Softcover ISBN:  9781470434700 
Product Code:  MEMO/257/1233 
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Electronic ISBN:  9781470449490 
Product Code:  MEMO/257/1233.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 257; 2019; 97 ppMSC: Primary 46;
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 nonAF, 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 TomsWinter conjecture. Inspired by homotopyrigidity 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

Chapters

Introduction

1. Preliminaries

2. A \texorpdfstring{$2\times 2$}2 x 2 matrix trick

3. Ultrapowers of trivial $\mathrm {W}^*$bundles

4. Property (SI) and its consequences

5. Unitary equivalence of totally full positive elements

6. $2$coloured equivalence

7. Nuclear dimension and decomposition rank

8. Quasidiagonal traces

9. Kirchberg algebras

Addendum


Additional Material

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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 nonAF, 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 TomsWinter conjecture. Inspired by homotopyrigidity 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.

Chapters

Introduction

1. Preliminaries

2. A \texorpdfstring{$2\times 2$}2 x 2 matrix trick

3. Ultrapowers of trivial $\mathrm {W}^*$bundles

4. Property (SI) and its consequences

5. Unitary equivalence of totally full positive elements

6. $2$coloured equivalence

7. Nuclear dimension and decomposition rank

8. Quasidiagonal traces

9. Kirchberg algebras

Addendum