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$$C^*$$-algebras and finite-dimensional approximations. (English) Zbl 1160.46001
Graduate Studies in Mathematics 88. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4381-9/hbk). xv, 509 p. (2008).
Finite-dimensional approximations play a central role in the study of operator algebras. Typically, they refer to approximation of von Neumann or C*-algebras by finite-dimensional (or more generally type I) subalgebras, or to approximation of morphisms between operator algebras by linear maps that factor through finite-dimensional algebras, preserving some important structure (e.g., positivity). Hyperfinite von Neumann algebras and AF (or, more generally, approximately sub-homogeneous) C*-algebras provide important classes of operator algebras arising from the former approximation type, while the notions of nuclear and exact C*-algebras emerge naturally from the latter. In many instances, the study of actions of groups on operator algebras makes significant use of such approximations.
The monograph under review provides a highly significant contribution to a broad area of research in operator algebras related to finite-dimensional approximation. The authors have succeeded in covering in detail a good number of topics where progress has been achieved in recent years, unveiling the main ideas and explaining how technical difficulties can be surpassed. This book is thoroughly, elegantly, and candidly written. A large number of themes are presented in a novel and ingenious way, illustrating the recent important contributions of the authors. Interesting exercises and useful comments and bibliographical references are present at the end of each chapter. Very importantly, a large number of open problems are mentioned, some being collected in Chapter 10. Among them, a central rôle is conferred to Connes’s embedding problem and its equivalent characterizations found by Kirchberg (Problem 10.4.4).
Prerequisites consist of basic knowledge of operator algebras, which can be provided by, say, the textbook [G. J. Murphy, “C*-algebras and Operator Theory”, Academic Press, Boston (1990; Zbl 0714.46041)]. There is very little (unavoidable) overlap with the tracts or textbooks that appeared in recent years, such as [S. Wassermann, “Exact C*-Algebras and Related Topics”, Seoul National University Lecture Notes 19 (1994; Zbl 0828.46054); K. R. Davidson, “C*-Algebras by Example”, Fields Institute Monographs 6 (1996; Zbl 0958.46029); P. Fillmore, “A User’s Guide to Operator Algebras”, Wiley, New York (1996; Zbl 0853.46053); M. Rørdam, F. Larsen, N. J. Laustsen, “An Introduction to K-theory for C*-Algebras”, London Mathematical Society Student Texts 49, Cambridge University Press (2000; Zbl 0967.19001); H. Lin, “An Introduction to the Classification of Amenable C*-Algebras”, World Scientific, Singapore (2001; Zbl 1013.46055)].
The self-declared aim of the authors was to serve, through this text, the needs of both the (second year) student and the veteran. In this reviewer’s opinion, their enterprise has been very successful and will vastly contribute to the advance and dissemination of research in operator algebras in the next decade (at least).
The monograph is divided into four parts. The extensive content deserves to be reproduced here.
Chapter 1: Fundamental Facts (Notation; C*-algebras; Von Neumann algebras; Double duals; Completely positive maps; Arveson’s extension theorem; Voiculescu’s theorem).
Part 1: Basic Theory.
Chapter 2: Nuclar and Exact $$C^*$$-Algebras (Nuclear maps; technicalities; Nuclear and exact C*-algebra; First examples; C*-algebras associated to discrete groups; Amenable groups; Type I C*-algebras).
Chapter 3: Tensor Products (Algebraic tensor products; Analytic preliminaries; The spatial and maximal C*-norms; Takesaki’s theorem; Continuity of tensor product maps; Inclusions and The Trick; Exact sequences; Nuclearity and tensor products; Exactness and tensor products).
Chapter 4: Constructions (Crossed products; Integer actions; Amenable actions; $$X\rtimes \Gamma$$-algebras; Compact group actions and graph C*-algebras; Cuntz-Pimsner algebras; Reduced amalgamated free products; Maps on reduced amalgamated free products).
Chapter 5: Exact Groups and Related Topics (Exact groups; Groups acting on trees; Hyperbolic groups; Subgroups of Lie groups; Coarse metric spaces; Groupoids).
Chapter 6: Amenable Traces and Kirchberg’s Factorization Property (Traces and the right regular representation; Amenable traces; Some motivation and examples; The factorization property and Kazhdan’s property (T)).
Chapter 7: Quasidiagonal C*-Algebras (The definition; Easy examples and obstructions; The representation theorem; Homotopy invariance; Two more examples; External approximation).
Chapter 8: AF Embeddability (Stable uniqueness and asymptotically commuting diagrams; Cones over exact RFD algebras; Cones over general exact algebras; Homotopy invariance; A survey).
Chapter 9: Local Reflexivity and Other Tensor Product Conditions (Local reflexivity; Tensor product properties; Equivalence of exactness and property C; Corollaries).
Chapter 10: Summary and Open Problems (Nuclear C*-algebras; Exact C*-algebras; Quasidiagonal C*-algebras; Open problems).
Part 2: Special Topics.
Chapter 11: Simple C*-Algebras (Generalized inductive limits; NF and strong NF algebras; Inner quasidiagonality; Excision and Popa’s technique; Connes’s uniqueness theorem).
Chapter 12: Approximation Properties for Groups (Kazhdan’s property (T); The Haagerup property; Weak amenability; Another approximation property).
Chapter 13: Weak Expectation Property and Local Lifting Property (The local lifting property; Tensorial characterizations of the LLP and WEP; The QWEP conjecture; Nonsemisplit extensions; Norms on $$B(\ell^2) \odot B(\ell^2)$$).
Chapter 14: Weakly Exact von Neumann Algebras (Definition and examples; Characterization of weak exactness).
Part 3: Applications.
Chapter 15: Classification of Group von Neumann Algebras (Subalgebras with noninjective relative commutants; On bi-exactness; Examples).
Chapter 16: Herrero’s Approximation Problem (Description of the problem; C*-preliminaries; Resolution of Herrero’s problem; Counterexamples).
Chapter 17: Counterexamples in K-Homology and K-Theory (BDF preliminaries; Property (T) and Kazhdan projections; Ext need not be a group; Topology on Ext).
Part 4: Appendices.
Appendix A: Ultrafilters and Ultraproducts.
Appendix B: Operator Spaces, Completely Bounded Maps and Duality.
Appendix C: Lifting Theorems.
Appendix D: Positive Definite Functions, Cocycles and Schoenberg’s Theorem.
Appendix E: Groups and Graphs.
Appendix F: Bimodules over von Neumann Algebras.

##### MSC:
 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46L35 Classifications of $$C^*$$-algebras 46L05 General theory of $$C^*$$-algebras 46L06 Tensor products of $$C^*$$-algebras 46L07 Operator spaces and completely bounded maps 46L10 General theory of von Neumann algebras 46L55 Noncommutative dynamical systems 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 19K33 Ext and $$K$$-homology 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 43A07 Means on groups, semigroups, etc.; amenable groups