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 otters 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. Besides being as an introduction to the theory of the classification of amenable C*-algebras, it is a comprehensive reference for those more familiar with the subject.
Author(s): Huaxin Lin
Edition: 1st
Publisher: World Scientific
Year: 2001
Language: English
Pages: 333
City: Singapore; River Edge, NJ
Preface......Page 8
Contents......Page 10
1.1 Banach algebras......Page 13
1.2 C*-algebras......Page 21
1.3 Commutative C*-algebras......Page 24
1.4 Positive cones......Page 28
1.5 Approximate identities, hereditary C*-subalgebras and quotients......Page 32
1.6 Positive linear functionals and a Gelfand-Naimark theorem......Page 37
1.7 Von Neumann algebras......Page 44
1.8 Enveloping von Neumann algebras and the spectral theorem......Page 50
1.9 Examples of C*-algebras......Page 54
1.10 Inductive limits of C*-algebras......Page 63
1.11 Exercises......Page 72
1.12 Addenda......Page 77
2.1 Completely positive linear maps and the Stinespring representation......Page 79
2.2 Examples of completely positive linear maps......Page 84
2.3 Amenable C*-algebras......Page 88
2.4 K-theory......Page 94
2.5 Perturbations......Page 101
2.6 Examples of K-groups......Page 109
2.7 K-theory of inductive limits of C*-algebras......Page 115
2.8 Exercises......Page 120
2.9 Addenda......Page 123
3.1 C*-algebras of stable rank one and their K-theory......Page 125
3.2 C*-algebras of lower rank......Page 132
3.3 Order structure of K-theory......Page 139
3.4 AF-algebras......Page 145
3.5 Simple C*-algebras......Page 152
3.6 Tracial topological rank......Page 158
3.7 Simple C*-algebras with TR(A) < 1......Page 166
3.8 Exercises......Page 172
3.9 Addenda......Page 174
4.1 Some basics about AT-algebras......Page 177
4.2 Unitary groups of C*-algebras with real rank zero......Page 182
4.3 Simple AT-algebras with real rank zero......Page 189
4.4 Unitaries in simple C*-algebra with RR(A) = 0......Page 194
4.5 A uniqueness theorem......Page 198
4.6 Classification of simple AT-algebras......Page 204
4.7 Invariants of simple AT-algebras......Page 208
4.8 Exercises......Page 216
4.9 Addenda......Page 220
5.1 Multiplier algebras......Page 223
5.2 Extensions of C*-algebras......Page 229
5.3 Completely positive maps to Mn(C)......Page 233
5.4 Amenable completely positive maps......Page 239
5.5 Absorbing extensions......Page 245
5.6 A stable uniqueness theorem......Page 255
5.7 K-theory and the universal coefficient theorem......Page 262
5.8 Characterization of KK-theory and a universal multi-coefficient theorem......Page 267
5.9 Approximately trivial extensions......Page 271
5.10 Exercises......Page 277
6.1 An existence theorem......Page 281
6.2 Simple AH-algebras......Page 291
6.3 The classification theorems......Page 300
6.4 Invariants and some isomorphism theorems......Page 307
Bibliography......Page 319
Index......Page 329
