13] shows that ε(D ⊂ B; n) ≤ 12n−1/4 . Also, while by [CEKP07] one has n(D ⊂ B; ε) ≥ ε−2 and by [PV14] one has ns (A ⊂ M ; ε) = n(Aω ⊂ M ω ; ε) ≥ ε−2 , for any MASA in a II1 factor A ⊂ M , it would be interesting to decide whether ε(D ⊂ B; n) and ε(A ⊂ M ; n) are in fact √ bounded from below by 2 n − 1/n, ∀n. For a singular MASA in a II1 factor, √ A ⊂ M , combining 4.1 with such a lower bound 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 equality holds indeed. Proposition 4.3. 1◦ Let M be a II1 factor and A ⊂ M a MASA. Assume v ∈ 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 any partition of 1 with projections p1 , . . . , pn ∈ Aω we have Σni=1√ pi vpi  ≥ 2 n − 1/n, with equality iff all pi have trace 1/n. Also, ε(A ⊂ M ; n) ≥ 2 n − 1/n, ∀n. the canonical generator 2◦ If M = L(Z∗(Z/2Z)), A = L(Z) and v = v ∗ denotes √ of 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, the are freely independent, and so by Proposition 2.2 one has projections pi and vpi v ∗ pi vpi  = pi vpi v ∗  = 2 τ (pi )(1 − τ (pi )). Thus,√if one of the projections pi has trace τ (pi ) > 1/n, then√Σj pj vpj  ≥ pi vpi  > 2 n − 1/n, while if τ (pi ) = 1/n, ∀i, then Σj pj vpj  = 2 n − 1/n. By applying 1◦ to part 2◦ , then using 4.1 and the fact that A = L(Z) is singular in M = L(Z ∗ (Z/2Z)) (cf. [P81]), proves the last part of the statement.  References 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 group C ∗ -algebra of Zn ∗ Zm , J. Operator Theory 26 (1991), no. 1, 3–23. MR1214917 (94c:46110) [B74] M. Bo˙zejko, 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 KadisonSinger problem, Oper. Matrices 1 (2007), no. 3, 391–408, DOI 10.7153/oam-01-23. MR2344683 (2009a:46105) [D54] J. Dixmier, Sous-anneaux ab´ eliens 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), Studia Math. 52 (1974), 149–158. MR0355480 (50 #7954) [MSS13] A. W. Marcus, D. A. Spielman, and N. Srivastava, Interlacing families II: Mixed characteristic 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 [AO74]

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[Pi92] [P81] [P13] [PV14] [Po75] [Vo86]

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) S. Popa, Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253–268. MR703810 (84h:46077) 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 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 R. T. Powers, Simplicity of the C ∗ -algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151–156. MR0374334 (51 #10534) 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 E-mail address: [email protected] Department of Mathematics, KU Leuven, Leuven (Belgium) E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13512

Matricial bridges for “Matrix algebras converge to the sphere” Marc A. Rieffel In celebration of the successful completion by Richard V. Kadison of 90 circumnavigations of the sun Abstract. In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. In the present paper, as preparation of discussing similar statements for convergence of “vector bundles” over matrix algebras to vector bundles over spaces, we introduce and study suitable matrix-norms for matrix algebras and spaces. Very recently Latr´emoli` ere introduced an improved quantum Gromov-Hausdorff-type distance between quantum metric spaces. We use it throughout this paper. To facilitate the calculations we introduce and develop a general notion of “bridges with conditional expectations”.

1. Introduction In several earlier papers [11, 12, 14] I showed how to give a precise meaning to 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 quantum physics literature given in [1, 2, 4, 5, 11, 13, 15].) I did this by introducing and developing a concept of “compact quantum metric spaces”, and a corresponding quantum Gromov-Hausdorff-type distance between them. The compact quantum spaces are unital C*-algebras, and the metric data is given by putting on the algebras seminorms that play the role of the usual Lipschitz seminorms on the algebras of continuous functions on ordinary compact metric spaces. The natural setting for “matrix algebras converge to the sphere” is that of coadjoint orbits of compact semi-simple Lie groups. But physicists need much more than just the algebras. They need vector bundles, gauge fields, Dirac operators, etc. So I now seek to give precise meaning to statements 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 grant DMS-1066368. c 2016 American Mathematical Society

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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 bundles on ordinary compact metric spaces for ordinary Gromov-Hausdorff distance. From that study it became clear that one needed Lipschitz-type seminorms on all the matrix algebras over the underlying algebras, with these seminorms coherent in the sense that they form a “matrix seminorm” (defined below). The purpose of this paper is to define and develop the properties of such matrix seminorms for the setting of coadjoint orbits, and especially to study how these matrix seminorms mesh with the non-commutative analogs of Gromov-Hausdorff distance. Very recently Latr´emoli`ere introduced an improved version of quantum GromovHausdorff distance [8] that he calls “the Gromov-Hausdorff propinquity”. We show that his propinquity works very well for our setting of coadjoint orbits, and so propinquity is the form of quantum Gromov-Hausdorff distance that we use in this paper. Latr´emoli`ere 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 need corresponding “matricial bridges”, and we show how to construct natural ones for the setting of coadjoint orbits. It is crucial to obtain good upper bounds for the lengths of the bridges that we construct. In the matricial setting the calculations become somewhat complicated. In order to ease the calculations we introduce a notion of “bridges with conditional expectations”, and develop their general theory, including the matricial case, and including bounds for their lengths in the matricial case. The main theorem of this paper, Theorem 6.10, states in a quantitative way that for the case of coadjoint orbits the lengths of the matricial bridges goes to 0 as the size of the matrix algebras goes to infinity. We also discuss a closely related class of examples coming from [12], for which we construct bridges between different matrix algebras associated to a given coadjoint orbit. This provides further motivation for our definitions and theory of bridges with conditional expectation. Contents 1. Introduction 2. The first basic class of examples 3. The second basic class of examples 4. Bridges with conditional expectations 5. The corresponding matricial bridges 6. The application to the first class of basic examples 7. The application to the second class of basic examples 8. Treks References

2. The first basic class of examples In this section we describe the first of the two basic classes of examples underlying 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 of examples.

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Let G be a compact group (perhaps even finite, at first). Let U be an irreducible unitary 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 by U , that is, αx (T ) = Ux T Ux∗ 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 the scalar multiples of the identity operator. Let P be a rank-one projection in B(H) (traditionally specified by giving a non-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(T αx (P )), where tr denotes the usual (un-normalized) trace on B. (When the αx (P )’s are viewed 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 subgroup of P for α. Then it is evident that σT can be viewed as a (continuous) function on G/H. We let λ denote the action of G on G/H, and so on A = C(G/H), by left-translation. If we note that tr is α-invariant, then it is easily seen that σ is a unital, 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. Then in 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 definition 4.1 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 evident analog of formula 2.1, except that we must permit LA to take the value ∞. It is shown in proposition 2.2 of [10] that the set of functions for which LA is finite (the Lipschitz functions) is a dense ∗-subalgebra of A. Also, LA is the restriction to A of the seminorm on C(G) that we get from  when we view C(G/H) as a subalgebra of C(G), as we will often do when convenient. From LA we can use equation 2.2 below to recover the usual quotient metric [16] on G/H coming from the metric on G determined by . One can check easily that LA in turn comes from this quotient metric. Thus (A, LA ) is the compact C*-metric-space associated to this ordinary compact 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 Latr´emoli`ere, which we will describe soon below. For any two unital C*-algebras A and B a bridge from A to B in the sense of Latr´emoli`ere [8] is a quadruple (D, πA , πB , ω) for which D is a unital C*-algebra, πA and πB are unital injective homomorphisms of A and B into D, and ω is a self-adjoint element of D such that 1 is an element of the spectrum of ω and ω = 1. Actually, Latr´emoli`ere only requires a looser but more complicated condition on ω, but the above condition will be appropriate for our examples. Following Latr´emoli`ere we will call ω the “pivot” for the bridge. We will often omit mentioning the injections πA and πB when it is clear what they are from the context, and accordingly we will often write as though A and B are unital subalgebras of D.

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For our first class of examples, in which A and B are as described in the paragraphs 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 defined similarly. From the many calculations done in [11, 14] it is not surprising that we define 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 it satisfies 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 seminorms LA and LB on A and B, Latr´emoli`ere [8] defines the “length” of the bridge in terms of these seminorms. For this he initially puts relatively weak requirements on the seminorms, but for the purposes of the matricial bridges that we will define later, we need somewhat different weak requirements. To begin with, Latr´emoli`ere only requires his seminorms, say LA on a unital C*-algebra A, to be defined on the subspace of self-adjoint elements of the algebra. However, we need LA to be defined 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 Latr´emoli`ere, our LA is permitted to take value +∞. Latr´emoli`ere also requires the subspace on which LA takes finite values to be dense in the algebra. We do not really need this here, but for us there would be no harm in assuming it, and all interesting examples probably will satisfy this. Finally, Latr´emoli`ere requires that the null space of LA (i.e where it takes value 0) be exactly R1A . We must loosen this to simply requiring that 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 of this 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 that L(1A ) = 0. Because of these weak requirements on LA , various quantities in this paper may be +∞, but most interesting examples will satisfy stronger requirements that will result in various quantities being finite. Latr´emoli`ere defines the length of a bridge, for given LA and LB , by first defining 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 a bridge 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 (L1A ω , ωL1B ), where HausD denotes the Hausdorff distance with respect to the norm of D, and where the product defining L1A ω and ωL1B is that of D.

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Latr´emoli`ere shows just before definition 3.14 of [8] that, under conditions that include the case in which (A, LA ) and (B, LB ) are C*-metric spaces, the reach of Π is finite. To define the height of Π we need to consider the state space, S(A), of A, and similarly 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 L1A .) Define ρB on S(B) similarly. Notation 2.3. We denote by S1A (ω) the restriction of the elements of S1 (ω) to A. We define S1B (ω) similarly. Definition 2.4. Let A and B be unital C*-algebras and let Π = (D, ω) be a bridge from A to B . Let LA and LB be slip-norms on A and B. The height of the bridge Π is given by height(Π) = max{HausρA (S1A (ω), S(A)), HausρB (S1B (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 representation of G, and for each of these two representations we pick a rank-one projection. The key feature that we require is that the stability subgroups of these two projections for the action of G coincide. The more concrete class of examples motivating this situation, but for which we will not need the details, is that in [11, 12, 14] in which G is a compact semi-simple Lie group, λ is a positive integral weight, and our two representations of G are the representations with highest weights mλ and nλ for positive integers m and n, m = n. Furthermore, the projections P are required to be those along highest weight vectors. The key feature of this situation that we do need to remember here is that the stability subgroups H for the two projections coincide. Accordingly, for our slightly more general situation, we will denote our two representations by (Hm , U m ) and (Hn , U n ), where now m and n are just labels. Our two C*-algebras will be B m = L(Hm ) and B n = L(Hn ). We will denote the action of G on these two algebras just by α, since the context should always make clear which algebra is being acted on. The corresponding projections will be P m

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and P n . The crucial assumption that we make is that the stability subgroups of these two projections coincide. We will denote this common stability subgroup by H as before. We construct a bridge from B m to B n as follows. We let A = C(G/H) as in our first class of examples, and we define D by D = B m ⊗ A ⊗ B n = C(G/H, B m ⊗ B n ). We view B m as a subalgebra of D by sending b ∈ B m to b ⊗ 1A ⊗ 1Bn , and similarly for B n . From the many calculations done in [12] it is not surprising that we define the pivot, ω, to be the function in C(G/H, B m ⊗ B n ) defined by ω(x) = αx (P m ) ⊗ αx (P n ). We let Lm be the Lip-norm defined on B m determined by the action α and the length function  as in formula (2.1), and similarly for Ln on B n . In terms of these Lip-norms the length of any bridge from B m to B n is defined. Thus the length of the bridge described above is defined. In Section 7 we will see how to obtain useful upper bounds on the length of this bridge. 4. Bridges with conditional expectations We will now seek a somewhat general framework for obtaining useful estimates for the lengths of bridges such as those of our two basic classes of examples. To discover this framework we will explore some properties of our two basic classes of examples. 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 left translation by elements of G. We denote the corresponding linear functional on A = C(G/H) by τA , and sometimes refer to it as the canonical tracial state on A. On B = L(H) there is a unique tracial state, which we denote by τB . These combine to form a tracial state, τD = τA ⊗ τB on D = A ⊗ B. Similarly, we have the unique tracial states τm and τn on B m and B n , which combine with τA to give a tracial state on D = Bm ⊗ A ⊗ Bn . For D = A ⊗ B, the tracial state τB determines a conditional expectation, E A , from D onto its subalgebra A, defined on elementary tensors by E A (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 expectation has the property that for any d ∈ D we have τA (E A (d)) = τD (d), and it is the unique conditional expectation with this property. (See corollary II.6.10.8 of [3].) In the same way the tracial state τA determines a canonical conditional expectation, E B from D onto its subalgebra B. For the case in which D = B m ⊗ A ⊗ B n , the tracial state τA ⊗ τn on A ⊗ B n determines a canonical conditional expectation, E m , from D onto B m in the same way as above, and the tracial state τm ⊗ τA determines a canonical conditional expectation, E n , from D onto B n .

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These conditional expectations relate well to the pivots of the bridges. For the case in which D = A ⊗ B we find that for any F ∈ D = C(G/H, B) we have E A (F ω)(x) = τB (F (x)αx (P )). In particular, for any T ∈ B we have E A (T ω)(x) = τB (T αx (P )) for all x ∈ G/H. Aside from the fact that we are here using the normalized trace instead of the standard trace on the matrix algebra B, the right-hand side is exactly the definition of the Berezin covariant symbol of T that plays such an important role 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 → E A (bω) might be of importance to us. For our specific first basic class of examples we note the following favorable properties: (1) Self-adjointness i.e. E A (F ∗ ω) = (E A (ωF ))∗ for all F ∈ D. (2) E A (F ω) = E A (ωF ) for all F ∈ D. (3) Positivity, i.e. if F ≥ 0 then E A (F ω) ≥ 0. (4) E A (1D ω) = r −1 1A where B is an r × r matrix algebra. However, if we consider E B instead E A , then for any F ∈ D we have ! E B (F ω) = F (x)αx (P ) dx, G/H

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 that properties 1-3 again hold. Even more, the expression ! f (x)αx (P ) dx G/H

is, except for normalization of the trace, the formula involved in the Berezin contravariant symbol that in [11] is denoted by σ ˘. For our second class of examples, in which D = B m ⊗ A ⊗ Bn , we find that for F ∈ D = C(G/H, B m ⊗ B n ) we have ! E m (F ω) = (ιA ⊗ τn )(F (x)(αx (P m ) ⊗ αx (P n ))) dx. G/H

Again we see that properties 1-3 above are not in general satisfied. But if we restrict F to be any T ∈ B n then the above formula becomes ! αx (P m )τn (T αx (P n )) dx, G/H

which up to normalization of the trace is exactly the second displayed formula in section 3 of [12]. It is not difficult to see that properties 1-3 above are again satisfied under this restriction. We remark that it is easily seen that the maps T → E m (T ω) from B n to B m and S → E n (Sω) from B m to B n are each other’s adjoints when they are viewed as being between the Hilbert spaces L2 (B m , τm ) and L2 (B n , τn ). A similar statement holds for our first basic class of examples. With these observations in mind, we begin to formulate a somewhat general framework. 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

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expectations E A and E B from D onto its subalgebras A and B. (We do not require that 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 ω 1/2 exists. Then the map d → E A (ω 1/2 dω 1/2 ) 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 (E A (ω 1/2 bω 1/2 )) = 0. But one of the conditions on a Lip-norm is that it takes value 0 exactly on the scalar multiples of the identity element, and the case of Lip-norms is important to us. For Lip-norms we see that the above requirement implies that E A (ω) ∈ C1A , and so E A (ω) = rω 1A for some positive real number rω . We require the same of E B with the same real number, so that we require that E A (ω) = rω 1A = E B (ω). We require further that rω = 0. We then define a map, ΦA , from D to A by ΦA (d) = rω−1 E A (ω 1/2 dω 1/2 ). In a similar way we define ΦB from D to B. We see that ΦA and ΦB are unital positive maps, and so are of norm 1 (as seen by composing them with states). Then the 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 when the above requirements are satisfied. Let b ∈ L1B be given. As an approximation to ωb by an element of the form aω for some a ∈ L1A we take a = ΦA (b). It is indeed in L1A by the requirements made just above. This prompts us to set (4.1)

γ B = sup{ΦA (b)ω − ωbD : b ∈ L1B },

and we see that ωb is then in the γ B -neighborhood of L1A ω. Note that without further assumptions on LB we could have γ B = +∞. Interchanging the roles of A and B, we define γ A similarly. We then see that reach(Π) ≤ max{γ A , γ B }. We will explain in Sections 6 and 7 why this upper bound is useful in the context of [11, 12, 14]. We now consider the height of Π. For this we need to consider S1 (ω) as defined in Section 2. Let μ ∈ S(A). Because ΦA is positive and unital, its composition with μ is in S(D). When we evaluate this composition at ω to see if it is in S1 (ω), we obtain μ(rω−1 E A (ω 2 )), and we need this to equal 1. Because μ(rω−1 E A (ω)) = 1, it follows that we need μ(rω−1 E A (ω − ω 2 )) = 0. If this is to hold for all μ ∈ S(A), we must have E A (ω − ω 2 ) = 0. If E A is a faithful conditional expectation, as is true for our basic examples, then because ω ≥ ω 2 it follows that ω 2 = ω so that ω is a 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 obtain an element, φμ , of S1 (ω), defined by φμ (d) = μ(rω−1 E A (ωdω)) = μ(ΦA (d)).

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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 S1A (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

δ A = sup{a − ΦA (a) : a ∈ L1A }.

Then we see that

ρLA (μ, φμ |A ) ≤ δ A . We define δ B in the same way, and obtain the corresponding estimate for the distances from elements of S(B) to the restriction of S1 (ω) to B. In this way we see that height(Π) ≤ max{δ A , δ B }. (Notice that δ A involves what ΦA does on A, whereas γ A involves what ΦB does on A.) While this bound is natural within this context, it turns out not to be so useful for our two basic classes of example. In Proposition 4.6 below we will give a different bound that does turn out to be useful for our basic examples. But perhaps other examples 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) be a bridge from A to B. We say that Π is a bridge with conditional expectations if conditional expectations E A and E B from D onto A and B are specified, satisfying the following properties: (1) The conditional expectations are faithful. (2) The pivot ω is a projection. (3) There is a constant, rω , such that E A (ω) = rω 1D = E B (ω). For such a bridge with conditional expectations we define ΦA on D by ΦA (d) = rω−1 E A (ωdω). We define ΦB similarly, with the roles of A and B reversed. We will often write Π = (D, ω, E A , E B ) for a bridge with conditional expectations. I should mention here that at present I do not see how the class of examples considered by Latr´emoli`ere that involves non-commutative tori [9] fits into the setting of bridges with conditional expectations, though I have not studied this matter carefully. It would certainly be interesting to understand this better. I also do not see how the general case of ordinary compact metric spaces, as discussed in theorem 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 A and B. We say that a bridge with conditional expectations Π = (D, ω, E A , E B ) is admissible for LA and LB if LA (ΦA (b)) ≤ LB (b) for all b ∈ B, and for all a ∈ A .

LB (ΦB (a)) ≤ LA (a)

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We define the reach, height and length of a bridge with conditional expectations (D, ω, E A , E B ) 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, E A , E B ) be a bridge with conditional expectations from A to B that is admissible for LA and LB . Then reach(Π) ≤ max{γ A , γ B }, where γ A = sup{aω − ωΦB (a)D : a ∈ L1A }, and similarly for γ B , while height(Π) ≤ max{δ A , δ B }, where δ A = sup{a − ΦA (a) : a ∈ L1A }, and similarly for δ B . Consequently length(Π) ≤ max{γ A , γ B , δ A , δ B }. (Consequently the propinquity between (A, LA ) and (B, LB ), as defined in [8], is no greater than the right-hand side above.) We could axiomitize the above situation in terms of just ΦA and ΦB , without requiring that they come from conditional expectations, but at present I do not know of examples for which this would be useful. It would not suffice to require that ΦA and ΦB just be positive (and unital) because for the matricial case discussed in 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 the quite special property that ω commutes with every element of A, or at least that E A (ωaω) = E A (aω) for all a ∈ A. Then ΦA (a) = a for all a ∈ A. Consequently δ A = 0, and the restriction of S1 (ω) to A is all of S(A). Proof. This depends on the conditional expectation property of E A . For a ∈ A we have ΦA (a) = rω−1 E A (aω) = arω−1 E A (ω) = a.  The following steps might not initially seem useful, but in Sections 6 and 7 we will see in connection with our basic examples that they are quite useful. Our notation is as above. Let ν ∈ S(B). Then as seen above, ν ◦ ΦB ∈ S1 (ω), and so its restriction to A is in S(A). But then ν ◦ ΦB ◦ ΦA ∈ S1 (ω). Let us denote it by ψν . Then the restriction of ψν to B can be used as an approximation to ν by an element 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 δˆB = sup{b − ΦB (ΦA (b)) : b ∈ L1B }.

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We note that LB (ΦB (ΦA (b))) ≤ LB (b) because of the admissibility requirements of Definition 4.2. It follows that ρLB (ν, ψν ) ≤ δˆB . We define δˆA in the same way, and obtain the corresponding estimate for the distances from elements of S(A) to the restriction of S1 (ω) to A. In this way we obtain: Proposition 4.6. For notation as above, height(Π) ≤ max{min{δ A , δˆA }, min{δ B , δˆB }}. 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 matrices with complex entries. For any C*-algebra A we let Mq (A) denote the C*-algebra of q × q matrices with entries in A. We often identify it in the evident way with the C*-algebra Mq ⊗ A. Let A and B be unital C*-algebras, and let Π = (D, ω) be a bridge from A to B. Then Mq (A) can be viewed as a subalgebra of Mq (D), as can Mq (B). Let ωq = 1q ⊗ ω, where 1q is the identity element of Mq , so ωq can be viewed as the diagonal matrix in Mq (D) with ω in each diagonal entry. Then it is easily seen that Πq = (Mq (D), ωq ) is a bridge from Mq (A) to Mq (B). Definition 5.1. The sequence {Πq } is called the matricial bridge corresponding to the bridge Π. B In order to measure the length of Πq we need slip-norms LA q and Lq on Mq (A) and Mq (B). It is reasonable to want these slip-norms to be coherent in some sense as q varies. The discussion that we will give just after Theorem 6.8 suggests that B the coherence requirement be that the sequences {LA q } and {Lq } form “matrix slipnorms”. To explain what this means, for any positive integers m and n we let Mmn denote the linear space of m × n matrices with complex entries, equipped with the norm obtained by viewing such matrices as operators from the Hilbert space Cn to the Hilbert space Cm . We then note that for any A ∈ Mn (A), any α ∈ Mmn , and any β ∈ Mnm the usual matrix product αAβ is in Mm (A). The following definition, for the case of Lip-norms, is given in definition 5.1 of [18] (and see also [6, 14, 17, 19]). A Definition 5.2. A sequence {LA n } is a matrix slip-norm for A if Ln is a ∗seminorm (with value +∞ permitted) on Mn (A) for each integer n ≥ 1, and this family of seminorms has the following properties: (1) For any A ∈ Mn (A), any α ∈ Mmn , and any β ∈ Mnm , we have A LA m (αAβ) ≤ αLn (A)β.

(2) For any A ∈ Mm (A) and any C ∈ Mn (A) we have " # A 0 A A Lm+n = max(LA m (A), Ln (C)). 0 C (3) LA 1 is a slip-norm.

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We remark that the properties above imply that for n ≥ 2 the null-space of LA n contains all of Mn , not just the scalar multiples of the identity. This is why our definition of slip-norms does not require that the null-space is exactly the scalar multiples of the identity. Now let Π = (D, ω, E A , E B ) be a bridge with conditional expectations. For any integer q ≥ 1 set EqA = ιq ⊗ E A , where ιq is the identity map from Mq onto itself. Define EqB similarly. Then it is easily seen that EqA and EqB are faithful conditional expectations from Mq (D) onto its subalgebras Mq (A) and Mq (B) respectively. Furthermore, EqA (ωq ) is the diagonal matrix each diagonal entry of which is E A (ω) = rω 1D , and from this we see that EqA (ωq ) = rω 1Mq (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, E A , E B ) be a bridge with conditional expectations from A to B. Then Πq = (Mq (D), ωq , EqA , EqB ) is a bridge with conditional expectations from Mq (A) to Mq (B). It has the same constant rω as does Π. A B q We can then set ΦA q = ιq ⊗ Φ , and similarly for Φq . Because Π has the same constant rω as does Π, we see that for any D ∈ Mq (D) we have −1 A ΦA q (D) = rω Eq (ωq Dωq ). B Suppose now that A and B have matrix slip-norms {LA n } and {Ln }. We remark A that a matrix slip-norm {Ln } is in general not at all determined by LA 1 . Thus a as in Definition 4.2 need not relate well to the bridge that is admissible for LA 1 for higher n. seminorms LA n B Definition 5.4. With notation as above, let {LA n } and {Ln } be matrix slipnorms on A and B. We say that a bridge with conditional expectations Π = B (D, ω, E A , E B ) is admissible for {LA n } and {Ln } if for all integers n ≥ 1 the bridge n A B Π is admissible for Ln and Ln ; that is, for all integers n ≥ 1 we have A B LA n (Φn (B)) ≤ Ln (B)

for all B ∈ Mn (B), where −1 A ΦA n (B) = rω En (ωn Bωn ),

and similarly with the roles of A and B reversed. B We assume now that Π = (D, ω, E A , E B ) is admissible for {LA n } and {Ln }. q A B Since for a fixed integer q the bridge Π is admissible for Lq and Lq , the length of Πq is defined. We now show how to obtain an upper bound for the length of Πq in terms of the data used in the previous section to get an upper bound on the length of Π. We consider first the reach of Πq . Set, much as earlier, ∗ L1q A = {A ∈ Mq (A) : A = A

and LA q (A) ≤ 1},

q and similarly for L1q B . Then the reach of Π is defined to be 1q HausMq (D) {L1q A ωq , ωq LB }.

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1q A Suppose that B ∈ L1q B . Then Φq (B) ∈ LA by the admissibility requirement. So we want to bound ΦA q (B)ωq − ωq BMq (D) .

I don’t see any better way to bound this in terms of the data used in Theorem 4.3 for Π than by using an entry-wise estimate as done in the third paragraph before lemma 14.2 of [14]). We use the fact that for a q × q matrix C = [cjk ] with entries in a C*-algebra we have C ≤ q maxjk {cjk } (as is seen by expressing C as the sum of the q matrices whose only non-zero entries are the entries cjk for which j − k is a given constant mod q). In this way, for B ∈ Mq (B) with B = [bjk ] we find that the last displayed term above is ≤ q max{ΦA (bjk )ω − ωbjk D }. jk

The small difficulty is that the bjk ’s need not be self-adjoint. But for any b ∈ B, if we denote its real and imaginary parts by br and bi , then because LB is a ∗-seminorm it follows that LB (br ) ≤ LB (b) and similarly for bi . Consequently ΦA (b)ω − ωbD ≤ ΦA (br )ω − ωbr D + ΦA (bi )ω − ωbi D ≤ γ B LB (br ) + γ B LB (bi ) ≤ 2γ B LB (b). Thus the term displayed just before is ≤ 2qγ B LB (bjk ). But {LB n } is a matrix slip-norm, and by the first property of such seminorms given in Definition 5.2, we have max{LB (bjk )} ≤ LB q (B). Thus for B ∈ L1q B we see that B ΦA q (B)ωq − ωq BMq (D) ≤ 2qγ ,

so that ωq B is in the 2qγ B -neighborhood of L1q A ωq . In the same way Aωq is in the 1q 1q A 2qγ -neighborhood of ωq LB for every A ∈ LA . We find in this way that reach(Πq ) ≤ 2q max{γ A , γ B }. 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 ΦA q is an element, φμ , of S1 (ωq ), specifically defined by −1 A φμ (D) = μ(ΦA q (D)) = μ(rω Eq (ωq Dωq )).

We take φμ |Mq (A) as an approximation to μ, and estimate the distance between these elements of S(Mq (A)). For A ∈ L1q A we calculate A |μ(A) − φμ (A)| = |μ(A) − μ(ΦA q (A))| ≤ A − Φq (A).

Again I don’t see any better way to bound this in terms of the data used in Theorem 4.3 for Π than by using an entry-wise estimates. For A ∈ Mq (A) with A = [ajk ] we find (by using arguments as above to deal with the fact that the ajk ’s need not be self-adjoint) that the last displayed term above is ≤ q max{ajk − ΦA (ajk )} ≤ 2qδ A max{LA (ajk )}. jk

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But again {LA n } is a matrix slip-norm, and so by the first property of such seminorms given in Definition 5.2 we have max{LA (ajk )} ≤ LA q (A). Since our assumption is that A ∈ L1q A , we see in this way that (μ, φμ |A ) ≤ 2qδ A . ρLA q Thus S(Mq (A)) is in the 2qδ A -neighborhood of the restriction to Mq (A) of S1 (ωq ). We find in the same way that S(Mq (B)) is in the 2qδ B -neighborhood of the restriction to Mq (B) of S1 (ωq ). Consequently, height(Πq ) ≤ 2q max{δ A , δ B }. We can instead use δˆA in the way done in Proposition 4.6. Using reasoning much like that used above, we find that for any B ∈ Mq (B) we have: A B A B − ΦB q (Φq (B)) ≤ q max{bjk − Φ (Φ (bjk )) jk

≤ 2q δˆB max{LB (bjk )} ≤ 2q δˆB LB q (B) Consequently we see that height(Πq ) ≤ 2q max{δˆA , δˆB }. We summarize what we have found by: B Theorem 5.5. Let {LA n } and {Ln } be matrix slip-norms on unital C*-algebras A B A and B, and let Π = (D, ω, E , E ) be a bridge with conditional expectations from B A to B that is admissible for {LA n } and {Ln }. For any fixed positive integer q let q Π be the corresponding bridge with conditional expectations from Mq (A) to Mq (B). Then

reach(Πq ) ≤ 2q max{γ A , γ B }, where as before γ A = sup{aω − ωΦB (a)D : a ∈ L1A }, and similarly for γ B ; while height(Πq ) ≤ 2q max{min{δ A , δˆA }, min{δ B δˆB }}, where as before δ A = sup{ΦA (a) − a : a ∈ L1A } and δˆA = sup{a − ΦA (ΦB (a)) : a ∈ L1A }, and similarly for δ B and δˆB . Consequently length(Πq ) ≤ 2q max{γ A , γ B , min{δ A , δˆA }, min{δ B δˆB }}.

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6. The application to the first class of basic examples We now apply the above general considerations to our first class of basic examples, described in Section 2, and we will use the same notation as is used in that section. We proceed to obtain an upper bound for the length of the bridge Π = (D, ω), where D = C(G/H, B) and ω(x) = αx (P ). We begin by considering its reach. As seen in Section 4 , for any F ∈ D = C(G/H, B) we have E A (ωF ω)(x) = τB (F (x)αx (P )). From this it is easily seen that rω−1 is the dimension of H, and so rω−1 τB is the usual unnormalized trace on B, which we now denote by trB . In particular, for any T ∈ B we have (6.1)

ΦA (T )(x) = rω−1 E A (ωT ω)(x) = trB (T αx (P ))

for all x ∈ G/H. But this is exactly the covariant Berezin symbol of T (for this general context) as defined early in section 1 of [11] and denoted there by σT . It is natural to put on D the action λ ⊗ α of G. One then easily checks that ΦA is equivariant 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 that part 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 our present example that γ B , as defined in equation (4.1), is given by (6.2) γ B = sup{ΦA (T )ω − ωT D : T ∈ L1B } = sup{P (tr(P T )1B − T )B : T ∈ L1B }. This last term is exactly the definition of γ B given in proposition 8.2 of [14]. We next consider γ A . For any f ∈ A we have ! f (x)αx (P ) dx, (6.3) ΦB (f ) = rω−1 E B (f ) = dH G/H

where dH = dim(H). But this is exactly the formula used for the Berezin contravariant symbol, as indicated in Section 4. Early in section 2 of [11] this ΦB is denoted by σ ˘f , that is, (6.4)

˘f ΦB (f ) = σ

for the present class of examples. One easily checks that ΦB is equivariant for λ ⊗ α and α. From this it is easy to verify that σf ) ≤ LA (f ) LB (ΦB (f )) = LB (˘ for all f ∈ A, as shown in section 2 of [11]. Thus we obtain:

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Proposition 6.1. The bridge with conditional expectations Π = (D, ω, E A , E B ) is admissible for LA and LB . Much as in the statement of proposition 8.1 of [14] set ! (6.5) γ˘ A = dH ρG/H (e, y)P αy (P )dy. In the proof of proposition 8.1 of [14] (given just before the statement of the proposition, and where γ˘A is denoted just by γ A ) it is shown, with different notation, that f ω − ω˘ σf D ≤ γ˘ A LA (f ).

(6.6) Thus if f ∈ L1A then It follows that γ A

f ω − ωΦB (f )D ≤ γ˘A . ≤ γ˘ A . We have thus obtained:

Proposition 6.2. For the present class of examples, with notation as above, we have γA, γB } reach(Π) ≤ max{γ A , γ B } ≤ max{˘ A B where γ˘ is defined in equation 6.5 and γ is defined in equation 6.2 (and 4.1) above. Even more, for the case mentioned at the beginning of Section 3 (and central to [11, 12, 14]) in which G is a compact semisimple Lie group and λ is a positive integral weight, for each positive integer m let (Hm , U m ) be the irreducible representation of G with highest weight mλ. Then let B m = L(Hm ) with action α of G, and let P m be the projection on the highest weight vector in Hm . All the P m ’s will have the same α-stability group, H. As before, we let A = C(G/H). Then for each m we can construct as in Section 4 the bridge with conditional expectations, A B , Em ). From a fixed length function  on G we will obtain LipΠm = (Dm , ω m , Em Bm norms {L } which together with {LA } give meaning to the lengths of the bridges m m A A A Πm . In turn the constants γm , γ˘m , γ B , δm , δ B will be defined. A A ≤ γ˘m for each m. But Now it follows from the discussion of γ˘ A above that γm A section 10 of [14] gives a proof that the sequence γ˘m converges to 0 as m goes to A converges to 0 as m goes to ∞. Then section 12 of [14] gives ∞. It follows that γm m a proof that the sequence γ B converges to 0 as m goes to ∞. Putting together m A and γ B , we obtain: these results for γm Proposition 6.3. The reach of the bridge Πm goes to 0 as m goes to ∞. We now consider the height of Π. For δ A something quite special happens. It is easily seen that A = C(G/H) is the center of D = A ⊗ B, and so all elements of A commute with ω. Thus we can apply Proposition 4.4 to conclude that δ A = 0. In order to deal with B we use δˆB of Notation 4.5 and the discussion surrounding it. For any T ∈ B we have ΦB (ΦA (T )) = rω−1 E B (ω(rω−1 E A (ωT ω))ω) ! = dH αx (P )(dH τB (αx (P )T αx (P ))αx (P )dx ! ˘ (σT ). = dH αx (P )(trB (αx (P )T )dx = σ

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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 δˆB of our Notation 4.5 is for the present class of examples exactly the δ B of notation 8.4 of [14]. For use in the next section we here denote it by δ˜B , that is: Notation 6.4. For G, A = C(G/H), B = L(H), σ, σ ˘ , etc as above, we set δ˜B = 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, we have height(Π) ≤ δ˜B . Consequently γ A , γ B , min{δ B , δ˜B }}. length(Π) ≤ max{γ A , γ B , min{δ B , δ˜B } ≤ max{˘ We will indicate in Section 8 Latr´emoli`ere’s definition of his propinquity between compact quantum metric spaces, but it is always no larger than the length of 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 we obtain: Corollary 6.6. With notation as above, Prpq((A, LA ), (B, LB )) ≤ max{˘ γ A , γ B , min{δ B , δ˜B }}. For the case of highest weight representations discussed just above, theorem B (in our notation) converges to 0 as 11.5 of [14] gives a proof that the sequence δ˜m m 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 ∞. Consem quently Prpq((A, LA ), (B m , LB )) goes to 0 as m goes to ∞. We now treat the matricial case, beginning with the general situation in which G is some compact group. We must first specify our matrix slip-norms. This is essentially done in example 3.2 of [18] and section 14 of [14]. As discussed in Section 2, we have the actions λ and α on A = C(G/H) and B = B(H) respectively. For any n 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 define B A seminorms LA n and Ln on Mn (A) and Mn (B). It is easily verified that {Ln } and B A A B B {Ln } are matrix slip-norms. Notice that here L1 = L and L1 = L are actually Lip-norms, and so, by property 1 of Definition 5.2, for each n the null-spaces of LA n and LB n are exactly Mn . Now fix q and take n = q. From our bridge with conditional expectations Π = (D, ω, E A , E B ) we define the bridge with conditional expectations Πq = (Mq (D), ωq , EqA , EqB ) from Mq (A) to Mq (B) in the way done in Proposition 5.3. A B We then set ΦA q = ιq ⊗ Φ , and similarly for Φq , as done right after Proposition 5.3.

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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 easily B seen that Πq is admissible for {LA q } and {Lq }. We are thus in position to apply Theorem 5.5. From it and Theorem 6.5 we conclude that: Theorem 6.9. With notation as above, we have γ A , γ B }, reach(Πq ) ≤ 2q max{γ A , γ B } ≤ 2q max{˘ where γ˘A is defined by formula 6.5. Furthermore height(Πq ) ≤ 2q min{δ B , δ˜B }, where δ˜B is defined in Notation 6.4. Thus γ A , γ B , min{δ B , δ˜B }}. length(Πq ) ≤ 2q max{˘ We remark that we could improve slightly on the above theorem by using a calculation given in section 14 of [14] in the middle of the discussion there of Wu’s results. Let F ∈ Mq (A) be given, with F = {fjk }, and set ! tjk = σ ˘fjk = dH fjk (y)αy (P )dy, and let T = {tjk }. Then

! (F ωq − ωq T )(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 γqA we need to take the supremum of the norm of this expression over all x and over all F with LA q (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))P αy (P )dy}} ! ≤ dH F (e) − F (y)P αy (P )dy ! A γA. ≤ Lq (F )dH ρG/H (e, y)P αy (P )dy = LA q (F )˘ In this way we see that γqA ≤ γ˘ A , with no factor of 2q needed. We can apply Theorem 6.9 to the situation considered before Proposition 6.3 in which G is a compact semisimple Lie group, λ is a positive integral weight, and (Hm , U m ) is the irreducible representation of G with highest weight mλ for each positive m, with B m = L(Hm ). We can then form the bridge with conditional A B , Em ) that is discussed there. For any positive expectations, Πm = (Dm , ω m , Em integer q we then have the matricial version involving Mq (A), Mq (Bm ), and the corresponding bridge Πqm . On applying Theorem 6.9 together with the results A B B , γm , and δ˜m to 0, we mentioned above about the convergence of the quantities γ˘m obtain one of the two main theorems of this paper:

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Theorem 6.10. With notation as above, we have A B B ˜B γm , γm , min{δm , δm }}, length(Πqm ) ≤ 2q max{˘ A B is defined as in formula 6.5, and where δ˜m is defined as in Notation 6.4. where γ˘m q Consequently length(Πm ) 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 above bound for length(Πqm ), we do not obtain convergence to 0 that is uniform in q. I do not have a counter-example to the convergence being uniform in q, but it seems to 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 much of the notation of Section 6, but now we have two representations, (Hm , U m ) and (Hn , U n ) (where for the moment m and n are just labels). We have corresponding C*-algebras B m and B n , and projections P m and P n . We let Lm be the Lip-norm defined on B m determined by the action α and the length function  as in equation 2.1, and similarly for Ln on B n . In terms of these Lip-norms the length of any bridge from Bm to B n is defined. As in Section 3 we consider the bridge Π = (D, ω) for which D = B m ⊗ A ⊗ Bn = C(G/H, B m ⊗ B n ), and the pivot, ω, in C(G/H, B m ⊗ B n ), is defined by ω(x) = αx (P m ) ⊗ αx (P n ). We view B m as a subalgebra of D by sending T ∈ B m to T ⊗1A ⊗1Bn , and similarly for B n . As seen in Section 4, the tracial state τA ⊗ τn on A ⊗ B n determines a canonical conditional expectation, E m , from D onto B m , and the tracial state τm ⊗ τA determines a canonical conditional expectation, E n , from D onto B n . We find that for any F ∈ D we have ! (ιm ⊗ τn )(F (x)) dx, E m (F ) = G/H

and similarly for E , where here ιm is the identity map from B m to itself. From this it is easily seen that rω−1 = dm dn where dm is the dimension of Hm and similarly for dn . Thus Π = (D, ω, E m , E n ) is a bridge with conditional expectations. Then Φm (F )(x) = rω−1 E m (ωF ω)(x) = dm dn E m (ωF ω)(x). But, if we set αx (P m ⊗ P n ) = αx (P m ) ⊗ αx (P n ), we have ! E m (ωF ω)(x) = (ιm ⊗ τn )(αx (P m ⊗ P n )F (x)αx (P m ⊗ P n ))dx. n

In particular, for any T ∈ B n we have ! E m (ωT ω)(x) = αx (P m )τn (T αx (P n ))dx,

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and so, since dn τBn is the usual unnormalized trace trn on B n , we have ! Φm (T ) = dm αx (P m )trn (T αx (P n ))dx. This is essentially the formula obtained in Section 4, and is exactly the second displayed 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 B m and B n etc, we see that we can write ˘ m (σTn ). Φm (T ) = ΦB (ΦA (T )) = σ m

(7.1)

We have a similar equation for Φn (T ), and we see that we depend on the context to make clear on which of the two algebras Bm and B n we consider ΦA to be defined. As in the proof of Proposition 6.1, we can use the fact that E m and E n are equivariant, 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 for Lm and Ln . The formula (7.1) suggests the following steps for obtaining a bound on the reach of Π in terms of the data of the previous section. Let S ∈ B m , f ∈ C(G/H), and T ∈ B n . Then, for the norm of B m ⊗ B m and for any x ∈ G/H, we have (Sω − ωT )(x) = (Sαx (P m )) ⊗ αx (P n ) − αx (P m ) ⊗ (αx (P n )T ) ≤ (Sαx (P m )) ⊗ αx (P n ) − f (x)(αx (P m ) ⊗ (αx (P n )) + f (x)(αx (P m ) ⊗ (αx (P n )) − αx (P m ) ⊗ (αx (P n )T ) = Sαx (P m ) − f (x)αx (P m ) + f (x)αx (P n ) − αx (P n )T .

(7.2)

Notice that the last two norms are in B m and B n respectively. We will also use the ΦB of equation (6.3), but now, to distinguish it from the m m Φ above, we indicate that it is defined on A (and maps to B m ) by writing ΦB A . For fixed T ∈ B n let = ΦA (T ) = trn (αx (P n )T ), and then let us  us set f (x) Bm m ˘fm by equation 6.4. When we set S = ΦA (f ) = dm f (x)αx (P ). Thus S = σ substitute these into the inequality (7.2), we obtain B m n (ΦB A (f )ω − ωT )(x) ≤ (ΦA (f )(x) − f (x))αx (P ) + αx (P )(f (x) − T ). m

m

In view of the definition of f , we recognize that the supremum over x ∈ G/H of the second term on the right of the inequality sign is the kind of term involved in the supremum in the right-hand side of equality (6.2). Consequently that second term n above is no greater than γ B Ln (T ). To indicate that this comes from equality (6.2) n n m B instead of just γ B . Because ΦB ˘fm , we also recognize that the we write γA A (f ) = σ supremum over x ∈ G/H of the first term above on the right of the inequality sign is exactly (after taking adjoints to get P m on the correct side) the left hand side of inequality (6.6), where the ω there is that of Section 6. Consequently that term is A A A L (f ), where the subscript m on γ˘m indicates that P m should no greater than γ˘m be used in equation (6.5). But from the admissibility in Proposition 6.1 involving m Bm and ΦA we have ΦB A =Φ A n Lm (S) = Lm (ΦB A (f )) ≤ L (f ) ≤ L (T ). m

Notice that it follows that if T ∈ L1Bn then S ∈ L1Bm . Anyway, on taking the supremum over x ∈ G/H, we obtain A B ΦB γm + γA )Ln (T ). A (f )ω − ωT D ≤ (˘ m

n

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A B We see in this way that the distance from ωT to L1Bm ω is no bigger than γ˘m + γA . m The role of A in Theorem 4.3 is here being played by B . So to reduce confusion we will here write γnm for the γ A of Theorem 4.3, showing also the dependence on n. Thus by definition n

γnm = sup{T ω − ωΦn (T )D : T ∈ L1B m }. n We define γm similarly. Then in terms of this notation, what we have found above is that n A Bn ≤ γ˘m + γA γm We now indicate the dependence of Π on m and n by writing Πm,n . The situation just above is essentially symmetric in m and n, and so, on combining this with the first inequality of Theorem 4.3, we obtain:

Proposition 7.2. With notation as above, we have n A B B reach(Πm,n ) ≤ max{γm , γnm } ≤ max{˘ γm + γA , γ˘nA + γA }, n

m

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 weights mλ and nλ. As mentioned in the previous section, section 10 of [14] gives a proof A converges to 0 as m goes to ∞, while section 12 of [14] gives that the sequence γ˘m Bm converges to 0 as m goes to ∞. We thus see that a proof that the sequence γA we 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 again use Proposition 4.6. We calculate as follows, using equation (7.1). For T ∈ B n we have σ m (σTn )) = σ ˘ n (σ m (˘ σ m (σTn ))) Φn (Φm (T )) = Φn (˘ Thus (7.3)

T − Φn (Φm (T )) ≤ T − σ ˘ n (σTn ) + ˘ σ n (σTn ) − σ ˘ n ((σ m ◦ σ ˘ m )(σTn ) n n ≤ δ˜B LB (T ) + σ n − σ m (˜ σ m (σ n )), A

T

T

Bn for where the first term of the last line comes from Notation 6.4 and we write δ˜A n the δ˜B there. But σTn is just an element of A, and in inequality 11.2 of [14] it is shown that for any f ∈ A we have f − σ m (˘ σ m (f ) ≤ δ˜A LA (f ), m

A is defined in equation 11.1 of [14] by where δ˜m ! A = ρG/H (e, x)dm tr(P m αx (P m )) dx. (7.4) δ˜m G/H A A is denoted just by δm . Also, σ m ◦ σ ˘ m is, within our (In equation 11.1 of [14] δ˜m setting, the usual Berezin transform.) Thus we see that the second term of the last n A A n L (σT ). But LA (σTn ) ≤ LB (T ). From line of inequality (7.3) is no bigger than δ˜m all of this we see that if T ∈ L1Bn then n T − Φn (Φm (T )) ≤ δ˜B + δ˜A . A

m

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Again, the role of B in Notation 4.5 is being played here by B n , and so to reduce n for the δˆB of Notation 4.5. We then see that for confusion we will here write δˆm our present class of examples, which depend on m and n, we have n Bn A δˆm ≤ δ˜A + δ˜m .

The situation is essentially symmetric in m and n, and so, combining this with Propositions 4.6 and 7.2, we obtain: Theorem 7.4. With notation as above, we have n ˆm Bn A ˜Bm height(Πm,n ) ≤ max{δˆm , δn } ≤ max{δ˜A + δ˜m , δA + δ˜nA }.

Consequently A Bn Bm ˜Bn A ˜Bm length(Πm,n ) ≤ max{˘ γm + γA , γ˘nA + γA , δA + δ˜m , δA + δ˜nA }.

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 weights mλ and nλ. As mentioned in the previous section, theorem 11.5 of [14] Bm (in our notation) converges to 0 as m goes to gives a proof that the sequence δ˜A A ∞, while theorem 3.4 of [11] shows that the sequence δm (where it was denoted by γm ) converges to 0 as m goes to ∞. Thus when we combine this with Proposition 7.3 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 n go to ∞ simultaneously, and thus Prpq((Bm , Lm ), (B n , Ln )) goes to 0 as m and n go to ∞ simultaneously. We now consider the matricial case. For any natural number q we apply the constructions of Section 5 to obtain the bridge with conditional expectations Πqm,n = (Mq (D), ωq , Eqm , Eqn ) from Mq (B m ) to Mq (B n ). From this we then obtain the corresponding maps Φm q and Φnq . We have the actions αq of G on Mq (B m ) and Mq (B n ), much as discussed after Theorem 6.8. From these actions and the length function  we obtain the slip-norms n q Lm q and Lq . As q varies, these result in matrix slip-norms. One shows that Πm,n is m n admissible for Lq and Lq 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 highestweight representations of a semisimple Lie group, the convergence to 0 indicated above for the various constants. We obtain the second main theorem of this paper: Theorem 7.6. With notation as above, we have A Bn Bm ˜Bn A ˜Bm length(Πqm,n ) ≤ 2q max{˘ γm + γA , γ˘nA + γA , δA + δ˜m , δA + δ˜nA } A B where γ˘m is defined as in formula 6.5 while γA is the γ B of equation (6.2), and m m B B A is defined in Notation 6.4 while δ˜m is defined by equation (7.4), where δ˜A = δ˜ and similarly for n. Consequently, for the case of highest-weight representations, length(Πqm,n ) converges to 0 as m and n go to ∞ simultaneously, for each fixed q. m

m

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8. Treks Latr´emoli`ere defines his propinquity in terms of “treks”. We will not give here the precise definition (for which see definition 3.20 of [8]), but the notion is quite intuitive. A trek is a finite “path” of bridges, so that the “range” of the first bridge should be the “domain” of the second, etc. The length of a trek is the sum of the lengths of the bridges in it. The propinquity between two quantum compact metric spaces is the infimum of the lengths of all the treks between them. Latr´emoli`ere shows in [8] that propinquity is a metric on the collection of isometric isomorphism classes of quantum compact metric spaces. Notably, he proves the striking fact that if the propinquity between two quantum compact metric spaces is 0 then they are isometrically isomorphic. There is an evident trek associated with our second class of examples. In this section we will briefly examine this trek. Let the notation be as in the early parts of the previous section. Thus we have A = C(G/H), and the operator algebras B m and B n . In Section 6 we have the bridge Πm = (A ⊗ B m , ωm ) from A to B m , and the corresponding bridge Πn from A to B n . But by reversing the roles of A and B m we obtain a bridge from B m to A. We do this by still viewing A and B m as subalgebras of Dm = C(G/H, B m ), but we now let A act on the right of Dm and −1 , which is consistent we let B m act on the left. We will denote this bridge by Dm with the notation of Latr´emoli`ere at the beginning of the proof of proposition 4.7 of m −1 has the “same” conditional expectations E A and E B as those [8]. Of course Dm A to distinguish it from the E A from Dn , which we of Πm . We will write E A as Em A −1 will denote by En . Then Dm is a bridge with conditional expectations, which is m easily seen to be admissible for LB and LA . It is then easily seen that −1 length(Dm ) = length(Dm ). −1 , Dn ) then forms a trek from B m to B n , and The pair Γm,n = (Dm −1 ) + length(Dn ). length(Γm,n ) = length(Dm

From Theorem 6.5 it follows that A γm , γ B , min{δ B , δ˜B }} length(Γm,n ) ≤ max{˘ n n n + max{˘ γ A , γ B , min{δ B , δ˜B }}. m

m

m

n

A Note that δ˜m and δ˜nA do not appear in the above expression, in contrast to their appearance in the estimate in Theorem 7.4 for length(Πm,n ). This opens the possibility that in some cases length(Γm,n ) gives a smaller bound for Prpq(B m , B n ) than does length(Πm,n ), and, even more, that this might give examples for which the lengths of certain multi-bridge treks are strictly smaller that the lengths of any single-bridge treks. But I have not tried to determine if this happens for the examples in this paper. We can view the situation slightly differently as follows. Although Latr´emoli`ere does not mention it, it is natural to define the reach of a trek as the sum of the reaches of the bridges it contains, and similarly for the height of a trek. One could then give a new definition of the length of a trek as simply the max of its reach and height. This definition is no bigger that the original definition, and might be smaller. I have not examined how this might affect the arguments in [8], but I imagine that the effect would not be very significant. Anyway, for the above

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examples we see from Proposition 6.2 that we would have A B B reach(Γm,n ) ≤ max{˘ γm , γA } + max{˘ γnA , γA }, m

n

so that the bound for reach(Πm,n ) given in Proposition 7.2 is no bigger than that above for reach(Γm,n ). But from Theorem 6.5 we see that Bm Bn + δ˜A height(Γm,n ) ≤ δ˜A m Bm (where δ˜A = δ˜B ), and this can clearly be less than the right-most bound for height(Πm,n ) given in Theorem 7.4.

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[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 947203840 E-mail address: [email protected]

Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13513

Structure and applications of 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 always worked over the complex numbers, and nobody paid much attention to real C ∗ -algebras. Over the last thirty years, that situation has changed, and it’s become apparent that real C ∗ -algebras have a lot of extra structure not evident from their complexifications. At the same time, interest in real C ∗ -algebras has been driven by a number of compelling applications, for example in the classification of manifolds of positive scalar curvature, in representation theory, and in the study of orientifold string theories. We will discuss a number of interesting examples of these, and how the real Baum-Connes conjecture plays an 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 Hilbert space. Remark 1.2. There is an equivalent abstract definition: a real C ∗ -algebra is a real Banach ∗-algebra A satisfying the C ∗ -identity a∗ a = a2 (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 have a slightly different emphasis from the one presented here. Theorem 1.3 (“Schur’s Lemma”). Let π be an irreducible representation of a real C ∗ -algebra A on a real Hilbert space H. Then the commutant π(A) of the representation 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, real Baum-Connes conjecture, assembly map, positive scalar curvature, Frobenius-Schur indicator. This work was partially supported by NSF grant DMS-1206159. The author would like to thank Jeffrey Adams and Ran Cui for helpful discussions about Section 3.2 and Patrick Brosnan for helpful discussions about Example 2.9. c 2016 American Mathematical Society

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Proof. Since π(A) is itself a real C ∗ -algebra (in fact a real von Neumann algebra), it is enough to show it is a division algebra over R, since by Mazur’s Theorem [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-adjoint element of π(A) . If p ∈ π(A) is a spectral projection of x, then pH and (1 − p)H are 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 · 1 with λ ∈ R, and similarly, yy ∗ is a real multiple of 1. Since the spectra of y ∗ y and yy ∗ must coincide except perhaps for 0, y ∗ y = yy ∗ = λ = y2 and either y = 0 or  else y is invertible (with inverse y−2 y ∗ ). So π(A) is a division algebra. Corollary 1.4. The irreducible ∗-representations of a real C ∗ -algebra can be classified 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 left translation.) Given a real C ∗ -algebra A, its complexification AC = A + iA is a complex C -algebra, and comes with a real-linear ∗-automorphism σ with σ 2 = 1, namely complex 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 with θ 2 = 1. Thus we can classify real C ∗ -algebras by classifying their complexifications and 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 a real C ∗ -algebra? Equivalently, does it admit a ∗-antiautomorphism θ with θ 2 = 1? The answer to this in general is no. For example, Connes [11, 12] showed that there are factors not anti-isomorphic to themselves, hence admitting no real form. Around the same time (ca. 1975), Philip Green (unpublished) observed that a stable continuous-trace algebra over X with Dixmier-Douady invariant δ ∈ H 3 (X, Z) cannot be anti-isomorphic to itself unless there is a self-homeomorphism of X sending δ to −δ. Since it is easy to arrange for this not to be the case, there are continuous-trace algebras not admitting a real form. By the way, just because a factor is anti-isomorphic to itself, that does not mean it has an self-antiautomorphism of period 2, and so it may not admit a real form. Jones constructed an example in [33]. Secondly we have: Problem 1.6. Given a complex C ∗ -algebra A that admits a real form, how many distinct such forms are there? Equivalently, how many conjugacy classes are there of ∗-antiautomorphisms θ with θ 2 = 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 this one can easily see that K(H) and B(H), the compact and bounded operators on a separable infinite-dimensional Hilbert space, each have two distinct real forms. For example, K(H) is the complexification of both K(HR ) and K(HH ). This makes the 1 Historical note: According to [37], Mazur presented this theorem in Lw´ ow in 1938. Because of space limitations in Comptes Rendus, he never published the proof, but his original proof is reproduced in [73] as well as in [41], which also includes a copy of Mazur’s original hand-typed manuscript, with the proof included.

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following theorem due independently to Størmer and to Giordano and Jones all the more surprising and remarkable. Theorem 1.7 ([20, 22, 67]). The hyperfinite II1 factor R has a unique real form. This has a rather surprising consequence: if RR is the (unique) real hyperfinite II1 factor, then RR ⊗R H ∼ = RR (since this is a real form of R ⊗ M2 (C) ∼ = R). In fact 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 and automorphisms. Thus we get the following classification theorem. Theorem 1.9 ([4, Theorem 9.1]). Commutative real C ∗ -algebras are classified by pairs consisting of a locally compact Hausdorff space X and a self-homeomorphism τ of X satisfying τ 2 = 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 becomes a ∗-automorphism of AC (since the order of multiplication is immaterial) and thus comes from a self-homeomorphism τ of X satisfying τ 2 = 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 a commutative real C ∗ -algebra.  One could also ask about the classification of real AF algebras. This amounts to answering Problems 1.5 and 1.6 for complex AF algebras A (inductive limits of finite dimensional C ∗ -algebras). Since complex AF algebras are completely classified 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 Giordano [21], but the answer is considerably more complicated than in the complex case. Of course this is hardly surprising, since we already know that even the simplest noncommutative finite dimensional complex C ∗ -algebra, M2 (C), has two two distinct real structures. Giordano also showed that his invariant is equivalent to one introduced by Goodearl and Handelman [26]. We will not attempt to give the precise 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 of M2 (C), by looking at KO2 .) Also, unlike the complex case, one usually has to deal with 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 a separable type I (complex) C ∗ -algebra, with primitive ideal space Prim A (equipped  → Prim A, sending the with the Jacobson topology), then the natural map A equivalence class of an irreducible representation π to its kernel ker π, 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 σ 2 = 1, or

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alternatively, with a linear ∗-antiautomorphism θ with θ 2 = 1. We can think of θ as an isomorphism θ : A → At (where At is the opposite algebra, the same complex vector space with the same involution ∗, but with multiplication reversed) such that the 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 a map A → B(H), and obviously, π induces a related map (which as a map of sets is exactly the same as π) π t : At → B(H)t . Composing with θ and with the standard ∗-antiautomorphism τ : B(H)t → B(H) (the “transpose map”) coming from the identification of B(H) as the complexification of B(HR ), we get a composite map A

θ

πt

/ At

/ B(H)t

/3 B(H) .

τ

θ∗ (π)

One can also see that doing this twice brings us back where we started, so we have seen: Proposition 1.10. If A is a complex C ∗ -algebra (for our purposes, separable 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 A

θ

/ At

θ∗ (π)t

/ B(H)t

τ

/3 B(H)

or θ∗ (π)t

A

θ

/ At

θ

/A

π

/ B(H)

τ

)

/ B(H)t

τ

but θ ◦ θ and τ ◦ τ are each the identity, so this is just π again.

/2 B(H) , 

 is just Note that in the commutative case A = C0 (X), the involution θ∗ on A the original involution on X. With these preliminaries out of the way, we can now begin to analyze the structure of (separable) real type I C ∗ -algebras. Some of this information is undoubtedly known to experts, but it is surprisingly hard to dig it out of the literature, so we will 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-dimensional real C ∗ -algebras, which are just semisimple Artinian algebras over R. The interest in this case comes from the real group rings RG of finite groups G, which are precisely of this type. A convenient reference for the real representation theory of finite groups is [64, §13.2]. A case which is not much harder is that of real representation theory of compact groups. In this case, the associated real C ∗ -algebra is infinite-dimensional in general, but splits as a (C ∗ -)direct sum of (finitedimensional) simple Artinian algebras over R. This case is discussed in great detail in [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 ∗ -algebra A are of three types. How does this type classification relate to the involution 0C ? The answer (which for the finite group case appears in of Proposition 1.10 on A [64, §13.2]) is given as follows:

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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 of real type, then we get an irreducible representation πC of AC on HC by complexifying, and the class of this irreducible representation πC is fixed by the involution of Proposition 1.10. If π is of complex type, then H can be made into a complex Hilbert space Hc (whose complex dimension is half the real dimension of H) either via the action of π(A) or via the conjugate of this action, and we get two distinct irreducible representations of AC on Hc which are interchanged under the involu0C . Finally, if π is of quaternionic type, then H can tion of Proposition 1.10 on A be made into a quaternionic Hilbert space via the action of π(A) . After tensoring with C, we get a complex Hilbert space HC whose complex dimension is twice the quaternionic dimension of H, and we get an irreducible representation πC of AC on HC 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 the compact operators on H, and in particular, there is an ideal m in A which maps onto the trace-class operators. Thus π has a well-defined “character” χ on m in the sense 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 χ2 . When π is of complex type, the two complex irreducible extensions of π have characters on m which are non-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 some examples. First let A = CR∗ (Z), the free real C ∗ -algebra on one unitary u. The trivial representation u → 1 is of real type and complexifies to the trivial reprethe sign representation u → −1 is of real sentation of AC = C ∗ (Z). Similarly   cos φ sin φ 2 type. The representation u → on R (φ not a multiple of π) − sin φ cos φ is of complex type. Note   that this representation is equivalent  tothe one given cos φ − sin φ 0 1 by u → since these are conjugate under . This represin φ cos φ 1 0 sentation class corresponds to a pair of inequivalent irreducible representations of AC = C ∗ (Z) on C, one given by u → eiφ and one given by u → e−iφ . The involution 0C sends one of these to the other. on A 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 to the quaternions with the same name) which is of quaternionic type. Complexifying gives two copies of the unique 2-dimensional irreducible complex representation of AC . Note incidentally that D8 , the dihedral group of order 8, and Q8 have the same complex representation theory. Keeping track of the types of representations enables us to distinguish the two groups. Proof of Theorem 1.11. A lot of this is obvious, so we will just concentrate on the parts that are not. If π is of real type, its commutant is R and its complexification πC has commutant C and is thus irreducible. The class of πC is fixed by the involution, since for a ∈ A, θ∗ (πC )(a) = τ ◦ πCt ◦ θ(a) = τ ◦ πCt (a∗ ) = τ (π t (a∗ )) = (π(a)t )t = π(a).

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If π is of complex type, we need to show that we get two distinct irreducible repre0C . If this were sentations of AC which are interchanged under the involution on A not the case, then π viewed as an irreducible representation on a complex Hilbert space Hc (via the identification of π(A) with C) would extend to an irreducible complex representation (let’s call it π c ) of AC which is isomorphic to θ∗ π c . Now if a + ib ∈ AC , where a, b ∈ A, then σ(a + ib) = a − ib and θ(a + ib) = a∗ + ib∗ . So θ∗ (π c )(a+ib) = τ ◦πCt (a∗ +ib∗ ) = π(a)+iπ(b), since for operators on Hc , τ (T ∗ ) = T , the conjugate operator. The complexification of H is canonically identified with Hc ⊕ Hc , so the complexification πC of π is thus identified with π c ⊕ θ∗ (π c ). If this were isomorphic to π c ⊕ π c , then its commutant would be isomorphic to M2 (C). But the commutant of πC must be the complexification of the commutant of π, so this is impossible. If π is of quaternionic type, its commutant is isomorphic to H, which complexifies to M2 (C). That means the complexification of π has commutant M2 (C), and thus the complexification of π is unitarily equivalent to a direct sum of two copies of an irreducible representation πC of AC . That the class of πC is fixed by 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 complexification and so the characters of πC and of π coincide on m. (Complexification of operators preserves traces.) If π is of quaternionic type, its complexification is equivalent to two copies of πC , so on m, the character χ of π is the character of the complexification of π and so coincides with twice the character χC of πC , which is 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 π c and of θ∗ (π c ) (on m) are complex conjugates of one another, and add up to the character of πC ∼ = π c ⊕ θ∗ (π c ). But complexification of an operator doesn’t change its trace, so πC and π have the same character on m, and the characters of π c and of θ∗ (π c ) add up to χ on m.  Remark 1.13. One can also phrase the results of Theorem 1.11 in a way more familiar from group representation theory. Let A be a real C ∗ -algebra and let π be an irreducible representation of AC on a complex Hilbert space HC such that the class of π is fixed under the involution of Proposition 1.10. Then π is associated to an irreducible representation of A of either real or quaternionic type. To tell which, observe that one of two possibilities holds. The first possibility is there is an A-invariant real structure on HC , i.e., HC is the complexification of a real Hilbert space H which is invariant under A, in which case π is of real type. This condition is equivalent to saying that there is a conjugate-linear map ε : HC → HC commuting with 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 on HC via the complex structure and let j act by ε, then since ε is conjugate-linear, i and j anticommute, and so we get an A-invariant structure of a quaternionic vector space on HC , whose dimension over H is half the complex dimension of HC . In this case, π clearly has quaternionic type. This point of view closely follows the presentation 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-Schur indicator χ(g 2 ) dg, which is 1 for representations of real type, 0 for representations

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of complex type, and −1 for representations of quaternionic type. But since this criterion is based on tensor products for representations for groups, it doesn’t seem to 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 of type I. Definition 2.1. Let A be a real C ∗ -algebra with complexification AC . We say A has continuous trace if AC has continuous trace in the sense of [13, §4.5], that 0C are is, if elements a ∈ (AC )+ for which π → Tr π(a) is finite and continuous on A dense in (AC )+ . Theorem 2.2. Any non-zero postliminal real C ∗ -algebra (this is equivalent to being type I, even in the non-separable case — see [49, Ch. 6] or [61, §4.6]) has a non-zero ideal of continuous trace. Proof. That AC has a non-zero ideal I of continuous trace is [13, Lemma 4.4.4]. 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 → Tr π(a) is finite and continuous, then π → Tr π(σ(a)) = Tr π(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 composition series (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 special attention on real C ∗ -algebras with continuous trace. To such an algebra A (which we will assume is separable to avoid certain pathologies, such as the possibility that the spectrum might not be paracompact) is associated a Real space (X, ι) in 0C and an the sense of Atiyah [5], that is, a locally compact Hausdorff space X = A involution ι on X defined by Proposition 1.10. The problem then arises of classifying all the real continuous-trace algebras associated to a fixed Real space (X, ι). There is always a unique such commutative real C ∗ -algebra, given by Theorem 1.9. When one considers noncommutative algebras, ∗-isomorphism is too fine for most purposes, and the most natural equivalence relation turns out to be Morita equivalence, which works for real C ∗ -algebras just as it does for complex C ∗ -algebras. Convenient references for the theory of Morita equivalence (in the complex case) are [53, 54]. A Morita equivalence between real C ∗ -algebras A and B is given by an A-B bimodule X with A-valued and B-valued inner products, satisfying a few simple axioms: (1) x, yA z = xy, zB and a · x, yB = x, a∗ · yB , x · b, yA = x, y · b∗ A for x, y, z ∈ X and a ∈ A, b ∈ B. (2) The images of the inner products are dense in A and in B. 1/2 1/2 (3) x, xA A = x, xB B is a norm on X, X is complete for this norm, and A and B act continuously on X by bounded operators.

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The real continuous-trace algebras with spectrum (X, ι) have been completely classified by Moutuou [46] up to spectrum-fixing Morita equivalence, at least in the 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). 0C , which is Hausdorff since A has continuous trace, and let In other words X = A ι be the involution on X defined by Proposition 1.10. The sign choice of A is the map α : X ι → {+, −} attaching a + sign to fixed points of real type and a − sign to fixed points of quaternionic type. (Of course, ι acts freely on X  X ι , and the orbits of this action correspond to the pairs of conjugate representations of complex type.) Note that if we give {+, −} the discrete topology, then it is easy to see that α is continuous2 , so it is constant on each connected component of X ι . Incidentally, the name sign choice for this invariant comes from a physical application we will see in Section 3, where it is related to the signs of O-planes in string theory. The other invariant of a (separable) real continuous-trace algebra is the DixmierDouady invariant. For a complex continuous-trace algebra with spectrum X, this is a class in H 2 (X, T ) (sheaf cohomology), where T is the sheaf of germs of continuous T-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 the short 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 sequence in sheaf cohomology coming from (2.1) gives H 2 (X, T ) ∼ = H 3 (X, Z), and indeed, the Dixmier-Douady invariant is usually presented as a class in H 3 . However, for purposes of dealing with real continuous-trace algebras, we need to take the involution ι on X (and on T ) into account. This will have the effect of giving a Dixmier-Douady invariant in an equivariant cohomology group Hι2 (X, T ) defined by Moutuou [45], who denotes it HR2 (X, T ) (with the ι understood). The HR• groups are similar to, but not identical with, the Z/2-equivariant cohomology groups H • (X; Z/2, F) as defined in Grothendieck’s famous paper [28, Ch. V]. The precise 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. Then if T is the sheaf of germs of T-valued continuous functions on X, equipped with the involution induced by the involution (x, z) → (ι(x), z) on X × T, then Moutuou’s HR2 (X, T ) coincides with H 2 (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 is an edge homomorphism Hι2 (X, T ) → H 2 (X; Z/2, T ) (which is not necessarily an isomorphism). 2 One way to see this is to apply the part of Theorem 1.11 about characters. If e ∈ A + is a local minimal projection near x ∈ X, then Tr π(e) = 1 if π is close to x and α(x) = + and Tr π(e) = 2 if π is close to x and α(x) = −.

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Proof. In order to deal with quite general topological groupoids, Moutuou’s ˇ construction. But in definition of Hι• (X, F) in [45] uses simplicial spaces and a Cech our 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 between ˇ Cech cohomology and sheaf cohomology for paracompact spaces [24, Th´eor`eme II.5.10.1], it reduces here to ordinary sheaf cohomology.  Grothendieck’s equivariant cohomology groups H • (X; Z/2, F) are the derived functors of the equivariant section functor X → Γ(X, F)ι . Moutuou’s groups are generally smaller. A few examples will clarify the notion, and also explain the difference between Grothendieck’s functor and Moutuou’s. (1) If ι is trivial on X, the involution on T is just complex conjugation, and Hι• (X, T ) can be identified with H • (X, T ι ) = H • (X, Z/2). Note, for example, 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 even degree, whereas Moutuou’s Hι• (pt, T ) is just H • (pt, Tι ) = Z/2, concentrated in degree 0. (2) If ι acts freely, so that π : X → Y is a 2-to-1 covering map, Hι• (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 associated to (2.1) and [28, Corollaire 3, p. 205]. Here Z is a locally constant sheaf obtained 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-Douady invariant of a separable real C ∗ -algebra A. Without loss of generality, we can tensor A with KR , which doesn’t change the algebra up to spectrum-fixing Morita equivalence. Then AC becomes stable, and is locally, but not necessarily globally, isomorphic to C0 (X, K). By paracompactness (here we use separability of A, which implies X is second countable and thus paracompact), there is a locally finite covering {Uj } of X such that AC is trivial over each {Uj }. We can also assume each Uj is ˇ ι-stable. The trivializations of AC over the Uj give a Cech cocycle in H 1 ({Uj }, PU), given by the “patching data” over overlaps Uj ∩ Uk . Here PU is the sheaf of germs of P U -valued continuous functions, since P U is the automorphism group of K. The image of this class in H 1 (X, PU) ∼ = H 2 (X, T ) is the complex Dixmier-Douady invariant. Here we use the long exact cohomology sequence associated to the sequence of sheaves (2.2)

1 → T → U → PU → 1,

where again the middle sheaf is fine since the infinite unitary group (with the strong or weak operator topology) is contractible. In our situation, there is a little more structure because AC was obtained by complexifying A. So we have the conjugation σ on AC , which induces the involution ι on X and on the sheaves U, PU , and T over X. Furthermore, the cocycle of the patching data must be ι-equivariant, and so defines the real Dixmier-Douady invariant, which is its coboundary in Hι2 (X, T ). Theorem 2.7 (Moutuou [46]). The spectrum-fixing Morita equivalence classes of real continuous-trace algebras over (X, ι) form a group (where the group operation

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comes from tensor product over X) which is isomorphic to H 0 (X ι , Z/2)⊕Hι2 (X, T ) via the map sending an algebra A to the pair consisting of its sign choice and real Dixmier-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 of how to translate it into this form, see [15, §3.3]. Example 2.9. Here are three examples, that might be relevant for physical applications, that show how one computes the Brauer group of Theorem 2.7 in practice. In all cases we will take X to be a K3-surface (a smooth simply connected complex projective algebraic surface with trivial canonical bundle) and the involution ι to be holomorphic (algebraic). (1) Suppose the involution ι is holomorphic and free. In this case the quotient Y = X/ι is an Enriques surface (with fundamental group Z/2) and ι reverses the sign of a global holomorphic volume form. (See for example [47, §1].) There is no sign choice invariant since the involution is free, and thus all representations must be of complex type. The Dixmier-Douady invariant lives in (twisted) 3-cohomology of the quotient space Y . By Poincar´e duality, H 3 (Y, Z) ∼ = H1 (Y, Z), but since X is 1-connected, the classifying map Y → BZ/2 = RP∞ is a 2-equivalence (an isomorphism on π1 and surjection on π2 ) and induces an isomorphism on twisted H1 . group So H 3 (Y, Z) ∼ (Z/2, Z) = 0. So the = H1 (Y, Z) ∼ = H1 (BZ/2, Z) ∼ = H1 Dixmier-Douady invariant is always trivial in this case. (2) If ι is a so-called Nikulin involution (see [44, 69]), then X ι consists of 8 isolated fixed points. Let Z = (X  X ι )/ι. By transversality, the complement X  X ι of the fixed-point set is still simply connected, so π1 (Z) ∼ = 2 (X  Z/2 and the map Z → BZ/2 is a 2-equivalence. We have Hι,c X ι, T ) ∼ = Hc3 (Z, Z), and by Poincar´e duality, Hc3 (Z, Z) ∼ = H1 (Z, Z) ∼ = H1 (BZ/2, Z) = 0. From the long exact sequence 2 (2.3) H 1 (X ι , Z/2) = 0 → Hι,c (X  X ι , T )

→ Hι2 (X, T ) → H 2 (X ι , T ) = 0, (see [24, Th´eor`eme II.4.10.1]) we see that Hι2 (X, T ) = 0 and the DixmierDouady invariant is always trivial in this case. However, there are many possibilities for the sign choice since H 0 (X ι , Z/2) ∼ = (Z/2)8 . (3) It is well known that there are K3-surfaces X with a holomorphic map f : 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 holomorphic involution ι having C as fixed-point set. We want to compute the Brauer group of real continuous-trace algebras over (X, ι). Since X ι = C is connected, there are only two possible sign choices, and algebras with sign choice − are obtained from those with sign choice + simply by tensoring with H. So we may assume the sign choice on the fixed set is a +. The calculation of the possible Dixmier-Douady invariants is complicated and uses Theorem 2.5. Theorem 2.10. Let X be a K3-surface and ι a holomorphic involution on X with fixed set X ι = C a smooth projective complex curve of genus 10 and with quotient space Y = X/ι = CP2 . Then Hι2 (X, T ) = 0.

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Proof. By Theorem 2.5, Hι2 (X, T ) ∼ = H 2 (CP2 , F), where the sheaf F is Z/2 over C and is locally isomorphic to T over the complement. By [24, Th´eor`eme II.4.10.1], we obtain an exact sequence 2 (X  C, T ) → Hι2 (X, T ) → H 2 (C, Z/2) (2.4) H 1 (C, Z/2) → Hι,c 3 (X  C, T ) → Hι3 (X, T ) → H 3 (C, Z/2) = 0. → Hι,c j Note that (X  X ι )/ι ∼ (X  C, T ) ∼ = CP2  C. Thus in (2.4), Hι,c = Hcj+1 (CP2  2 2 ∼ C, Z) = H3−j (CP  C, Z). Since C ⊂ CP is a curve of degree 6, the map H 2 (CP2 , Z) → H 2 (C, Z) induced by the inclusion is multiplication by 6, and we find from the long exact sequence 6

→ H 2 (C, Z) → Hc3 (CP2  C, Z) → 0 H 2 (CP2 , Z) − that Hc3 (CP2  C, Z) ∼ = H1 (CP2  C, Z) ∼ = Z/6. This implies that for j ≤ 1, 2 Hj (CP  C, Z) will coincide with the Z-homology of a lens space with fundamental group Z/6, or with Hjgroup (Z/6, Z) = Z/2, j even, and 0, j odd. Hence (2.4) reduces to δ 0 → Hι2 (X, T ) → Z/2 − → Z/2 → Hι3 (X, T ) → 0, and Hι2 (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 Hι2 (X, T ) with I2,0 2 in the spectral sequence p q p+q Ip,q (X; Z/2, T ) 2 = H (Y, H (Z/2, T )) ⇒ H

of [28, Th´eor`eme 5.2.1]. We will examine this spectral sequence as well as the other one in that theorem, p q p+q IIp,q (X; Z/2, T ). 2 = H (Z/2, H (X, T )) ⇒ H

First consider I•,• 2 . We have a short exact sequence of sheaves (2.5)

1 → (T )XC → T → (T )C → 1,

and ι acts trivially on C and freely on X C. Thus H q (Z/2, (T )XC ) = 0 for q > 0 [28, Corollaire 3, p. 205]. So from the long exact cohomology sequence derived from (2.5), H q (Z/2, T ) = H q (Z/2, (T )C ) is supported on C for q > 0. On C, the action of ι is by complex conjugation, and so one easily sees that H q (Z/2, T ) = H q (Z/2, (T )C ) = (Z/2)C for q > 0 even, 0 for q odd. So for q > 0, Ip,q 2 vanishes for q odd and is H p (C, Z/2) for q even, which is Z/2 for p = 0 or 2, (Z/2)20 for p = 1, 0,1 2,0 and 0 for p > 2. In particular, I0,1 2 = 0, so d2 : I2 → I2 vanishes and so the edge 2 2 homomorphism Hι (X, T ) → H (X; Z/2, T ) is injective. Furthermore, I1,1 2 = 0 and 0,2 2 I2 = Z/2, so H (X; Z/2, T ) is finite and   2   H (X; Z/2, T ) ≤ 2 · Hι2 (X, T ) . Equality will hold if and only if the map d3 : I0,2 = Z/2 → I3,0 = Hι3 (X, T ) is 3 3 trivial. 0 Now consider the other spectral sequence II•,• 2 . We have H (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 the involution ι on X and complex conjugation, which corresponds to multiplication

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by −1 on R and Z. Thus as a Z/2 module, the group on the left in (2.6) is really Z, Z with the non-trivial action. On the q = 0 row, we use equation (2.6) and the fact that higher cohomology of a finite group with coefficients in a real vector space has to vanish to obtain that  p odd, p,0 p p+1 ∼H ∼ 0, II2 = H (Z/2, C(X, T)) = (Z/2, Z) = Z/2, p even > 0. For q > 0, we know that H q (X, T ) ∼ = H q+1 (X, Z), which will be nonzero (and torsion-free) only for q = 1 and q = 3. Again, since the action on Z/2 on the constant sheaf Z in (2.1) is by multiplication by −1, the action of Z/2 on H 3 (X, T ) ∼ = H 4 (X, Z) ∼ = Z is by multiplication by −1. The case of H 1 (X, T ) ∼ = H 2 (X, Z) ∼ = Z22 is more complicated because we also have the action of ι on H 2 (X, Z), which has fixed set of rank 1 [47, p. 595]. So we need to determine the structure of H 2 (X, Z) as a Z/2-module. The action of ι on H 2 has to respect the intersection pairing, with respect to which H 2 (X, Z) splits (non-canonically) as E8 ⊕ E8 ⊕ H ⊕ H ⊕  H, where  0 1 2 E8 is the E8 lattice and H is a hyperbolic plane (Z with form given by ). 1 0 2 ι ∼ Since H (X, Z) = Z, one can quickly see that the only possibility is that ι acts by −1 on both E8 summands and on two of the H summands, and interchanges the generators of the other H summand. Our action here of Z/2 is reversed from this, so as Z/2-module, H 1 (X, T ) ∼ = Z20 ⊕ H. Since H p (Z/2, Z) is non-zero only p for p even and H (Z/2, H) = 0 for p > 0 (by simple direct calculation, or else by Shapiro’s Lemma, since H as a Z/2-module is induced from Z as a module for = H 1 (Z/2, Z20 ⊕ H) = 0. Since we already the trivial group), we find that II1,1 2 0,2 2,0 computed that II2 = 0 and II2 = Z/2, we see that |H 2 (X; Z/2, T )| ≤ 2. It will 2,0 be 0 only if d2 : II0,1 is non-zero. Putting everything together, we finally 2 → II2 see that the only possibilities for the two spectral sequences are as in Figures 1 and 2. Comparing the (dotted) diagonal lines of total degrees 2 and 3 in the two figures,  we conclude that Hι2 (X, T ) and Hι3 (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 curvature and to orientifold string theories in physics. We should mention that other physical applications have appeared in the theory of topological insulators in condensed matter 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-Connes conjecture. The (topological) K-theory of real C ∗ -algebras is of course a special case of topological K-theory of real Banach algebras. As such it has all the usual properties, such as homotopy invariance and Bott periodicity of period 8. A convenient reference is [63]. A nice feature of the K-theory of real continuous-trace C ∗ -algebras is that it unifies all the variants of topological K-theory (for spaces) that have appeared in the literature. This includes of course real K-theory KO, complex K-theory K, and symplectic K-theory KSp, but also Atiyah’s “Real” K-theory KR [5], Dupont’s symplectic analogue of KR [16], sometimes called KH, and the self-conjugate

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qO 3

0

0

0

0

2

Z/2

(Z/2)20

Z/2

0

1

0

0

0

0

0

?

Z20

Hι2 (X, T )

Hι3 (X, T )

·

0

1

2

3

/p

Figure 1. The first Grothendieck spectral sequence I•,• 2 qO 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 Ktheory of the commutative real C ∗ -algebra C0 (X, τ ) of Theorem 1.9. KSC • (X) is KR• (X ×S 1 ), where S 1 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 (but vanishing Dixmier-Douady invariant) is “KR-theory with a sign choice” as defined in [14], and the K-theory of a stable real continuous-trace with no sign choice but

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a nontrivial Dixmier-Douady invariant is what has generally been called “twisted K-theory” (of either real or complex type, depending on the types of the irreducible representations of the algebra). See [55, 57] for some of the original treatments, as well as [6, 34] for more modern approaches. 3.1. Positive scalar curvature. A first area where real C ∗ -algebras and their K-theory plays a significant role is the classification of manifolds of positive scalar curvature. The first occurrence of real C ∗ -algebras in this area is implicit in an observation of Hitchin [31], that if M is a compact Riemannian spin manifold of dimension n with positive scalar curvature, then the KOn -valued index of the Dirac operator on M has to vanish. For n divisible by 4, this observation was not new and goes back to Lichnerowicz [40], but for n ≡ 1, 2 mod 2, a new torsion obstruction shows up that cannot be “seen” without real K-theory. The present author observed that there is a much more extensive obstruction theory when M is not simply connected. Take the fundamental group π of M , a countable discrete group. Complete the real group ring Rπ in its greatest C ∗ -norm to get the real group C ∗ -algebra A = CR∗ (π). (Alternatively, one could use the ∗ (π), the completion of the group ring for reduced real group C ∗ -algebra Ar = CR,r 2 its left action on L (π). For present purposes it doesn’t much matter.) Coupling  ×π A over M , one the Dirac operator on M to the universal flat CR∗ (π)-bundle M gets a Dirac index with values in KOn (A), which must vanish if M has positive scalar curvature. Thus we have a new source of obstructions to positive scalar curvature. As shown in [56, 58], this KOn (A)-valued index obstruction can be computed to be μ ◦ f∗ (αM ), where αM ∈ KOn (M ) is the “Atiyah orientation” of M , i.e., the KO-fundamental class defined by the spin structure, f : M → Bπ is a classifying  → M , and μ : KOn (Bπ) → KOn (A) is the “real map for the universal cover M assembly 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 reduced assembly map μ (what one gets after pulling out the contribution from the trivial group, i.e., the Lichnerowicz and Hitchin obstructions) is precisely the image in KO• (Rπ) of the 2-torsion in KO• (Rπ2 ), π2 ⊆ π a Sylow 2-subgroup. This lives in degrees 1 and 2 mod 4 and comes from the irreducible representations of π2 of real and quaternionic type, in the sense that we explained in Theorem 1.11. So far the obstructions detected by μ are the only known obstructions to positive scalar curvature on closed spin manifolds of dimension > 4 with finite fundamental group. The problem of existence or non-existence of positive scalar curvature on a spin manifold can be split into two parts, one “stable” and one “nonstable.” Stability here refers to taking the product with enough copies of a “Bott manifold” Bt8 , a simply connected closed Ricci-flat 8-manifold representing the generator of Bott periodicity. Since index obstructions in K-theory of real C ∗ -algebras live in groups which 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 it was shown in [60] that the stable conjecture (M × Btk admits a metric of positive 3 The relationship is this. Let Eπ denote the universal proper π-space and let Eπ denote the universal free π-space. These coincide if and only if π is torsion-free. The Baum-Connes assembly map is defined on KO•π (Eπ) whereas our map is defined on KO•π (Eπ) = KOn (Bπ). Since Eπ is a proper π-space, there is a canonical π-map Eπ → Eπ (unique up to equivariant homotopy) and so our μ factors through the Baum-Connes assembly map, and agrees with it if π has no torsion.

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scalar curvature for sufficiently large k if and only if μ◦f∗ (αM ) vanishes) holds when π is finite. Stolz has extended this theorem as follows: for a completely general closed spin manifold M n with fundamental group π, M × Btk admits a metric of positive 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-Connes ∗ )) holds for π. In fact, Baum-Connes can assembly map μ : KO•π (Eπ) → KO• (CR,r be weakened here in two ways — one only needs injectivity of μ, not surjectivity, ∗ and one can replace CR,r by the full C ∗ -algebra CR∗ . Since the full C ∗ -algebra surjects onto the reduced C ∗ -algebra, injectivity of the assembly map for the full C ∗ -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 to play a fundamental role here, it is worth remarking that the real and complex versions of the Baum-Connes conjecture are actually equivalent [8, 62]. Thus the real Baum-Connes conjecture holds in the huge number of cases where the complex Baum-Connes conjecture has been verified.

3.2. Representation theory. Since we have already mentioned the real Baum-Connes conjecture, it is worth mentioning that this, as well as the general theory of real C ∗ -algebras, has some relevance to representation theory. Suppose G is a locally compact group (separable, say, but not necessarily discrete). The real group C ∗ -algebra CR∗ (G) is the completion of the real L1 -algebra (the convolution 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 ∗ -algebra ∗ (G) inside the reduced C ∗ -algebra Cr∗ (G). ComC ∗ (G), and similarly we have CR,r ∗ ∗ puting the structure of CR (G) or of CR,r (G) gives us more information than just computing the structure of their complexifications. For instance, it gives us the type classification of the representations, as we saw in Theorem 1.11 and Example 1.12. The real Baum-Connes conjecture, when it’s known to hold, gives us at least ∗ (G) (its K-theory). partial information on the structure of CR,r Here are some simple examples (where the real structure is not totally uninteresting) to illustrate these ideas. Example 3.1. Let G = SU (2). As is well known, this has (up to equivalence) irreducible complex representation of each positive integer dimension. It is 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 dim Vk = 2k + 1. The character χk of Vk is given on a maximal torus by , which is real-valued, and thus all the representations must have eiθ → sin(2k+1)θ sin θ real or quaternionic type. In fact, Vk is of real type if k is an integer and is of quaternionic type if k is a half-integer. (That’s because V1 is the complexification of the covering map SU (2) → SO(3), while V1/2 acts on the unit quaternions, and all the other representations can be obtained from these by taking tensor products and decomposing. A tensor product of real representations is real, and a tensor product of a real representation with a quaternionic one is quaternionic.) Indeed  if one computes the Frobenius-Schur indicator G χk (g 2 ) dg for Vk using the Weyl

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integration formula, one gets ! !  1 2π  4kiθ χk (g 2 ) dg = + e4(k−1)iθ + · · · + e−4kiθ sin2 θ dθ e π 0 G ! 2π    1 e4kiθ + e4(k−1)iθ + · · · + e−4kiθ 2 − e2iθ − e−2iθ dθ. = 4π 0 If k is an integer, we get ! 2π    1 (· · · + e4iθ ) + 1 + (e−4iθ + · · · ) 2 − e2iθ − e−2iθ dθ = 1, 4π 0 where the terms in small parentheses are missing if k = 0, while if k is a half-integer, we get ! 2π    1 2 − e2iθ − e−2iθ dθ = −1. · · · + e2iθ + e−2iθ + · · · 4π 0 This confirms the type gave earlier. * classification we* Thus CR∗ (G) ∼ = k∈N M2k+1 (R) ⊕ k= 12 , 32 ,··· Mk+ 12 (H). In particular, we see that KO• (CR∗ (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 appear only in degrees divisible by 4. Conversely, if one had some independent method of computing KO• (CR∗ (G)), it would immediately tell us that G has no irreducible representations of complex type, and infinitely many representations of both real and of quaternionic type. Example 3.2. Let H be the compact group T ∪ j T, where j is an element with j 2 = −1, jzj −1 = z for z ∈ T. This is a nonsplit extension of Gal(C/R) ∼ = Z/2 by 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 H are 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 = IndH T σn , n ∈ Z  {0} of two-dimensional representations, where σn (z) = z n , z ∈ T. These representations are all of quaternionic type since they come from complexifying the representation H → H× given by z → z n , j → j. ∼ R ⊕ R ⊕ (H)∞ . We immediately conclude that CR∗ (H) = Thus ∗ KO• (CR (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 compute KO• (CR∗ (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 simple complex Lie group with K = SU (2) as maximal compact subgroup. The reduced dual of G is Hausdorff, and the complex reduced C ∗ -algebra Cr∗ (G) is a stable continuous-trace algebra with trivial Dixmier-Douady invariant, i.e., it is  r ). All the irreducible complex representations of Cr∗ (G) Morita equivalent to C0 (G are principal series representations, and all unitary principal series are irreducible. 0/W ), where M is a Cartan Thus we see that Cr∗ (G) is Morita equivalent to C0 (M subgroup, which we can take to be C× , and W = {±1} is the Weyl group, which

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0× ∼ acts on C =*Z × R by −1 (on both factors). So Cr∗ (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 isoμ morphism K•G (G/K) ∼ → K• (Cr∗ (G)), sending the generator of the = R(K) ⊗ K•−3 − representation ring R(K) associated to Vk (in the notation of Example 3.1) to the generator of the Z summand in K0 (Cr∗ (G)) associated to the principal series with discrete parameter ±(2k + 1). There is no contribution to K0 (Cr∗ (G)) from the spherical principal series (corresponding to the fixed point n = 0 of W on Z) since R/{±1} ∼ = [0, ∞) is properly contractible. Now let’s analyze the real structure. We can start by looking at K-types. The group SL(2, C) is the double cover of the Lorentz group SO(3, 1)0 . Representations that 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 these representations are also of real type, at least when restricted to K. The genuine representations of SL(2, C) that do not descend to SO(3, 1)0 must have K-types with half-integral spin, and these representations are of quaternionic type, at least when 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 is simply IndG B 1, where B is a Borel subgroup. Since the trivial representation of B is of real type, we get a real form for the complex induced representation by using induction with real Hilbert spaces instead. And thus we get an irreducible real representation on L2R (G/B) ∼ = L2R (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 since none of the other one-dimensional unitary characters of C× are of real type. If we look at a principal series representation of G with discrete parameter ±n ∈ Z, its restriction to K can be identified with IndK T χn , which by Frobenius reciprocity contains Vk with multiplicity equal to the multiplicity of χn in Vk . This is 0 if 2k and n have opposite parity or if |n| > 2k, and is 1 otherwise. So this principal series representation 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 PropoG  does to the principal series. Clearly it sends IndG sition 1.10) on G B χ to IndB χ, if χ is a one-dimensional representation of M viewed as a representation of B. But since W = {±1}, χ = w · χ, for w the generator of W , and we get an equivalent  r is trivial. One can also check this very representation. Thus the involution on G easily 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. Adams has studied this property in much greater generality and proved: Theorem 3.4 (Adams [2, Theorem 1.8]). If G is a connected reductive algebraic group over R with maximal compact subgroup K, if −1 lies in the Weyl group of the complexification of G, and if every irreducible representation of K is of real or quaternionic type, then every unitary representation of G is also of real or quaternionic type.  r is trivial and that Cr∗ (G) is a real Thus we know that the involution on G C -algebra of continuous trace, with spectrum a countable union of contractible components, all but one of which are homeomorphic to R, with the exceptional ∗

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component homeomorphic to [0, ∞). The real Dixmier-Douady invariants must all  r , Z/2) = 0. The sign choice invariants are now determined by vanish since H 2 (G the K-types, since all the K-types have multiplicity one and thus a representation of 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 continuous-trace algebra, Morita equivalent to % % C0R ([0, ∞)) ⊕ C0R (R) ⊕ C0H (R). n>0 even

n>0 odd

Schick’s proof in [62] that the complex Baum-Connes conjecture implies the real Baum-Connes conjecture is stated only for discrete groups, but it goes over without difficulty to general locally compact groups, at least in the case of trivial coefficients. Since the complex Baum-Connes conjecture (without coefficients) is known for all connective reductive Lie groups [38, 71], the real Baum-Connes μ ∗ → KO• (CR,r (G)). This by itself gives some map is an isomorphism KO•G (G/K) − ∗ information on the real structure of the various summands in CR,r (G). Since G/K has a G-invariant spin structure in this case, by the results of [35, §5], KO•G (G/K) ∼ = KO•K (G/K) ∼ = KO•K (p), which by equivariant Bott periodicity ∗ is KO•−3 (CR (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 is isomorphic as a K-module to the adjoint representation of K.) On the other side of the isomorphism, we have KO• (C0R (R)) ∼ = KO −• (R) ∼ = KO −•−1 (pt) ∼ = KO•+1 , H −• −•−5 ∼ ∼ and similarly KO• (C0 (R)) = KSp (R) = KO (pt) ∼ = KO•+5 . So the torsion∗ (G)) are all in degrees 3 mod 4. Since there are no free summands in KO• (CR,r ∗ torsion-free summands in KO• (CR,r (G)) in degrees 1 mod 4, we immediately conclude that no unitary principal series (except perhaps for the spherical principal series, which don’t contribute to the K-theory) are of complex type, and that there are infinitely many lines of principal series of both real and quaternionic type. This is 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. ∗ (G)) is a sum of copies of Since KO•G (G/K) ∼ = KO•−3 (CR∗ (K)), while KO• (CR,r 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 representations of K of integral spin on the left match with those of half-integral spin on the right, and those of half-integral spin on the left match with those of integral spin on the right. This is due to the “ρ-shift” in Dirac induction. If we were to replace SL(2, C) by the adjoint group G = P SL(2, C), the maximal compact subgroup would 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) would be given by genuine representations of the double cover of K, i.e., representations only of half-integral spin. Example 3.6. The techniques we used in Example 3.3 and Theorem 3.5 can also be used to compute the reduced real C ∗ -algebras of arbitrary connected complex reductive Lie groups. A useful starting point is [50, Proposition 4.1], which  r ) ⊗ K is a  r is Hausdorff and Cr∗ (G) ∼ states that for such a group G, G = C0 ( G stable continuous-trace algebra with trivial Dixmier-Douady class. The cases of

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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 real type, 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 ∗  r ) ⊗ KR Cr,R (G) ∼ = C0R (G

is a stable real continuous-trace algebra with trivial Dixmier-Douady class.  r is trivial, and the sign choice Proof. By Theorem 3.4, the involution on G invariant has to be constant on each connected component. Let M be a Cartan subgroup of G, let W be its Weyl group, let B = M N be a Borel subgroup, let K 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 the reduced dual are all principal series IndG B χ, where χ is a character of M extended to a character of B by taking it to be trivial on N . When restricted to K, this  is the same as IndK T χ|T . If ρ ∈ K is a K-type, then by Frobenius reciprocity, ρ appears in this induced representation with multiplicity equal to the multiplicity of χ|T in ρ|T . So given χ, take ρ to have highest weight in the W -orbit of χ|T , G and we see that ρ occurs in IndG B χ with multiplicity 1. Since ρ is real and IndB χ is of either real or quaternionic type, we see that its being of real type is the only possibility. (Otherwise the invariant skew-symmetric form on the representation ∗ (G) is a stable would restrict to a skew-symmetric invariant form on ρ.) Thus Cr,R real continuous-trace algebra with all irreducible representations of real type. The Dixmier-Douady invariant has to vanish since as pointed out in [50, p. 277], all  r are contractible.  connected components of G In a similar fashion we have Theorem 3.8. Let G = Sp(n, C), let M be a Cartan subgroup, and let W be its Weyl group. Let K = Sp(n), a maximal compact subgroup, and let T = M ∩ K, ∗ a maximal torus in K Then Cr,R (G) is a stable real continuous-trace algebra which is Morita equivalent to a direct sum of pieces of the form C0R (Y ) and C0H (Y ). Here 0/W . Infinitely many pieces of each type (real Y ranges over the components of M 0 and χ|T is its “discrete parameter”, then the or quaternionic) occur. If χ ∈ M ∗ (G) is of real type if and only if the representation of associated summand in Cr,R K 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 difference 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 and their K-theory seem to play a significant role is in the study of orientifold string theories in physics. (See for example [51, §8.8] and [52, Ch. 13].) Such a theory is based on a spacetime manifold X equipped with an involution ι, and the theory is based on a “sigma model” where the fundamental “strings” are equivariant maps from a “string worldsheet” Σ (an oriented Riemann surface) to X, equivariant with respect to the “worldsheet parity operator” Ω, an orientation-reversing involution on Σ, and ι, the involution on X. Restricting attention only to equivariant strings is

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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 assumes that the spacetime manifold (except for a flat Minkowski space factor, which we can ignore here) is chosen to be a Calabi-Yau manifold, that is, a complex K¨ ahler manifold X with vanishing first Chern class, and that the involution of X is either holomorphic or anti-holomorphic. If we choose X to be compact, then in low dimensions there are very few possibilities — if X has complex dimension 1, then it is an elliptic curve, and if X has complex dimension 2, then it is either a complex torus or a K3 surface. In the papers [14, 15], the case of an elliptic curve was treated in great detail. Example 2.9 and Theorem 2.10 were motivated by the case of K3 surfaces, which should also be of great physical interest. Now we need to explain the connection with real C ∗ -algebras and their Ktheory. An orientifold string theory comes with two kinds of important submanifolds of the spacetime manifold X: D-branes, which are submanifolds on which “open” strings — really, compact strings with boundary — can begin or end (where we specify boundary conditions of Dirichlet or Neumann type), and O-planes, which are the connected components of the fixed set of the involution ι. There are charges attached 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 the string theory, and should be a variant of KR-theory for orientifold theories. The O-planes have ± signs which determine whether the Chan-Paton bundles restricted to them have real or symplectic type. These sign choices result in “twisting” of the KR-theory, such as appeared above in Definition 2.4. In addition, there is a further twisting of the KR-theory due to the “B-field” which appears in the Wess-Zumino term in the string action. It would be too complicated to explain the physics involved, but mathematically, the B-field gives rise to a class in Moutuou’s Hι2 (X, T ). But in short, the effect of the O-plane charges and the B-field is to make the D-brane charges live in twisted KR-theory, i.e., in the K-theory of a real continuous-trace algebra 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 algebras, which is most interesting from the point of view of physics when (X, ι) is a Calabi-Yau manifold with a holomorphic or anti-holomorphic involution [14, 15]. An important aspect of string theories is the existence of dualities between one theory and another. These are cases where two seemingly different theories predict the same observable physics, or in other words, are equivalent descriptions of the same physical system. The most important examples of such dualities are T-duality, or target-space duality, where the target space X of the model is changed by replacing tori by their duals, and the very closely related mirror symmetry of Calabi-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 the twisting (sign choice and/or Dixmier-Douady class). The possible theories with X an 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. All of the theories in a single group are related to one another by dualities, and theories in two different groups can never be related by dualities. One way to see this is via the twisted KR-theory classifying the D-brane charges. Theories which are dual to

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one another must have twisted KR-groups which agree up to a degree shift, while if the KR-groups are non-isomorphic (even after a degree shift), the two theories cannot possibly be equivalent. Thus calculations of twisted KR-theory provide a methodology for testing conjectures about dualities in string theory. In the case where X 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 (ι the identity, ι anti-holomorphic with a fixed set with two components and with trivial sign choice, and ι holomorphic with four isolated fixed points), the KR-theory turned out to be KO −• (T 2 ), up to a degree shift. In the next group (ι holomorphic and free, ι anti-holomorphic and free, ι holomorphic with four fixed points with sign choice (+, +, −, −), and ι anti-holomorphic with a fixed set with two components and sign choice (+, −)), the groups turned out to be KSC −• ⊕ KSC −•−1 up to a 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 turned out to be KO −• ⊕ KO −• ⊕ K −•−1 up to a degree shift. The KR-groups in one T-duality group are not isomorphic to those in another, so there cannot be any additional dualities between theories. Rather curiously, it turns out (as was shown in [59]) that all of the isomorphisms of twisted KR-groups associated to dualities of elliptic curve orientifold theories arise from the real Baum-Connes isomorphisms for certain solvable groups with Z or Z2 as a subgroup of finite index. This suggests a rather mysterious connection between representation theory and duality for string theories, which we intend to explore further. It will be especially interesting to study dualities between orientifold theories compactified on abelian varieties of dimension 2 or 3 and on 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), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291, DOI 10.1090/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 Br¨ ocker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the German manuscript; Corrected reprint of the 1985 translation. MR1410059 (97i:22005)

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Contemporary Mathematics Volume 671, 2016 http://dx.doi.org/10.1090/conm/671/13514

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 matrices from entangled ones. The results obtained are then applied to study the SPA of a positive map.

1. Introduction The theory of positive maps on C*-algebras goes back to the pioneering work of 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 around 1990 that it was very useful in the mathematical formalism of quantum information theory. The present paper will be an extension of some of the ideas arising from the 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 invariant S(a) defined via maximally entangled states, which gives necessary conditions for separability of a. We then develop some of the theory for S(a) and show that 0 ≤ S(a) ≤ 1 for a separable, that S(a) ≤ n for all a, and we show some results for when S(a) = 1. In the last section we shall use the invariant to study the structural physical approximations, the SPA, of positive maps, and in particular show that the SPA 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 done in [13], using essentially the same example, but using different methods. In this paper we follow the treatment in [13] closely. The author is indebted to H. Christandl for bringing the paper [7] to his attention. 2. The invariant S(a) Recall, see e.g. [11] Lemma 4.1, that each vector x ∈ Cn ⊗ Cn has a Schmidt decomposition  x= ci gi ⊗ hi , i

where the gi , (respective hi ) form an orthonormal basis in Cn , and the norm vectors 2 2 | ci | = 1. x = c 2016 American Mathematical Society

259

260

ERLING STØRMER

A pure state ωx on Mn ⊗ Mn is called maximally entangled if x can be written 1 in the form above with ci = n− 2 for all i ≤ n. For simplicity of notation we shall call a vector x as above with ci = 1 for all i for a maximally entangled vector. Then n−1 ωx 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 the paper [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 that S(a) ≤ n  a . We shall mostly consider S(a) when a is a density matrix. Then we have the following description of matrices a with maximal value for S(a). We denote 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 iff a = [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 mutually orthogonal rank 1 projections pi ∈ Mn ⊗ Mn and λi ≥ 0 with sum 1 such that  a= λi pi . By compactness of the set of maximally entangled vectors there exists a maximally entangled vector x such that    n = S(a) = (ax, x) = λi (pi x, x) ≤ λi S(pi ) ≤ λi n = n. Thus (pi x, 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 eij gm = δjm gi and similarly for fkl . Then each matrix a ∈ Mn ⊗ Mn can be written in the form  (1) a= a(ij)(kl) eij ⊗ fkl . n Lemma 3. If x = 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) =



a(ij)(kl) (eij ⊗ fkl

= =

  

gr ⊗ hr ,

r

ijkl

=



a(ij)(kl) δjr δlr (gi ⊗ hk ,





gs ⊗ hs )

s

gs ⊗ hs )

a(ij)(kl) δjr δlr δis δks a(ij)(ij) .

The proof is complete. Recallthat an operator a ∈ Mn ⊗ Mn is separable if a can be written in the form a = bi ⊗ ci with bi , ci ≥ 0.

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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 ) with respect to the basis (gi ) and c = (ckl ) with respect to (hj ). Then a(ij)(kl) = bij ckl . Hence by Lemma 3, if  . 2 denotes the Hilbert-Schmidt norm and  . 1 the trace norm, then  (ax, x) = bij cij   1 ≤ ( | bij |2 | cij |2 ) 2 =  b 2  c 2 ≤  b 1  c 1 = 1. Since thisholds for all maximally entangled vectors x, S(a) ≤ 1.In the general λk = 1. Thus case a = λk bk ⊗ ck with bk , ck density operators and λk ≥ 0 and by the above applied to bk ⊗ ck ,   (ax, x) = λk (bk ⊗ ck x, x) ≤ λk = 1, proving the theorem. Corollary 5. Let a = b ⊗ c ∈ Mn ⊗ Mn with b and c density matrices. Then S(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 and similarly for c. This is possible only if rank b = 1 and rank c = 1 with norms equal to 1, hence that they are rank 1 projections. Conversely if b = [y] and c = [z] let  (yi ) respectively (zi ) be orthonormal bases yi ⊗ zi is a maximally entangled vector for Cn with y1 = y, z1 = z. Then x = 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 = [ bi yi ] and c = [ ci zi ], 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 of Theorem 4  1= bij cij =< 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 we therefore have  Lemma 6. With  the above notation, if x = yi ⊗ zi is a maximally entangled vector, and b = [ bi yi ], c = [ ci zi ] 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 a unit vector in Cn ⊗ Cn . Then S(q) ≥ 1. Furthermore, S(q) = 1 iff q = e ⊗ f for rank 1 projections e and f . Proof. The vector y has a Schmidt decomposition y=

i=k 

λi gi ⊗ hi ,

i=1

with λ ≥ 0, where (gi ), (hi ) orthonormal systems, see [11], Lemma 4.1. Extend  them to orthonormal bases , so y = i=n i=1 λi gi ⊗ hi with λi = 0 for i ≥ k. The  vector x = gi ⊗ hi is maximally entangled. We have S(q) ≥

(qx, x)

=

((x, y)y, x)

=

| (x, y) |2   |( gi ⊗ hi , λk gk ⊗ hk ) |2  | (gi ⊗ hi , λi gi ⊗ hi ) |2  | λ i |2  | λ2i |2

= = = ≥

= 1,  2 since y is a unit vector, so λi = 1, proving the part.   first If S(q) = 1 the above shows that λi = λ2i , hence, since λi ≥ 0, λi = λ2i for all i, hence λi = 1 or 0. Hence there is a unique i such that y = gi ⊗ hi , hence q = [y] = e ⊗ f where e = [gi ], f = [hi ], proving one implication in the proposition. The converse follows from Corollary 5.  It is clear from thedefinition of S(a) that if a is a sum a = ai of positive matrices, then S(a) ≤ S(ai ). Since S(ai ) is obtained as S(ai ) = (ai xi , xi ) with ai xi a maximally entangled vector, and the vectors xi may be different for different  , 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) = 1  n n ⊗z unit vectors iff there exist a maximally entangled vector x = y i i ∈ C ⊗C and  n uk = aki yi ∈ C such that if ek = [uk ], then a is a convex sum a = λk ek ⊗ fk , where fk = etk . Furthermore ((ek ⊗ fk )x, x) = 1 for all k.  Proof. Since a is separable a = λk bk ⊗ck with b k , ck density matrices. If S(a) = 1 then there is a maximally entangled vector x = yi ⊗ zi such that   λk = 1, 1 = (ax, x) = λk ((bk ⊗ ck )x, x) ≤ using Theorem 4. Thus ((bk ⊗ ck )x, x) = 1 for  all k. By Corollary 5 ek = bk and aki yi such that ek = [uk ]. Then fk = ck are rank 1 projections. Let uk = ek = (aki akj ), so by Corollary 6 and its proof fk = etk , and 1 = S(ek ⊗ fk ) = ((ek ⊗ fk )x, x)

MAXIMALLY ENTANGLED STATES AND POSITIVE MAPS

263

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 orthogonal. However, if a = μi pi ⊗ 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 a state, where ι denotes the identity map and t the transpose map. This means that if a is a density matrix for ρ, then ι ⊗ t(a) is also a density matrix. We next show that S(a) is preserved under this transformation. Theorem 9. Let a ∈ Mn ⊗ Mn be a density matrix such that ι ⊗ t(a) ≥ 0, then S(a) = S(ι ⊗ t(a)). Proof. Let (ei ) be the standard basis for Cn . Let   J( xi ei ) = xi ei .   yi ei then Then bt = Jb∗ J for b ∈ Mn . If x = xi ei , y =   xi y i = (Jx, y) = xi yi = (x, Jy) = (Jy, x). Thus if b, c ∈ Mn , then (ι ⊗ t(b ⊗ c)x ⊗ y, x ⊗ y)

= (bx, x)(ct y, 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 , then ι ⊗ J(x) is maximally entangled. It thus follows from the above computation and the 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 invariant S(a) to verify whether a state is separable or not. We end this section with an open problem, namely; Let a be a density operator with 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. Then φ is positive, written φ ≥ 0, if φ(a) ≥ 0 for all a ≥ 0. φ is completely positive if φ ⊗ ι : Mn ⊗ Mn → Mn ⊗ Mn is positive. φ is optimal if φ − ψ ≥ 0 for ψ completely positive, implies ψ = 0. Let (eij ) be a complete set of matrix units for Mn . Then the Choi matrix Cφ for φ is the matrix  eij ⊗ φ(eij ) ∈ Mn ⊗ Mn . Cφ = Then φ is completely positive iff Cφ ≥ 0, see [12], Theorem 4.1.8, and φ ≥ 0 iff T r ⊗ T r(Cφ a ⊗ b) ≥ 0 for all a, b ≥ 0 in Mn , see [12], Theorem 4.1.11, where T r is the usual trace on Mn . φ is said to be super-positive or entanglement breaking if T r ⊗ T r(Cφ Cψ ) ≥ 0 for all positive maps ψof Mn into itself. We then have that φ ai ωi , where ωi are states on Mn , and is super-positive iff Cφ is separable iff φ =

264

ERLING STØRMER

ai ≥ 0 in Mn , as follows from [12], Lemma 4.2.3 and Proposition 5.1.4. If φ is a positive map of Mn into itself then the map a −→ φ(1)T r(a) + φ(a)

(2)

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 that such 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 . Furthermore Cφ is separable iff S(Cφ ) = V 22 . Proof. Let (eij ) be the usual  matrix units corresponding to the usual  orthonormal basis ei for Cn . Let P = eij ⊗ eij . Then P = n[xo ], where xo = ei ⊗ ei . Then  Cφ = eij ⊗ V ∗ eij V = (1 ⊗ V )∗ P (1 ⊗ V ), so Cφ is a positive rank 1 matrix. Its trace is T r ⊗ T r(Cφ )

 = T r ⊗ T r( eij ⊗ V ∗ eij V )  = T r( V ∗ eii V ) = T r(V ∗ V ) = V 22 ,

Cφ is a rank 1 projection [y] for a unit vector y. Hence by so that  V −2 2 Proposition 7  V −2 2 S(Cφ ) = S([y]) ≥ 1, i.e. S(Cφ ) ≥ V 22 proving 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 case we can chose the maximal entangled vector as x = (1⊗V ∗ )xo , with xo as above and thus assume V = 1. Then Cφ = P , and (P xo , xo ) = n(xo , xo ) = n2 . Furthermore T r ⊗ T r(Cφ ) = T r ⊗ T r(P ) = n, hence 1/nCφ is a density matrix such that (1/nCφ xo , 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 a density 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 they are not physical, and then cannot be directly implemented. It is therefore challenging to try to find a physical way to approximate the action of a positive map. This is the goal of the structural approximation, SPA, see e.g. [1, 10]. The idea is to mix a positive map φ with a simple completely positive map making the mixture completely positive. The resulting map can then be realised in the laboratory, and its action characterizes entanglement of the states detected by φ.

MAXIMALLY ENTANGLED STATES AND POSITIVE MAPS

265

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 . Then T r ⊗ T r(W ) = 1. We put for 0 ≤ t ≤ 1  (t) = 1 − t 1 ⊗ 1 + tW. W n2  (t) has trace 1. Note that CT r = 1 ⊗ 1, and n−2 1 ⊗ 1 is a density matrix. Then W 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  (t) ≥ 0. The SPA of φ, SP A(φ) is defined as is the maximal t for which W  (t∗ ) SP A(φ) = W and is the Choi matrix of the completely positive map on the line segment between φ and the tracial state 1/nT r which is nearest to φ. A straightforward computation, see e.g. [13], page 2202, shows  (t∗ ) = SP A(φ) = W

1 ( Cφ−  1 ⊗ 1 + Cφ ). n + n2  Cφ− 

We now easily obtain the following sufficient condition for separability for SP A(φ). For a special class of maps this was also obtained in [4]. Proposition 12. When φ is a unital positive map then SP A(φ) is the density matrix for a state, which is separable if  Cφ− = 1. Proof. Since  Cφ−  1 ⊗ 1 + Cφ ≥ 0, SP A(φ) ≥ 0, and the above formula implies T r ⊗ T r(SP A(φ)) = 1, it follows that SP A(φ) is the density matrix for a state. By equation (2) T r + φ is super-positive, hence if  Cφ− = 1, then 1 ⊗ 1 + Cφ = CT r+φ is separable. We also have a necessary condition for separability of SP A(Cφ ). This will be an application of Theorem 4. Proposition 13. If φ is unital and SP A(φ) separable then S(Cφ ) ≤ n + n(n − 1)  Cφ−  . Proof. By Proposition 12 SP A(φ) is the density matrix for a state, which by assumption is separable. Hence by Theorem 4, 0 ≤ S(SP A(φ)) ≤ 1. Since S(1 ⊗ 1) = n, it follows from the formula stated just before Proposition 12 that  Cφ−  n + S(Cφ )

=  Cφ−  S(1 ⊗ 1) + S(Cφ ) = S( Cφ−  1 ⊗ 1 + Cφ ) = (n + n2  Cφ− )S(SP A(φ)) ≤ n + n2  Cφ−  .

Hence S(Cφ ) ≤ n + n(n − 1)  Cφ− , proving the proposition.

266

ERLING STØRMER

The definition of SP A(φ) is only of interest when φ is not separable, or even not majoring a completely positive map, hence one studies SP A(φ) when φ is an optimal map. By checking the standard examples of optimal maps physicists conjectured that SP A(φ) of an optimal map is always separable, see [2, 10]. This conjecture was shown to be false by Ha and Kye in [6], and the author shortly afterwords. 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], of M3 into itself. Let a, b, c be non-negative real numbers, −π ≤ θ ≤ π. Define the map φ(a, b, c, θ) : M3 → M3 by φ(a, b, c, θ)((xij )) = ⎛ ax11 + bx22 + cx33 ⎝ −e−iθ x21 −eiθ x31 Then the Choi matrix for φ is given ⎛ a 0 0 ⎜ 0 c 0 ⎜ ⎜ 0 0 b ⎜ ⎜ 0 ⎜ −iθ 0 0 0 0 Cφ = ⎜ ⎜−e ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎜ ⎝ 0 0 0 −eiθ 0 0

−eiθ x12 cx11 + ax22 + bx33 −e−iθ x32

⎞ −e−iθ x13 ⎠. −eiθ x23 bx11 + cx22 + ax33

by 0 −eiθ 0 0 0 0 b 0 0 a 0 0 0 0 0 0 0 −e−iθ

0 0 0 0 0 c 0 0 0

0 0 0 0 0 0 c 0 0

⎞ 0 −e−iθ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 −eiθ ⎟ ⎟. 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ b 0 ⎠ 0 a

Let pθ = max{2Reei(θ−2π/3) , 2Reeiθ , 2Reei(θ+2π/3)} . Then 1 ≤ pθ ≤ 2. The main properties 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 is close 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 i or 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 for a 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 (Cφ xo , xo ) = 3a − 3eiθ − 3e−iθ = 3(a − 2 cos θ). As θ is close to π, cos θ is close to −1, and  Cφ−  is close to 0. Thus (Cφ xo , xo ) is close to 3(1 + 2) = 9 > 3 + 6  Cφ− , so by Proposition 13 SP A(φ) cannot be separable. For more details see [13].

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Operator Algebras: A Tribute to R. V. Kadison • Doran and Park, Editors

This volume contains the proceedings of the AMS Special Session Operator Algebras and Their Applications: A Tribute to Richard V. Kadison, held from January 10–11, 2015, in San Antonio, Texas. Richard V. Kadison has been a towering figure in the study of operator algebras for more than 65 years. His research and leadership in the field have been fundamental in the development of the subject, and his influence continues to be felt though his work and the work of his many students, collaborators, and mentees. Among the topics addressed in this volume are the Kadison-Kaplanksy conjecture, classification of C ∗ -algebras, connections between operator spaces and parabolic induction, spectral flow, C ∗ -algebra actions, von Neumann algebras, and applications to mathematical physics.