**Memoirs of the American Mathematical Society**

2019;
97 pp;
Softcover

MSC: Primary 46;

Print ISBN: 978-1-4704-3470-0

Product Code: MEMO/257/1233

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-4949-0
Product Code: MEMO/257/1233.E**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

# Covering Dimension of C*-Algebras and 2-Coloured Classification

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*Joan Bosa; Nathanial P. Brown; Yasuhiko Sato; Aaron Tikuisis; Stuart White; Wilhelm Winter*

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.

#### Table of Contents

# Table of Contents

## Covering Dimension of C*-Algebras and 2-Coloured Classification

- Cover Cover11
- Title page i2
- Introduction 110
- Chapter 1. Preliminaries 918
- Chapter 2. A 2 x 2 matrix trick 2332
- Chapter 3. Ultrapowers of trivial 𝑊*-bundles 2736
- Chapter 4. Property (SI) and its consequences 4352
- Chapter 5. Unitary equivalence of totally full positive elements 5766
- Chapter 6. 2-coloured equivalence 6776
- Chapter 7. Nuclear dimension and decomposition rank 7382
- Chapter 8. Quasidiagonal traces 7988
- Chapter 9. Kirchberg algebras 8392
- Addendum 91100
- Bibliography 93102
- Back Cover Back Cover1112