### Abstract

We introduce the concept of nitely 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 classication 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 nite topological covering dimension, this conrms 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 nite nuclear dimension.

\homotopy equivalence implies isomorphism" result for large classes of C-algebras with nite nuclear dimension.

Original language | English |
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Number of pages | 97 |

Journal | Memoirs of the American Mathematical Society |

Volume | 257 |

Issue number | 1233 |

Early online date | 10 Jan 2019 |

DOIs | |

Publication status | Published - 30 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*(1233). https://doi.org/10.1090/memo/1233