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Covering Dimension of C$^{*}$-Algebras and 2-Coloured Classification
About this Title
Joan Bosa, School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, Scotland., Nathanial P. Brown, Department of Mathematics, The Pennsylvavia State University, University Park, State College, Pennsylvania 16802, Yasuhiko Sato, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan., Aaron Tikuisis, Institute of Mathematics, School of Natural and Computing Sciences, University of Aberdeen, AB24 3UE, Scotland., Stuart White, School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, Scotland —and— Mathematisches Institut der WWU Münster, Einsteinstraße 62, 48149 Münster, Germany. and Wilhelm Winter, Mathematisches Institut der WWU Münster, Einsteinstraße 62, 48149 Münster, Germany
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 257, Number 1233
ISBNs: 978-1-4704-3470-0 (print); 978-1-4704-4949-0 (online)
DOI: https://doi.org/10.1090/memo/1233
Published electronically: January 10, 2019
MSC: Primary 46L05, 46L35
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
Abstract
We introduce the concept of finitely coloured equivalence for unital $^*$-homomorphisms between $\mathrm C^*$-algebras, for which unitary equivalence is the $1$-coloured case. We 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 we 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, we derive a “homotopy equivalence implies isomorphism” result for large classes of $\mathrm {C}^{*}$-algebras with finite nuclear dimension.
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