**Contemporary Mathematics**

Volume: 167;
1994;
399 pp;
Softcover

MSC: Primary 22; 46; 58; 19;
Secondary 55

Print ISBN: 978-0-8218-5175-3

Product Code: CONM/167

List Price: $83.00

Individual Member Price: $66.40

**Electronic ISBN: 978-0-8218-7758-6
Product Code: CONM/167.E**

List Price: $83.00

Individual Member Price: $66.40

# $C^{*}$-Algebras: 1943–1993

Share this page *Edited by *
*Robert S. Doran*

This volume contains the proceedings of an AMS Special Session held at the Joint Mathematics Meetings in San Antonio in January 1993 to celebrate the first fifty years of \(C^*\)-algebra theory. The book contains carefully written expository and research articles by leaders in the field. Also included is a reprinting of the original 1943 paper on \(C^*\)-algebras by Gelfand and Neumark, which has had such a profound influence on the field. The volume covers a broad spectrum of topics, including the Gelfand-Neumark theorems, \(C^*\)-algebras and quantization, projections in \(C^*\)-algebras, Mackey's theory of group representations and their relation to \(C^*\)-algebras, transformation group \(C^*\)-algebras, the influence of algebraic topology on \(C^*\)-algebras, K-theory and index theory in operator algebras, exponential rank in \(C^*\)-algebras, and a survey of the development of type III von Neumann algebras. With historical perspectives and up-to-date overviews to orient readers new to the field, this book will interest mathematicians, physicists, and mathematical historians.

#### Table of Contents

# Table of Contents

## $C^{*}$-Algebras: 1943-1993

- Contents vii8 free
- Preface ix10 free
- On the imbedding of normed rings into the ring of operators in Hilbert space 314 free
- Notes on the Gelfand-Neumark theorem 2132
- C*-algebras and quantization 5566
- Quantization and C*-algebras 6778
- Some quantizations and reflections inspired by the Gelfand-Naimark theorem 99110
- The role of C*-algebras in infinite dimensional numerical linear algebra 115126
- Projections in C*-algebras 131142
- C*-algebras and Mackey's theory of group representations 151162
- Transformation group C*-algebras: A selective survey 183194
- Algebraic topology and C*-algebras 219230
- 1. The initial early contacts 220231
- 2. Kasparov's bivariant homology theory 222233
- 3. Digression: A missed opportunity in topology 223234
- 4. K-Theory enters operator theory 224235
- 5. Torsion and Universal Coefficient Theorems 225236
- 6. The Kasparov Spectral Sequence 227238
- 7. The Limits of Algebraic Topology 228239
- 8. Applications to algebraic topology 229240

- Twenty-five years in the theory of type III von Neumann algebras 233244
- Classifying space for proper actions and K-theory of group C*-algebras 241252
- 0. Introduction 242253
- 1. Proper Actions 245256
- 2. Examples of EG 247258
- 3. Equivariant K-Homology 250261
- 4. Lie Groups 254265
- 5. Cosheaf Homology 264275
- 6. p-adic Groups 265276
- 7. Discrete Groups 273284
- 8. An Equivariant Novikov Conjecture 281292
- 9. The Conjecture with Coefficients 283294
- A1. Infinite Join Construction of EG 284295
- A2. Axioms for EG 285296
- A3. What Does BG Classify? 286297

- Odd index theorems and operator algebras 293304
- Index theory without symbols 305316
- A survey of exponential rank 353364
- 0. Introduction 354365
- 1. Definitions and motivation for exponential rank 354365
- 2. Some historical remarks 357368
- 3. Algebras of matrix valued functions and stable C*-algebras 360371
- 4. Real rank zero, weak (FU), and weak (FN) 364375
- 5. Simple C*-algebras with real rank greater than zero 370381
- 6. Exponential length and multiplier algebras 371382
- 7. Exponential rank of Banach algebras 375386
- 8. Projective length and rank 377388
- 9. Commutators 380391
- 10. Other factorization problems 384395
- 11. Real and imaginary relations between the various ranks 389400
- 12. Applications 392403

- Appendix A-1412

#### Readership

Advanced graduate students, professional mathematicians and physicists.

#### Reviews

The articles are of an impressively high standard and wide-ranging in their scope …best picture of the subject since the Proceedings of the 1980 Kingston Symposium. All operator algebraists will want a copy of this book on their shelves, as will workers in those areas of mathematics and physics where \(C^*\)-algebra theory has an influence. The editor … is to be highly congratulated on a highly successful enterprise.

-- Bulletin of the London Mathematical Society