### Abstract

© 2019 American Mathematical Society. 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.

Original language | English |
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Pages (from-to) | 1-112 |

Journal | Memoirs of the American Mathematical Society |

Volume | 257 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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## Cite this

Bosa, J., Brown, N. P., Sato, Y., Tikuisis, A., White, S., & Winter, W. (2019). Covering dimension of C∗-algebras and 2-coloured classification.

*Memoirs of the American Mathematical Society*,*257*, 1-112. https://doi.org/10.10.1090/memo/1233