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

TL;DR

This paper introduces a new classification method for *-homomorphisms between certain C*-algebras using coloured equivalence, leading to results on nuclear dimension and confirming parts of the Toms-Winter conjecture.

Contribution

It develops finitely coloured equivalence for *-homomorphisms, enabling classification up to 2-coloured equivalence based on trace behavior, and calculates nuclear dimension for specific C*-algebras.

Findings

Nuclear dimension of certain Z-stable C*-algebras is 1.

Classification up to 2-coloured equivalence based on traces.

Confirms the Toms-Winter conjecture for algebras with finite topological covering dimension.

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.