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

Authors :
Joan Bosa
Nathanial P. Brown
Yasuhiko Sato
Aaron Tikuisis
Stuart White
Wilhelm Winter
Joan Bosa
Nathanial P. Brown
Yasuhiko Sato
Aaron Tikuisis
Stuart White
Wilhelm Winter
Publication Year :
2019

Abstract

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.

Details

Language :
English
ISBNs :
9781470434700 and 9781470449490
Volume :
00257
Database :
eBook Index
Journal :
Covering Dimension of C*-Algebras and 2-Coloured Classification
Publication Type :
eBook
Accession number :
2042357