| eBook ISBN: | 978-1-4704-4949-0 |
| Product Code: | MEMO/257/1233.E |
| List Price: | $81.00 |
| MAA Member Price: | $72.90 |
| AMS Member Price: | $48.60 |
| eBook ISBN: | 978-1-4704-4949-0 |
| Product Code: | MEMO/257/1233.E |
| List Price: | $81.00 |
| MAA Member Price: | $72.90 |
| AMS Member Price: | $48.60 |
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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 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.
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Table of Contents
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Chapters
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Introduction
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1. Preliminaries
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2. A \texorpdfstring{$2\times 2$}2 x 2 matrix trick
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3. Ultrapowers of trivial $\mathrm {W}^*$-bundles
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4. Property (SI) and its consequences
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5. Unitary equivalence of totally full positive elements
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6. $2$-coloured equivalence
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7. Nuclear dimension and decomposition rank
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8. Quasidiagonal traces
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9. Kirchberg algebras
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Addendum
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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 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.
-
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
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Addendum
