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

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

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arXiv · 1506.03974

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

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

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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.

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Joan Bosa, Nathanial P. Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White, Wilhelm Winter. 2015-06-12. Covering dimension of C*-algebras and 2-coloured classification. https://arxiv.org/abs/1506.03974

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