**CBMS Regional Conference Series in Mathematics**

Volume: 124;
2017;
240 pp;
Softcover

MSC: Primary 46;

Print ISBN: 978-1-4704-3490-8

Product Code: CBMS/124

List Price: $52.00

AMS Member Price: $41.60

MAA member Price: $46.80

**Electronic ISBN: 978-1-4704-4192-0
Product Code: CBMS/124.E**

List Price: $52.00

AMS Member Price: $41.60

MAA member Price: $46.80

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#### Supplemental Materials

# From the Basic Homotopy Lemma to the Classification of \(C^{*}\)-algebras

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*Huaxin Lin*

A co-publication of the AMS and CBMS

This book examines some recent developments in the theory of \(C^*\)-algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable \(C^*\)-algebras, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, \(KK\)-theory related rotation maps are introduced. The book is based on lecture notes from the CBMS lecture sequence at the University of Wyoming in the summer of 2015.

A co-publication of the AMS and CBMS.

#### Readership

Graduate students and researchers interested in C*-algebras.

#### Table of Contents

# Table of Contents

## From the Basic Homotopy Lemma to the Classification of $C^{*}$-algebras

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Chapter 0. An overview of the Elliott program 18
- Chapter 1. An introduction to the Basic Homotopy Lemma 714
- Chapter 2. Maps to finite dimensional πΆ*-algebras 2128
- Chapter 3. Stable Homotopy Lemmas 5158
- Chapter 4. The Basic Homotopy Lemma, finite dimensional cases 7380
- Chapter 5. πΆ*-algebras of generalized tracial rank one 109116
- 5.1. Elliott-Thomsen building blocks 109116
- 5.2. A uniqueness theorem for subhomogeneous πΆ*-algebras 117124
- 5.3. πΆ*-algebras with πππ (π΄)\le1 129136
- 5.4. A uniqueness theorem for πΆ*-algebras with πππ (π΄)\le1 139146
- 5.5. The range theorem and the isomorphism theorem 149156
- 5.6. Exercises 154161

- Chapter 6. More Basic Homotopy Lemmas 157164
- Chapter 7. Asymptotic unitary equivalence 189196
- Chapter 8. Classification of simple πΆ*-algebras of finite rank 215222
- Bibliography 233240
- Index 239246
- Back Cover Back Cover1249