**Graduate Studies in Mathematics**

Volume: 88;
2008;
509 pp;
Hardcover

MSC: Primary 46; 05; 22; 43;

Print ISBN: 978-0-8218-4381-9

Product Code: GSM/88

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

**Electronic ISBN: 978-1-4704-2118-2
Product Code: GSM/88.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

#### Supplemental Materials

# \(C^{*}\)-Algebras and Finite-Dimensional Approximations

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*Nathanial P. Brown; Narutaka Ozawa*

\(\mathrm{C}^*\)-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications—written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of \(\mathrm{C}^*\)-approximation theory.

#### Readership

Graduate students and research mathematicians interested in \(\mathrm{C}^*\)-algebras and operator algebras.

#### Reviews & Endorsements

This exciting book takes its readers through a wide palette of topics of current interest within operator algebras and operator space theory. ... These authors have succeeded very well in writing a book that is at the same time a textbook for (graduate) students and a research monograph for experts. ...the entire book makes for enjoyable reading for both types of readers thanks to its clear, informal and witty style. Each section ends with a list of appetizing exercises. ...no other book so far has taken this particular charming path through the landscapes of operator algebras.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## $C^{*}$-Algebras and Finite-Dimensional Approximations

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface xi12 free
- Chapter 1. Fundamental Facts 118 free
- Part 1. Basic Theory 2340
- Chapter 2. Nuclear and Exact C*-Algebras: Definitions, Basic Facts and Examples 2542
- Chapter 3. Tensor Products 5976
- §3.1. Algebraic tensor products 5976
- §3.2. Analytic preliminaries 6683
- §3.3. The spatial and maximal C*-norms 7289
- §3.4. Takesaki's Theorem 7794
- §3.5. Continuity of tensor product maps 8299
- §3.6. Inclusions and The Trick 85102
- §3.7. Exact sequences 92109
- §3.8. Nuclearity and tensor products 99116
- §3.9. Exactness and tensor products 105122
- §3.10. References 112129
- Chapter 4. Constructions vi7

- Chapter 4. References 115132
- §4.1. Crossed products 115132
- §4.2. Integer actions 121138
- §4.3. Amenable actions 124141
- §4.4. X xi T-algebras 129146
- §4.5. Compact group actions and graph C*-algebras 133150
- §4.6. Cuntz-Pimsner algebras 136153
- §4.7. Reduced amalgamated free products 154171
- §4.8. Maps on reduced amalgamated free products 157174
- §4.9. References 165182

- Chapter 5. Exact Groups and Related Topics 167184
- Chapter 6. Amenable Traces and Kirchberg's Factorization Property 211228
- Chapter 7. Quasidiagonal C*-Algebras 237254
- Chapter 8. AF Embeddability 261278
- Chapter 9. Local Reflexivity and Other Tensor Product Conditions 283300
- Chapter 10. Summary and Open Problems 301318

- Part 2. Special Topics 311328
- Part 3. Applications 405422
- Part 4. Appendices 443460
- Appendix A. Ultrafilters and Ultraproducts 445462
- Appendix B. Operator Spaces, Completely Bounded Maps and Duality 449466
- Appendix C. Lifting Theorems 459476
- Appendix D. Positive Definite Functions, Cocycles and Schoenberg's Theorem 463480
- Appendix E. Groups and Graphs 471488
- Appendix F. Bimodules over von Neumann Algebras 479496

- Bibliography 493510
- Notation Index 503520
- Subject Index 505522
- Back cover Back Cover1530