| Description |
1 online resource (xi, 320 pages) |
| Physical Medium |
polychrome |
| Description |
text file |
| Bibliography |
Includes bibliographical references (pages 307-316) and index. |
| Summary |
The theory and applications of C*-algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C*-algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C*-algebras (up to isomorphism) by their K-theoretical data. It started with the classification of AT-algebras with real rank zero. Since then great efforts have been made to classify amenable C*-algebras, a class of C*-algebras that arises most naturally. For example, a large class of simple amenable C*-algebras is discovered to be classifiable. The application of these results to dynamical systems has been established. This book introduces the recent development of the theory of the classification of amenable C*-algebras - the first such attempt. The first three chapters present the basics of the theory of C*-algebras which are particularly important to the theory of the classification of amenable C*-algebras. Chapter 4 offers the classification of the so-called AT-algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C*-algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH-algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C*-algebras. As well as providing an introduction to the theory of the classification of amenable C*-algebras, it is a comprehensive reference for those more familiar with the subject. |
| Contents |
Ch. 1. The Basics of C*-algebras. 1.1. Banach algebras. 1.2. C*-algebras. 1.3. Commutative C*-algebras. 1.4. Positive cones. 1.5. Approximate identities, hereditary C*-subalgebras and quotients. 1.6. Positive linear functionals and a Gelfand-Naimark theorem. 1.7. Von Neumann algebras. 1.8. Enveloping von Neumann algebras and the spectral theorem. 1.9. Examples of C*-algebras. 1.10. Inductive limits of C*-algebras. 1.11. Exercises. 1.12. Addenda -- ch. 2. Amenable C*-algebras and K-theory. 2.1. Completely positive linear maps and the Stinespring representation. 2.2. Examples of completely positive linear maps. 2.3. Amenable C*-algebras. 2.4. K-theory. 2.5. Perturbations. 2.6. Examples of K-groups. 2.7. K-theory of inductive limits of C*-algebras. 2.8. Exercises. 2.9. Addenda -- ch. 3. AF-algebras and ranks of C*-algebras. 3.1. C*-algebras of stable rank one and their K-theory. 3.2. C*-algebras of lower rank. 3.3. Order structure of K-theory. 3.4. AF-algebras. 3.5. Simple C*-algebras. 3.6. Tracial topological rank. 3.7. Simple C*-algebras with TR(A) [symbol] 1. 3.8. Exercises. 3.9. Addenda -- ch. 4. Classification of simple AT-algebras. 4.1. Some basics about AT-algebras. 4.2. Unitary groups of C*-algebras with real rank zero. 4.3. Simple AT-algebras with real rank zero. 4.4. Unitaries in simple C*-algebra with RR(A) = 0. 4.5. A uniqueness theorem. 4.6. Classification of simple AT-algebras. 4.7. Invariants of simple AT-algebras. 4.8. Exercises. 4.9. Addenda -- ch. 5. C*-algebra extensions. 5.1. Multiplier algebras. 5.2. Extensions of C*-algebras. 5.3. Completely positive maps to Mn(C). 5.4. Amenable completely positive maps. 5.5. Absorbing extensions. 5.6. A stable uniqueness theorem. 5.7. K-theory and the universal coefficient theorem. 5.8. Characterization of KK-theory and a universal multi-coefficient theorem. 5.9. Approximately trivial extensions. 5.10. Exercises -- ch. 6. Classification of simple amenable C*-algebras. 6.1. An existence theorem. 6.2. Simple AH-algebras. 6.3. The classification theorems. 6.4. Invariants and some isomorphism theorems. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
C*-algebras.
|
|
C*-algebras. |
|
K-theory.
|
|
K-theory. |
| Genre/Form |
Electronic books.
|
|
Electronic books.
|
| Added Title |
Amenable C*-algebras |
| Other Form: |
Print version: Lin, Huaxin, 1956- Introduction to the classification of amenable C*-algebras. Singapore ; River Edge, NJ : World Scientific, ©2001 9810246803 9789810246808 (DLC) 2002282557 (OCoLC)49195508 |
| ISBN |
9789812799883 (electronic book) |
|
9812799885 (electronic book) |
|
1281951439 |
|
9781281951434 |
|
9789810246808 |
|
9810246803 |
|
