Hardcover ISBN:  9780821843819 
Product Code:  GSM/88 
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Electronic ISBN:  9781470421182 
Product Code:  GSM/88.E 
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Book DetailsGraduate Studies in MathematicsVolume: 88; 2008; 509 ppMSC: Primary 46; 05; 22; 43;
\(\mathrm{C}^*\)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 userfriendly 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, welltrained 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.
ReadershipGraduate students and research mathematicians interested in \(\mathrm{C}^*\)algebras and operator algebras.

Table of Contents

Chapters

Chapter 1. Fundamental facts

Part 1. Basic theory

Chapter 2. Nuclear and exact $\textrm {C}^*$algebras: Definitions, basic facts and examples

Chapter 3. Tensor products

Chapter 4. Constructions

Chapter 5. Exact groups and related topics

Chapter 6. Amenable traces and Kirchberg’s factorization property

Chapter 7. Quasidiagonal $\textrm {C}^*$algebras

Chapter 8. AF embeddability

Chapter 9. Local reflexivity and other tensor product conditions

Chapter 10. Summary and open problems

Part 2. Special topics

Chapter 11. Simple $\textrm {C}^*$algebras

Chapter 12. Approximation properties for groups

Chapter 13. Weak expectation property and local lifting property

Chapter 14. Weakly exact von Neumann algebras

Part 3. Applications

Chapter 15. Classification of group von Neumann algebras

Chapter 16. Herrero’s approximation problem

Chapter 17. Counterexamples in $\textrm {K}$homology and $\textrm {K}$theory

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


Additional Material

Reviews

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


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

Chapters

Chapter 1. Fundamental facts

Part 1. Basic theory

Chapter 2. Nuclear and exact $\textrm {C}^*$algebras: Definitions, basic facts and examples

Chapter 3. Tensor products

Chapter 4. Constructions

Chapter 5. Exact groups and related topics

Chapter 6. Amenable traces and Kirchberg’s factorization property

Chapter 7. Quasidiagonal $\textrm {C}^*$algebras

Chapter 8. AF embeddability

Chapter 9. Local reflexivity and other tensor product conditions

Chapter 10. Summary and open problems

Part 2. Special topics

Chapter 11. Simple $\textrm {C}^*$algebras

Chapter 12. Approximation properties for groups

Chapter 13. Weak expectation property and local lifting property

Chapter 14. Weakly exact von Neumann algebras

Part 3. Applications

Chapter 15. Classification of group von Neumann algebras

Chapter 16. Herrero’s approximation problem

Chapter 17. Counterexamples in $\textrm {K}$homology and $\textrm {K}$theory

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

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