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

Nathanial P. Brown Pennsylvania State University, State College, PA
Narutaka Ozawa University of California, Los Angeles, Los Angeles, CA and University of Tokyo, Tokyo, Japan
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Hardcover ISBN: 978-0-8218-4381-9
Product Code: GSM/88
List Price: $92.00 MAA Member Price:$82.80
AMS Member Price: $73.60 Electronic ISBN: 978-1-4704-2118-2 Product Code: GSM/88.E List Price:$86.00
MAA Member Price: $77.40 AMS Member Price:$68.80
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List Price: $138.00 MAA Member Price:$124.20
AMS Member Price: $110.40 Click above image for expanded view$\textrm{C}^*$-Algebras and Finite-Dimensional Approximations Nathanial P. Brown Pennsylvania State University, State College, PA Narutaka Ozawa University of California, Los Angeles, Los Angeles, CA and University of Tokyo, Tokyo, Japan Available Formats:  Hardcover ISBN: 978-0-8218-4381-9 Product Code: GSM/88  List Price:$92.00 MAA Member Price: $82.80 AMS Member Price:$73.60
 Electronic ISBN: 978-1-4704-2118-2 Product Code: GSM/88.E
 List Price: $86.00 MAA Member Price:$77.40 AMS Member Price: $68.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$138.00 MAA Member Price: $124.20 AMS Member Price:$110.40
• Book Details

Volume: 882008; 509 pp
MSC: 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 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.

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

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 882008; 509 pp
MSC: 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 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.

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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.