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Introduction To The Classification Of Amenable C*-algebras, An - Huaxin Lin

Introduction To The Classification Of Amenable C*-algebras, An

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Published: 12th November 2001
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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.

..". this book is a well-readable and comprehensive guide for those who want to study and work with this theory."

Preface
The Basics of C*-algebrasp. 1
Banach algebrasp. 1
C*-algebrasp. 9
Commutative C*-algebrasp. 12
Positive conesp. 16
Approximate identities, hereditary C*-subalgebras and quotientsp. 20
Positive linear functionals and a Gelfand-Naimark theoremp. 25
Von Neumann algebrasp. 32
Enveloping von Neumann algebras and the spectral theoremp. 38
Examples of C*-algebrasp. 42
Inductive limits of C*-algebrasp. 51
Amenable C*-algebras and K-theoryp. 67
Completely positive linear maps and the Stinespring representationp. 67
Examples of completely positive linear mapsp. 72
Amenable C*-algebrasp. 76
K-theoryp. 82
Perturbationsp. 89
Examples of K-groupsp. 97
K-theory of inductive limits of C*-algebrasp. 103
AF-algebras and Ranks of C*-algebrasp. 113
C*-algebras of stable rank one and their K-theoryp. 113
C*-algebras of lower rankp. 120
Order structure of K-theoryp. 127
AF-algebrasp. 133
Simple C*-algebrasp. 140
Tracial topological rankp. 146
Simple C*-algebras with TR(A) [less than or equal to] 1p. 154
Classification of Simple AT-algebrasp. 165
Some basics about AT-algebrasp. 165
Unitary groups of C*-algebras with real rank zerop. 170
Simple AT-algebras with real rank zerop. 177
Unitaries in simple C-algebra with RR(A) = 0p. 182
A uniqueness theoremp. 186
Classification of simple AT-algebrasp. 192
Invariants of simple AT-algebrasp. 196
C*-algebra Extensionsp. 211
Multiplier algebrasp. 211
Extensions of C*-algebrasp. 217
Completely positive maps to M[subscript n](C)p. 221
Amenable completely positive mapsp. 227
Absorbing extensionsp. 233
A stable uniqueness theoremp. 243
K-theory and the universal coefficient theoremp. 250
Characterization of KK-theory and a universal multi-coefficient theoremp. 255
Approximately trivial extensionsp. 259
Classification of Simple Amenable C*-algebrasp. 269
An existence theoremp. 269
Simple AH-algebrasp. 279
The classification theoremsp. 288
Invariants and some isomorphism theoremsp. 295
Bibliographyp. 307
Indexp. 317
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9789810246808
ISBN-10: 9810246803
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 332
Published: 12th November 2001
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 21.59 x 15.88  x 2.54
Weight (kg): 0.56