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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
Covering Dimension of C*-Algebras and 2-Coloured Classification
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
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
Softcover ISBN:  978-1-4704-3470-0
eBook: ISBN:  978-1-4704-4949-0
Product Code:  MEMO/257/1233.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
Covering Dimension of C*-Algebras and 2-Coloured Classification
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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
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
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
Softcover ISBN:  978-1-4704-3470-0
eBook ISBN:  978-1-4704-4949-0
Product Code:  MEMO/257/1233.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $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.

  • 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
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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
  • Addendum
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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