## 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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Publisher | American Mathematical Society |

Number of pages | 97 |

Volume | 257 |

Edition | 1233 |

ISBN (Electronic) | 978-1-4704-4949-0 |

ISBN (Print) | 978-1-4704-3470-0 |

DOIs | |

Publication status | Published - 30 Jan 2019 |

### Publication series

Name | Memoirs of the American Mathematical Society |
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Publisher | American Mathematical Society |

ISSN (Print) | 0065-9266 |

### Bibliographical note

Research partially supported by EPSRC (grant no. I019227/1-2), by NSF (grant no.DMS-1201385), by JSPS (the Grant-in-Aid for Research Activity Start-up 25887031), by NSERC (PDF, held by AT), by an Alexander von Humboldt foundation fellowship (held by SW) and by the DFG (SFB 878).