eBook

Covering Dimension of C-Algebras and 2-Coloured Classification

White, Stuart, Brown, Nathanial P., Bosa, Joan, Sato, Yasuhiko, Winter, Wilhelm, Tikuisis, Aaron

2019

Overview

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.

• Publisher: American Mathematical Society
• Subject: C-algebras / Extremal problems (Mathematics) / Homomorphisms (Mathematics)
• ISBN: 9781470434700, 1470434709