Hilbert C*-Modules

✍ Scribed by Vladimir Markovich Manuǐlov, Evgeniĭ Vadimovich Troit︠s︡kiĭ

• Publisher: American Mathematical Society
• Year: 2005
• Tongue: English
• Leaves: 214
• Series: Translations of Mathematical Monographs volume 226
• Category: Library

✦ Synopsis

Based on lectures delivered by the authors at Moscow State University, this volume presents a detailed introduction to the theory of Hilbert $C^$-modules. Hilbert $C^$-modules provide a natural generalization of Hilbert spaces arising when the field of scalars $\mathbf{C}$ is replaced by an arbitrary $C^$-algebra. The general theory of Hilbert $C^$-modules appeared more than 30 years ago in the pioneering papers of W. Paschke and M. Rieffel and has proved to be a powerful tool in operator algebras theory, index theory of elliptic operators, $K$- and $KK$-theory, and in noncommutative geometry as a whole. Alongside these applications, the theory of Hilbert $C^*$-modules is interesting on its own. In this book, the authors explain in detail the basic notions and results of the theory, and provide a number of important examples. Some results related to the authors' research interests are also included. A large part of the book is devoted to structural results (self-duality, reflexivity) and to nonadjointable operators. Most of the book can be read with only a basic knowledge of functional analysis; however, some experience in the theory of operator algebras makes reading easier.

✦ Table of Contents

Cover......Page 1
Title: Hilbert C-Modules......Page 3
512'.556—dc22......Page 4
Contents......Page 6
Preface......Page 8
1.1. C- Algebras......Page 10
1.2. Pre-Hilbert modules......Page 12
1.3. Hubert C-Modules......Page 13
1.4. The standard Hubert module HA......Page 17
1.5. Hubert C-Bimodules and strong Morita equivalence......Page 20
2.1. Bounded and adjointable operators......Page 24
2.2. Compact operators in Hubert modules......Page 27
2.3. Complementable submodules and projectionsin Hubert C-Modules......Page 31
2.4. Full Hubert C-Modules......Page 33
2.5. Dual modules. Self-duality......Page 36
2.6. Banach-compact operators......Page 41
2.7. C-Fredholm operators and index. Mishchenko's approach......Page 42
2.8. Representations of groups on Hubert modules......Page 51
2.9. Equivariant Fredholm operators......Page 61
3.1. W-Algebras......Page 64
3.2. Inner product on dual modules......Page 67
3.3. Hubert W-Modules and dual Banach spaces......Page 70
3.4. Properties of Hubert W'-modules......Page 71
3.5. Topological characterization of self-dual Hubert W-Modules......Page 74
3.6. Fredhohm operators over W-Algebras......Page 75
3.7. Dupré — Fillmore theorem for Hubert moduleso ver finite W-Algebras......Page 78
4.1. Inner product on bidual modules......Page 84
4.2. Ideals and bidual modules......Page 88
4.3. Reflexivity of Hubert modules over K+......Page 91
4.4. Reflexivity of modules over C(X)......Page 93
4.5. Hubert modules related to conditional expectations of finite index......Page 95
5.1. Extension of a Hubert C-module by the enveloping W-algebra......Page 108
5.2. Multipliers and centralizers......Page 110
5.3. Multipliers of A-compact operators......Page 116
5.4. Quasi-multipliers of A-compact operators......Page 119
5.5. Strict topology......Page 123
5.6. Multipliers and Hubert modules. The commutative case......Page 127
5.7. Inner products on Hilbert C-Modules......Page 135
6.1. Problem of diagonalizing operators in Hubert C-Modules......Page 142
6.2. Quadratic forms on H'a related to selfadjoint operators......Page 145
6.3. Diagonalizing operators in the W-Case......Page 147
6.4. Continuity of "eigenvalues"......Page 152
6.5. Case of infinite W-Algebras......Page 154
6.6. Case of C-Algebras of real rank zero......Page 155
6.7. Case of continuous fields of trace C-Algebras......Page 157
6.8. Schrödinger operator as an operator acting on a Hubert C*-Module......Page 163
6.9. Example: A continuous field of' rotation algebras......Page 166
7.1. Technical lemmas......Page 168
7.2. Proof of the Cuntz—Higson theorem......Page 173
7.3. The case A subst of K......Page 175
7.4. Some other cases......Page 178
7.5. Dixmier-Douady Theorem for L2(A)......Page 182
7.6. Some generalizations......Page 184
7.7. Neubauer type homotopy......Page 185
8.1. Tensor products......Page 190
8.2. Main definitions......Page 191
8.3. Cuntz's approach......Page 193
8.4. Generalized Kasparov bimodules......Page 197
8.5. Classifying spaces for some K- and KK-groups......Page 198
Bibliography......Page 202
Notation Index......Page 208
Index......Page 210
Titles in This Series......Page 212
Back Cover......Page 214