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: Con- temporary mathematics ; volume 671 | Includes bibliographical references. Identifiers: LCCN 2015043280 | ISBN 9781470419486 (alk. paper) Subjects: LCSH: Operator algebras–Congresses. — AMS: Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – General theory of C∗- algebras. msc | Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – General theory of von Neumann algebras. msc | Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – Classifications of C∗-algebras. msc | Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neu- mann (W ∗-) algebras, etc.) – Noncommutative dynamical systems. msc | Functional analysis – Selfadjoint operator algebras (C∗-algebras, von Neumann (W ∗-) algebras, etc.) – Noncommu- tative differential geometry. msc | K-theory – K-theory and operator algebras – Index theory. msc | Topological groups, Lie groups – Lie groups – Representations of Lie and linear 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
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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 235
vii
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 alge- bras 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,heldonJanuary 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 Ses- sion 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 George Elliott San Jose State University University of Toronto Joe Ball Adam Fuller Virginia Tech University University of Nebraska, Lincoln Paul Baum Liming Ge Pennsylvania State Unversity University of New Hampshire and Alex Bearden Chinese Academy of Sciences University of Houston Elizabeth Gillaspy David P Blecher University of Colorado, Boulder University of Houston James Glimm R´emi Boutonnet Stony Brook University University of California, San Diego Jan Gregus Michael Brannan Abraham Baldwin Agricultural College 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 Magnus Landstad Ronald Douglas Norwegian University of Science and Texas A&M University Technology Ken Dykema David Larson Texas A&M University Texas A&M University Edward Effros Zhe Liu University of California, Los Angeles University of Central Florida Søren Eilers Jireh Loreaux University of Copenhagen University of Cincinnati
xi
xii LIST OF PARTICIPANTS
Terry Loring Mikael Røordam University of New Mexico University of Copenhagen Martino Lupini Jonathan Rosenberg York University (Canada) University of Maryland Ellen Maycock Christopher Schafhauser American Mathematical Society University of Nebraska, Lincoln Azita Mayeli Mohamed W. Sesay City University of New York Howard University Matt McBride Juhhao Shen University of Oklahoma University of New Hampshire Niels Meesschaert Fred Shultz KU Leuven (Belgium) Wellesley College Ramis Movassagh Roger Smith MIT and Northeastern University Texas A&M University Paul Muhly Baruch Solel University of Iowa Technion (Israel) Magdalena Musat Myungsin-Sin Song University of Copenhagen Southern Illinois University Erling Størmer Pieter Naaijkens University of Oslo Leibniz Univerit¨at Hannover Wo jciech Szymansk Judith Packer University of Southern Denmark University of Colorado, Boulder Mark Tomforde Efton Park University of Houston Texas Christian University John Vastola Geoffry Price University of Central Florida United States Naval Academy Henry Warchall Sorin Popa National Science Foundation University of California, Los Angeles Gary Weiss Ian Putnam University of Cincinnati University of Victoria Alan Wiggins Timothy Rainone University of Michigan, Dearborn Texas A&M University and University of Waterloo Wei Zhang Purdue University Kamran Reihani Texas A&M University Marc A Rieffel University of California, Berkeley Min Ro University of Oregon
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 con- jecture 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 Baum- Connes 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 counterexam- ple 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 Baum- Connes 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 nec- essary 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 ex- panders 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 Kadison- Kaplansky conjecture for the 1-algebra of a group.
2. Exact groups and crossed products Throughout, G will be a countable discrete group. Much of what follows 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 ∗-automor- phisms. The natural representations for G-C∗-algebras are the covariant repre- sentations: these consist of a C∗-algebra representation of A as bounded linear operators on a Hilbert space H, together with a unitary representation of G on the same Hilbert space, π : A →B(H)andu : G →B(H), satisfying the covariance condition π(αg(a)) = ugπ(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 G- ∗ action 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: −1 ∗ −1 ∗ f1 f2(g)= f1(h)αg(f2(h g)) and f (t)=αg(f(g ) ). h∈G
The algebra A alg G is functorial for G-equivariant ∗-homomorphisms in the ob- vious way. We shall refer to A alg G as the algebraic crossed product of A by G. Finally, a covariant representation integrates to a ∗-representation of A alg G accordingtotheformula π u(f)= π(f(g))ug . 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
fmax =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)) and λ : G →B(H⊗2(G)) by
π (a)(v ⊗ δg)=π(αg−1 (a))v ⊗ δg and λh(v ⊗ δg)=v ⊗ δhg.
The reduced crossed product A red G is the image of A max G under the integrated form of this covariant representation. In other words, A red G is the completion of A alg G for the norm
fred = π λ(f). 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 / / / / 0 I red G A red G B red G 0 is itself short exact. Several remarks are in order here. First, the map to B 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.Bythiswe ∗ 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 A algG (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| =0iffg = 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−1h|.Thismetrichasbounded
6 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 →Hto 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 ∈Hand notice that g · v − h · v = g−1h · v − v∼g−1h · 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,Problems3and4]ofGromovand[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].
1Gromov 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 as- sembly map with coefficients in any G-C∗-algebra A top K∗ (G; A) → K∗(A red G) is an isomorphism if G is a-T-menable, and is split injective if G is coarsely em- beddable. The class of a-T-menable groups is large: it contains all amenable groups as well as free groups, and classical hyperbolic groups; see [22]forasurvey.A- T-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]andallGromov 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 de- scribe 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)=1andk(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 aiaj 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(g1g, g1h)=k(g, h) for every g1, g and h ∈ G. 3.6. Theorem. Let X be a bounded geometry, uniformly discrete metric space. Then X has Property A if and only if for every R (large) and ε (small),thereexists a positive definite kernel k : X × X → C such that (i) |1 − k(x, y)| < whenever d(x, y) 8 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 con- dition: (ii)’ for every δ>0 the set { d(x, y) ∈ [0, ∞): |k(x, y)|≥δ } is bounded. A countable discrete group G is amenable if and only if for every R and ε 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 +3Property A ≡ exactness a-T-menability +3coarse embeddability . The class of groups with Property A covers all the examples of coarsely embed- dable groups mentioned earlier. Indeed, proving the existence of groups without Property A is as difficult as proving the existence groups that do not coarsely em- bed. Nonetheless, Osajda has recently shown the existence of coarsely embeddable (and even a-T-menable) groups without Property A [56]. In particular, there 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 equiv- alent: (i) G has Property A; (ii) G admits an amenable action on a compact space; (iii) G is an exact group; ∗ ∗ ∗ (iv) the group C -algebra Cred (G) is an exact C -algebra. Thereadercanseethesurvey[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 Theo- rem 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. Ex- panders are the easiest examples of metric spaces that do not coarsely embed. They are also connected to K-theory through the construction of Kazhdan projec- tions; this construction is at the root of the counterexamples to the Baum-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 Δf(x)=deg(x)f(x) − f(y)= f(x) − f(y). 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 =0andx ∼ 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 ex- pander 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 con- struction 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 10 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 2 2 acting on (Y ), identified with the direct sum of the spaces (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 ≡ 0on[c, ∞)thenp = 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)hasfinite propagation if there exists R>0 such that T cannot propa- gate 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)asanideal. 4.2. Proposition. Let Y be the box space of an expander sequence. The Kazh- dan projection p is not compact, and belongs to C∗(Y ). Proof. The Kazhdan projection has infinite rank; it projects onto the space 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,aghost is an element T ∈ C∗(X) whose ‘matrix entries tend to 0 at infinity’; precisely, the suprema sup |Txz| and sup |Tzx| z∈X 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 ⎛ ⎞ 11... 1 ⎜ ⎟ 1 ⎜11... 1⎟ ⎜ ⎟ pn = ⎝. . . .⎠ . card(Yn) . . .. . 11... 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 12 PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT entries of T ∗T are given by ∗ 2 (4.3) (T T )xx = |Tzx| ; z ∗ thus they are non-negative so their limit is as well. Finally, we define C∞(X)tobe 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;thisisaquotientofC∗(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)wehave 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 ∗ C∞(X). ∗ ∗ Conversely, suppose that T maps to zero in C∞(X), so that T T does as well. ∗ Hence the limit of the on-diagonal matrix entries (T T )xx is zero against every free ultrafilter, so that they converge to zero in the ordinary sense as x →∞.Now, according to (4.3) we have ∗ 2 2 (T T )xx = |Tzx| ≥ sup |Tzx| , z z | |2 → →∞ ∗ so that supz Tzx 0asx as well. Applying the same argument to TT | |2 →∞ showsthatsupz Txz 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 ) → 2 B( (Yn)) defined by cutting down by the appropriate projection. These are asymp- totically multiplicative on the algebra of finite propagation operators, and we obtain a ∗-homomorphism B(2(Y )) ∗ → n n (4.5) C (Y ) 2 . ⊕nB( (Yn)) EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE 13 Taking the rank in each block (equivalently, taking the map on K-theory in- duced by the canonical matrix trace on B(2(Y ))) gives a homomorphism n 2 (4.6) K0 B( (Yn)) → Z. n n 2 As K1(⊕nB( (Yn)) = 0, the six term exact series in K-theory specializes to an exact sequence B(2(Y )) ⊕ B 2 / B 2 / n n / K0( n ( (Yn))) K0( ( (Yn))) K0 2 0 . n ⊕nB( (Yn)) Z → Composing the ‘rank homomorphism’ in line (4.6) with the quotient map n Z ⊕ Z ⊕ B 2 n / n clearly annihilates the image of K0( n ( (Yn))), and thus from the sequence above gives rise to a homomorphism B(2(Y )) Z n n → n K0 2 . ⊕nB( (Yn)) ⊕nZ Finally, combining with the K-theory map induced by the ∗-homomorphism in line (4.5) gives a group homomorphism Z ∗ n K0(C (Y )) → . ⊕nZ ∗ 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 2 a rank one projection on each (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]forthe 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 Baum- Connes conjecture began in earnest with the provocative remarks found in the last section of [52]. There, Gromov describes a model for a random presentation 14 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],andinthefurther 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 →∞.Thisis 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)intoabox 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 pro- duces two labelings of the edges in the individual Yn with letters from a finite alpha- bet: one satisfies a small cancellation condition for pieces from different blocks and the other for pieces from a common block. He then combines these in a straightfor- ward way to obtain a labeling that globally satisfies the C(1/24) small 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 fun- damental 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]leadsto 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 16 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 consid- ered it as a higher index map → ∗ (6.1) K∗(M) K∗(Cred(G)), 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 coeffi- cients 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 prop- erties, 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]itwas 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,definedin[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 EG(A, B). These are abelian groups, and it follows that for a fixed G-C∗-algebra B, the assignments A → EG(A, B)andA → EG(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 G- invariant 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 top G K0 (G; A) := lim E (C0(X); A), X⊆EG X cocompact and similarly for K1 using suspensions. The universal property of EG together top with homotopy invariance of the E-theory groups implies that K∗ (G; A)doesnot 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 con- struction of the equivariant E-theory category to the trivial group gives the E- theory category, which we shall denote by E. The objects in this category 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. G 6.1. Theorem. There is a (maximal) descent functor max : E → E which agrees with the usual maximal crossed product functor both on objects and on mor- phisms in EG coming from equivariant ∗-homomorphisms. To complete the definition of the Baum-Connes assembly map, we need to 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) maxG)=E(C,C0(X) max G) has the following properties: (i)[pX ] depends only on X (and not on choices made in the definition of pX ); (ii)[pX ] is functorial for equivariant maps. Here, functoriality means that if X → Y is an equivariant map of spaces as in the statement of the proposition, then the classes [pX ]and[pY ] correspond under the functorially induced map on K-theory. 18 PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT 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 G E (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(C0(X) max G, A max G) E(C,A max G) / E(C0(Y ) max G, A max G) E(C,A max G) commutes. Here, the horizontal arrows are given by composition with the appro- priate basic projections, and the left hand vertical arrow is composition with the ∗-homomorphism C0(Y ) max G → C0(X) max G induced by the inclusion X → Y . top Hence the assembly maps are compatible with the direct limit defining K0 (G; A), and give a well-defined homomorphism top → C K0 (G; A) E( ,A red G)=K0(A red G). Everything works similarly on the level of K1 using suspensions, and thus we get a homomorphism top μ : K∗ (G; A) → K∗(A red G), which is, by definition, the Baum-Connes assembly map.TheBaum-Connes con- jecture 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 as- sembly map top μ : K∗ (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 top / K∗ (G; A) K∗(A red G) top / K∗ (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 /I /A /B /0 be a short exact sequence of separable G-C∗-algebras. There is a commutative dia- gram of Baum-Connes assembly maps top / top / top K0 (G; I) K0 (G; A) K0 (G; B) / / K0(I red G) K0(A red G) K0(B red G), in which the horizontal arrows are the functorially induced ones. Moreover, the 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 conjec- ture for G with coefficients in all of I, A,andB 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)gen- erated by the functions supported in Y ;asthisC∗-algebra is commutative, we may 20 PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT as well write it as C0(W )whereW is its spectrum, a locally compact G-space. Note that as G acts on itself by isometries on the left,theright 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)andM the multipli- cation action of ∞(G)on2(G). Then the pair (M,ρ)isacovariantrepresentation of ∞(G)fortherightG-action. Moreover, it is well-known (compare for exam- ple [17, Proposition 5.1.3]) that this pair integrates to a faithful representation of ∞ 2 (G) red G on (G)thattakesC0(G) red G onto the compact operators. As the reduced crossed product preserves inclusions, it makes sense to restrict this repre- sentation to C0(W ) red G, thus giving a faithful representation of C0(W ) red G on 2(G). The key facts we need to build our counterexamples are contained in the fol- lowing lemma. To state it, let C∗(Y ) denote the uniform Roe algebra of Y and ∗ ∗ 2 C∞(Y ) the quotient as in Section 4. Represent C (Y )on (G) by extending by zero on the orthogonal complement 2(G \ Y )of2(Y ). ∗ 2 7.3. Lemma. The faithful representations of C0(W ) red G and C (Y ) on (G) defined above give rise to a commutative diagram 2 / ∗ / ∗ K( (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 ,consideredasanelement ∗ 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 fρg,wheref ∈ C0(W )andg ∈ G. The compression of such an operator 2 2 (7.1) χ(fρg)χ : (Y ) → (Y ) has matrix coefficients f(x),y= xg (7.2) δx,χfρgχ(δy) = δx,fρg(δy) =(fδ −1 )(x)= yg 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,Tδxg ,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 ,andan elementary check of matrix coefficients using line (7.2) shows that T is given by the (finite) sum T = χ(fgρg)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 )aswell. ∗ The C -algebra C0(∂W) red G is handled by analogous computations, regard- ing χ as an element of C0(∂W). Finally, under the identification of C0(G) red G 2 2 with K( (G)), it is clear that K( (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 fρg where f is a bounded function with support in the set Yh,forsomeh ∈ G. Thus, it suffices to show that each such operator belongs to the ideal of C0(W ) red G generated by χ.Now,the 2 ∗ characteristic function of Yh,viewedasanoperatoron (G), is ρhχρh. It follows that ∗ fρg = f(ρhχρh)ρg =(fρh−1 )χρhg belongs to the ideal generated by χ, and we are through. In a similar way, the image of χ is a full projection in C0(∂W) red G.Fi- nally, any non-zero projection on 2(G),andinparticularχ, is a full multiplier of 2 C0(G) red G = K( (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 Baum- ∗ Connes with coefficients fails for G,astheC -algebras C0(W )andC0(∂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 G- ∗ invariant C -subalgebra of C0(W ) that contains C0(G) and the characteristic func- tion χ of the expander Y .Then (i) the crossed product C0(Z) red G contains the Kazhdan projection associ- ated to Y ; (ii) the sequence / / K0(C0(G) red G) K0(C0(Z) red G) K0(C0(∂Z) red G) is not exact in the middle; (iii) the Baum-Connes conjecture with coefficients is false for G. Proof. For (i), let d ∈ N be the degree of all the vertices in Y . The Laplacian on Y (compare line (4.2) above) is then given by Δ=dχ − χρgχ g∈G, |g|=1 andisthusinC0(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,andthecommu- tative diagram / / K0(C0(G) red G) K0(C0(Z) red G) K0(C0(∂Z) red G) / / K0(C0(G) red G) K0(C0(W ) red G) K0(C0(∂W) red G) , where the vertical arrows are all induced by the canonical inclusions. Part (iii) 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 top K∗ (G; A) → K∗(A max G) → K∗(A τ G) of the maximal assembly map, and the map induced on K-theorybythequotient 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 a- T-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 follow- ing desirable properties of a crossed product functor τ that might be used to ‘fix’ the Baum-Connes conjecture. Exactness. It should fix the exactness problems: that is, for any short exact se- quence 0 /I /A /B /0 of G-C∗-algebras, the induced sequence of C∗-algebras / / / / 0 I τ G A τ G B τ G 0 should be exact. Compatibility with Morita equivalences. Two G-C∗-algebras are equivariantly sta- bly isomorphic if A⊗KG is equivariantly ∗-isomorphic to B⊗KG. Here, KG 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 as- sembly 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. G Existence of descent. There should be a descent functor τ : E → 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 KKG-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 ∗ C (see [67]or[39, Section 3.7]) in Cmax(G)= 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 Baum- Connes conjecture should have the following properties: (i) it is exact; (ii) it is Morita compatible; (iii) it has a descent functor in E-theory; (iv) it annihilates Kazhdan projections. Such crossed products do indeed exist! In order to prove this, we introduced in [10, Section 3] a partial order on crossed product functors by saying that τ ≥ σ if the norm on A alg G coming from A τ G is at least as large as that 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 top K∗ (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, G andinparticularwhether E admits a descent functor E : KK → 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)andC0(∂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)andC0(∂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)fromtheE-Baum- Connes conjecture for C0(G)andC0(∂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 0 /I /A /B /0 be a short exact sequence of separable G-C∗-algebras. There is a commutative dia- gram of six term sequences Ktop(G; I) /Ktop(G; A) /Ktop(G; B) 0 O NNN 0 NNN 0 NNN NNN NNN NNN NNN NNN NNN NN& NN& NN& / / K0(I OE G) K0(A E G) K0(B E G) Ktop(G; B) o Ktop(G; A) o Ktop(G; I) 1 NNN 1 NNN 1 NNN NNN NNN NNN NNN NNN NNN NN& NN& NN& o o K1(B E G) K1(A E G) K1(I E G), in which the front and back rectangular six term sequences are exact, and the maps from the back sequence to the front are E-Baum-Connes assembly maps. In partic- ular, if the Baum-Connes conjecture holds with coefficients in two out of three of I, A,andB, 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). { } { } R (iii) For any finite subset g1,...,gn of G, any finite subset t1,...,tn of n ∈ such that i=1 ti =0,andanyx X,wehavethat n −1 ≤ titj h(xgi,gi 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 ,andletW and ∂W be as in Section 7(and as explained at the beginning of this section).Letπ : 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 ,thenY ‘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,letNr(Y )={g ∈ G | d(g, Y ) ≤ r} denote the r-neighborhood of Y in G.IfNr(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,letχYg denote the characteristic function of the right trans- late of Y by g, which is in C0(Z) by definition of this algebra. Hence f = |g|≤r χYg −1 −1 is in C0(Z). The closure of Nr(Y )isequaltof (0, ∞)andtof [1, ∞), and is thus compact and open as f is an element of C0(Z). Finally, note that finitely sup- ported elements of ∞(G) and translates of χ by the right action of G are supported in Nr(Y )forsomer>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).Notethatd(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.Foreachm ∈{1,...,n},let Em = {x ∈ Nr(Y ) | xgm ∈ Y } · and note that the characteristic function χEm of Em is equal to χYgm 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 andisthusalso 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},letχk,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 | −1 | d(π(x),π(xg)) = d(xgk,xggl)= gk ggl . Hence n n | −1 | hr,g = gk 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 E- Baum-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 ).LetC0(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 conjec- ture 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 1 ∗ Kadison-Kaplansky conjecture. As (G)isasubalgebraofCred(G), the Kadison- Kaplansky conjecture implies that 1(G) contains no idempotents other than 0 or 1. In this section, we show that the E-Baum-Connes conjecture, and in fact any ‘ex- otic’ Baum-Connes conjecture, implies that 1(G) has no non-trivial idempotents. Thus the E-Baum-Connes conjecture implies a weak form of the Kadison-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.Iftheσ-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 ∗ → C τ : Cred(G) ,τ(a)= δe,aδe . The trace τ is well known to be faithful in the sense that a non-zero positive element ∗ of Cred(G)hasstrictly positive trace: see, for example [17, Proposition 2.5.3]. One has the following standard C∗-algebraic lemma. Lemma. ∈ ∗ 10.2. Let e Cred (G) be an idempotent. If τ(e) is an integer then e =0or e =1. Proof. ∗ The idempotent e is similar in Cred(G) to a projection p [15,Propo- sition 4.6.2] so that τ(p)=τ(e) ∈ Z. Positivity of τ and the operator inequality EXACTNESS AND THE KADISON-KAPLANSKY CONJECTURE 29 0 ≤ p ≤ 1 imply that 0 ≤ τ(p) ≤ 1andsoτ(p)=0orτ(p) = 1. Since τ is faithful, we conclude p =0orp = 1, and the same for e. ∗ → R Recall that the trace τ defines a map τ∗ : K0(Cred(G)) which sends the ∈ ∗ K-theory class of an idempotent e Cred(G)toτ(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. Proposition. ∈ ∗ 10.3. Let G be a countable torsion free group. If x K0(Cred (G)) is in the image of the Baum-Connes assembly map with trivial coefficients then τ∗(x) is an integer. Proof. This is a corollary of Atiyah’s covering index theorem [6] (see also [20]). The most straight-forward way to connect Atiyah’s covering index theorem to the Baum-Connes conjecture, and thus to prove the proposition, is via the Baum- Douglas geometric model for K-homology [9, 11]: this is explained in [69, 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)isa ∗ ∗ C subalgebra of both Cred(G)andCσ(G):= σ G, we may consider the K-theory ∗ classes [e]red and [e]σ defined by e for each of these C -algebras. The usual (reduced) ∗ → ∗ Baum-Connes assembly map factors through the quotient map Cσ(G) Cred(G), 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 top 1 K∗ (G, A) → K∗( (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 1 induced by the inclusion (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 1 A redG, and in fact that the inclusion (G, A) → A redG induces an isomorphism. While the Bost conjecture may seem more natural because it has the appro- priate functoriality in the group G, it does not have the important implications 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 30 PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT Gromov-Lawson-Rosenberg conjecture about existence of positive scalar curvature metrics on spin manifolds. 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MR1728880 (2000j:19005) Department of Mathematics, Pennsylvania State University, University Park, Penn- sylvania 16802 E-mail address: [email protected] University of Hawai‘i at Manoa,¯ Department of Mathematics, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273 E-mail address: [email protected] University of Hawai‘i at Manoa,¯ Department of Mathematics, 2565 McCarthy Mall, Honolulu, HI 96822-2273 E-mail address: [email protected] 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 Ses- sion “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 35 36 DAVID P. BLECHER 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) pos- itivity 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 posi- tivity 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 Ba- nach 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]). An- other reason is that we welcome this opportunity to add some additional details and complements to [21](andto[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 el- ements 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 abun- dance 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 ‘pos- itive 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 un- til 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 conjec- ture is raised. Successful applications so far include for example noncommutative C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 37 topology (eg. noncommutative Urysohn and Tietze theorems for general 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 prop- erties of self-adjoint operators individually and with respect to con- taining operator algebras”. This was from the paper [49], which appeared in 1950. In the early 1950s the 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 immedi- ately recommend it for publication. Your results are very interesting. You are, of course, very right: I am and I have been for a long time strongly interested in a “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 pro- jections) 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 generalityforthesecondone...”. It is remarkable here to recall that von Neumann invented the abstract def- inition of Hilbert spaces, the theory of unbounded operators (as well as much of the bounded theory), ergodic theory, the mathematical formulation of quantum mechanics, many fundamental concepts associated with groups (like amenability), 38 DAVID P. BLECHER 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 Kadi- son entered the world with a bang: a spate of amazing papers at von Neumann’s ‘second level’. Indeed Kadison soon took leadership of the American school of op- erator algebras. Some part of his early work was concerned with positive cones 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 op- erators individually and with respect to containing operator algebras’. His memoir “A representation theory for commutative topological algebra” [51] soon followed, one small aspect of which was the introduction and study of positive cones, states, and square roots in Banach algebras. In the 1951 Annals paper [50], Kadison gen- eralized the “Banach-Stone” theorem, characterizing surjective isometries between C∗-algebras. This result has inspired very many functional analysts and innumer- able 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 contin- uing) 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 non- selfadjoint 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 deter- minant 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 us- ing 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],Arvesonwentontowritemanyother seminal papers on nonselfadjoint operator algebras, perhaps most notably [5], in C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 39 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, in- troduced strictly positive elements in a C∗-algebra A,namelyx ∈ 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 equiv- alent: (1) A has a strictly positive element. (2) A has a countable increasing contractive approximate identity. (3) A = zAz for some positive z ∈ A. (4) The positive cone A+ has an element z of full support (that is, the support projection s(z) is 1). The approximate identity in (2) may be taken to be commuting, indeed it may be 1 ∗ taken to be (z n ) for z as in (3).IfA 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 characteri- zation 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 (topo- logically) principal, that is of the form zAz (resp. zA)forsomez ∈ 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 compari- son 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 40 DAVID P. BLECHER selfadjoint operator algebras (or perhaps even Banach algebras). In fact one of- ten can, as we have shown in joint work with Charles Read, Matt Neal, Narutaka Ozawa, and others. In the Banach algebra literature of course there are many gen- eralizations of C∗-algebra results, but as far as we are aware there is no ‘positivity’ approach like ours (although there is a trace of it in [37]). In particular we 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 ap- proximate 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.Inthiswork,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+t2x2 for all t>0. C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 41 (3) exp(−tx)≤1 for all t>0. −1≤ 1 (4) (t + x) t for all t>0. (5) 1 − tx≤1 − t2x2 for all t>0. Proof. For the equivalence of (1) and (3), see [27, p. 17]. Clearly (5) implies (2). That (2) implies (1) follows by applying a state ϕ to see |1 − tϕ(x)|≤1+Kt2, ≥ 1 which forces Re ϕ(x) 0) (see [63, Lemma 2.1]). Given (4) with t replaced by t , we have 1 − tx = (1 + tx)−1(1 + tx)(1 − tx)≤1 − t2x2. This gives (5). Finally (1) implies (4) by e.g. Stampfli and Williams result [76, Lemma 1] that the norm in (4) is dominated by the reciprocal of the distance from −t to the numerical range of x. (We mention another equivalent condition: given >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.Itisreal completely positive,orRCP 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]forFA.) Theorem 3.2. A (not necessarily unital) linear map T : A → B between C∗- algebras or operator systems is completely positive in the usual sense iff it is RCP. We say that an algebra is approximately unital if it has a contractive approxi- mate identity (cai). Theorem 3.3 (Extension and Stinespring-type Theorem). AlinearmapT : 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).HereC∗(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. 42 DAVID P. BLECHER 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.WesaythatP is a conditional expectation if P (P (a)bP (c)) = P (a)P (b)P (c)fora, b, c ∈ A. Proposition 3.5. A real completely positive completely contractive map (resp. projection) on an approximately unital operator algebra A, extends to a unital com- pletely contractive map (resp. projection) on the unitization A1. Much earlier, we studied completely contractive projections P and 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,iffP is real completely positive. Proof. For the ‘forward direction’ note that P ∗∗ is a unital complete contrac- tion, and hence is real completely positive as we said in above Theorem 3.4. For the ‘backward direction’ the following proof, due to the author and Neal, was originally aremarkin[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)de- fines a bilinear completely contractive product on Ran(P ) with ‘unit’ P (1). Let I(A) be the injective envelope of A.WemayextendP 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 ). C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 43 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)=0wherez = 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 (z2) ≤ P (z)=0. Thus P (P (1)x)=x if x ∈ P (A). Similarly, P (xP (1)) = x as desired. Thus P (xy) defines a bilinear completely contractive product on Ran(P ) with ‘unit’ P (1). The main thrust of [20] is the investigation of the completely contractive pro- jections and conditional expectations, and in particular the ‘symmetric projection problem’ and the ‘bicontractive projection problem’, in the category of operator al- gebras, 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), concern- ing 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)ifI − 2P is completely contractive (resp. if P and I − P are completely contractive). ‘Completely symmetric’ implies ‘com- pletely bicontractive’. The two problems mentioned at the start of this 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 projec- 1 ∗ tions are also symmetric, and are precisely 2 (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 setoffixedpointsofθ,isaC∗-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 bi- contractive’ 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).Thereexistsa 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 44 DAVID P. BLECHER D → D.Also,P is the zero map on q⊥A + Aq⊥ + q⊥Aq⊥,and 1 P = (I + θ)onD. 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 1 P = 2 (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 com- pletely 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,whereq = 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,where1A ∈ B = P (A),C =(I − P )(A), and we have the relations C2 ⊂ B,CB + BC ⊂ C (see [20, Lemma 4.1] and its proof). The period 2 map θ : x + y → x − y for x ∈ B,y ∈ C is a homomorphism (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 → ± ≤ 1 θ : A A such that I θ cb 2 and P = 2 (I + θ).TherangeofP 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. C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 45 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,andP 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 1 spectral radius). So P = 2 (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 π is the cone consisting of elements of A which are ‘sectorial’ of angle θ< 2 .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 iρ Sθ consisting of complex numbers re with r ≥ 0and|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≤1thenf → f(I − x)isacontractive 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 < .Thatis,x has sectorial angle < arcsin .In an operator algebra any contraction x with such a sectorial angle is accretive and satisfies x−Re x≤ ,sox 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) 46 DAVID P. BLECHER n-th roots of a have spectrum and numerical radius within a sector S π , and hence 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 C∗-METHODS FOR OPERATOR AND BANACH ALGEBRAS 47 unless the last quantity is < 1inwhichcase1 − x(A1)∗∗ =1.Heree 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)forA, (ii) a contractive functional ϕ on A with ϕ(et) → 1 for all cai (et)forA, and (iii) a norm 1 functional on A that extends to a state on A1,whereA1 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,ifA is unital, be confused with the real positive (i.e. accretive) elements in A∗∗.Itisshownin[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∗∗ .HenceA has 1 acaiin 2 FA. 48 DAVID P. BLECHER Applied to approximately unital operator algebras (which as we said are all M- approximately unital) the last assertion of Theorem 4.1 becomes Read’s 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 =1andy − 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 ∗ afixedcaie =(et)forA define Se(A)={ϕ ∈ Ball(A ) : limt ϕ(et)=1} (a subset e { ∈ ≥ ∈ } of S(A)). Define rA = x A :Reϕ(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 e rA contains rA. On the other hand, [21, Theorem 6.5] (or a minor variant of the ∗∗ e ∗∗ proofofit)showsthatifA is unital then rA is never contained in rA (or in the ∗∗ e accretive elements in A ) unless rA = rA. Theorem 4.3. [21] A Banach algebra A withasequentialcaie and with Qe(A) e weak* closed, has a sequential cai in rA. 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)existsforeveryx ∈ 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,forallg ∈ conv{fn},iff ∈ conv{fn} − − 3 { } with f g < 4 ,thereisapointx in K with g(x) < 4 .SoA = 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