Covering dimension of C*-algebras and 2-coloured classification / Joan Bosa [and five others].

Format:

Book

Author/Creator:

Bosa, Joan, 1985- author.

Series:

Memoirs of the American Mathematical Society ; Volume 257, Number 1233.

Memoirs of the American Mathematical Society ; Volume 257, Number 1233

Language:

English

Subjects (All):

C*-algebras.

Homomorphisms (Mathematics).

Extremal problems (Mathematics).

Physical Description:

1 online resource (112 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2019]

Summary:

The authors introduce the concept of finitely coloured equivalence for unital ^*-homomorphisms between \mathrm C^*-algebras, for which unitary equivalence is the 1-coloured case. They use this notion to classify ^*-homomorphisms from separable, unital, nuclear \mathrm C^*-algebras into ultrapowers of simple, unital, nuclear, \mathcal Z-stable \mathrm C^*-algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application the authors calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, \mathcal Z-stable \mathrm C^*-algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, the authors derive a "homotopy equivalence implies isomorphism" result for large classes of \mathrm C^*-algebras with finite nuclear dimension.

Contents:

Cover

Title page

Introduction

From classification to nuclear dimension

Outline of the proof of Theorem D

Structure of the paper

Acknowledgements

Chapter 1. Preliminaries

1.1. Order zero maps

1.2. Traces and Cuntz comparison

1.3. Ultraproducts and the reindexing argument

Chapter 2. A 2 x 2 matrix trick

Chapter 3. Ultrapowers of trivial *-bundles

3.1. Continuous *-bundles

3.2. Tensor products, ultraproducts and McDuff bundles

3.3. Strict comparison of relative commutant sequence algebras for McDuff bundles

3.4. Traces on a relative commutant

3.5. Unitary equivalence of maps into ultraproducts

Chapter 4. Property (SI) and its consequences

4.1. Property (SI)

4.2. Proof of Theorem 4.1

Chapter 5. Unitary equivalence of totally full positive elements

5.1. Proof of Theorem 5.1

5.2. Theorem D

Chapter 6. 2-coloured equivalence

Chapter 7. Nuclear dimension and decomposition rank

Chapter 8. Quasidiagonal traces

Chapter 9. Kirchberg algebras

Addendum

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4949-8