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

Joan Bosa University of Glasgow, Glasgow, Scotland, United Kingdom
Nathanial P. Brown The Pennsylvania State University, University Park, Pennsylvania
Yasuhiko Sato Kyoto University, Kyoto, Japan
Aaron Tikuisis University of Aberdeen, Aberdeen, Scotland, United Kingdom
Stuart White University of Glasgow, Glasgow, Scotland, United Kingdom and University of Münster, Münster, Germany
Wilhelm Winter University of Münster, Münster, Germany
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Softcover ISBN: 978-1-4704-3470-0
Product Code: MEMO/257/1233
List Price: $81.00 MAA Member Price:$72.90
AMS Member Price: $48.60 Electronic 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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AMS Member Price: $72.90 Click above image for expanded view Covering Dimension of C*-Algebras and 2-Coloured Classification Joan Bosa University of Glasgow, Glasgow, Scotland, United Kingdom Nathanial P. Brown The Pennsylvania State University, University Park, Pennsylvania Yasuhiko Sato Kyoto University, Kyoto, Japan Aaron Tikuisis University of Aberdeen, Aberdeen, Scotland, United Kingdom Stuart White University of Glasgow, Glasgow, Scotland, United Kingdom and University of Münster, Münster, Germany Wilhelm Winter University of Münster, Münster, Germany Available Formats:  Softcover ISBN: 978-1-4704-3470-0 Product Code: MEMO/257/1233  List Price:$81.00 MAA Member Price: $72.90 AMS Member Price:$48.60
 Electronic 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 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$121.50 MAA Member Price: $109.35 AMS Member Price:$72.90
• Book Details

Memoirs of the American Mathematical Society
Volume: 2572019; 97 pp
MSC: 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.

• 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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Volume: 2572019; 97 pp
MSC: 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.

• 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