Covering dimension of C*-algebras and 2-coloured classification

Bosa, J., Brown, N. P., Sato, Y., Tikuisis, A., White, S. and Winter, W. (2016) Covering dimension of C*-algebras and 2-coloured classification. Memoirs of the American Mathematical Society, (Accepted for Publication)

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Abstract

We introduce the concept of finitely coloured equivalence for unital *-homomorphisms between C*-algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *- homomorphisms from separable, unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear, Z-stable 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, Z-stable 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 C*-algebras with finite nuclear dimension.

Item Type:Articles
Status:Accepted for Publication
Refereed:Yes
Glasgow Author(s) Enlighten ID:White, Professor Stuart and Bosa, Dr Joan
Authors: Bosa, J., Brown, N. P., Sato, Y., Tikuisis, A., White, S., and Winter, W.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Memoirs of the American Mathematical Society
Publisher:American Mathematical Society
ISSN:0065-9266
ISSN (Online):1947-6221
Copyright Holders:Copyright © 2016 American Mathematical Society
First Published:First published in Memoirs of the American Mathematical Society 2016
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
596411The Cuntz Semigroup and the Fine Structure of Nuclear C*-AlgebrasStuart WhiteEngineering & Physical Sciences Research Council (EPSRC)EP/I019227/1M&S - MATHEMATICS
626331The Cuntz Semigroup and the Fine Stucture of Nuclear C*-AlgebrasJoachim ZachariasEngineering & Physical Sciences Research Council (EPSRC)EP/I019227/2M&S - MATHEMATICS