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

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2018Copyright date: ©2019Edition: 1st edDescription: 1 online resource (112 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470449490
Subject(s): Genre/Form: Additional physical formats: Print version:: Covering Dimension of C*-Algebras and 2-Coloured ClassificationDDC classification:
  • 512.556
LOC classification:
  • QC20.7.C14 .B673 2019
Online resources:
Contents:
Cover -- Title page -- Introduction -- From classification to nuclear dimension -- Outline of the proof of Theorem D -- Structure of the paper -- Acknowledgements -- Chapter 1. Preliminaries -- 1.1. Order zero maps -- 1.2. Traces and Cuntz comparison -- 1.3. Ultraproducts and the reindexing argument -- Chapter 2. A 2 x 2 matrix trick -- Chapter 3. Ultrapowers of trivial *-bundles -- 3.1. Continuous *-bundles -- 3.2. Tensor products, ultraproducts and McDuff bundles -- 3.3. Strict comparison of relative commutant sequence algebras for McDuff bundles -- 3.4. Traces on a relative commutant -- 3.5. Unitary equivalence of maps into ultraproducts -- Chapter 4. Property (SI) and its consequences -- 4.1. Property (SI) -- 4.2. Proof of Theorem 4.1 -- Chapter 5. Unitary equivalence of totally full positive elements -- 5.1. Proof of Theorem 5.1 -- 5.2. Theorem D -- Chapter 6. 2-coloured equivalence -- Chapter 7. Nuclear dimension and decomposition rank -- Chapter 8. Quasidiagonal traces -- Chapter 9. Kirchberg algebras -- Addendum -- Bibliography -- Back Cover.
Summary: 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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Cover -- Title page -- Introduction -- From classification to nuclear dimension -- Outline of the proof of Theorem D -- Structure of the paper -- Acknowledgements -- Chapter 1. Preliminaries -- 1.1. Order zero maps -- 1.2. Traces and Cuntz comparison -- 1.3. Ultraproducts and the reindexing argument -- Chapter 2. A 2 x 2 matrix trick -- Chapter 3. Ultrapowers of trivial *-bundles -- 3.1. Continuous *-bundles -- 3.2. Tensor products, ultraproducts and McDuff bundles -- 3.3. Strict comparison of relative commutant sequence algebras for McDuff bundles -- 3.4. Traces on a relative commutant -- 3.5. Unitary equivalence of maps into ultraproducts -- Chapter 4. Property (SI) and its consequences -- 4.1. Property (SI) -- 4.2. Proof of Theorem 4.1 -- Chapter 5. Unitary equivalence of totally full positive elements -- 5.1. Proof of Theorem 5.1 -- 5.2. Theorem D -- Chapter 6. 2-coloured equivalence -- Chapter 7. Nuclear dimension and decomposition rank -- Chapter 8. Quasidiagonal traces -- Chapter 9. Kirchberg algebras -- Addendum -- Bibliography -- Back Cover.

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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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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