### Abstract

\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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*Memoirs of the American Mathematical Society*,

*257*(1233). https://doi.org/10.1090/memo/1233

**Covering Dimension of C*-Algebras and 2-Coloured Classification.** / Bosa, Joan; Brown, Nathanial P.; Sato, Yasuhiko; Tikuisis, Aaron; White, Stuart; Winter, Wilhelm.

Research output: Contribution to journal › Article

*Memoirs of the American Mathematical Society*, vol. 257, no. 1233. https://doi.org/10.1090/memo/1233

}

TY - JOUR

T1 - Covering Dimension of C*-Algebras and 2-Coloured Classification

AU - Bosa, Joan

AU - Brown, Nathanial P.

AU - Sato, Yasuhiko

AU - Tikuisis, Aaron

AU - White, Stuart

AU - Winter, Wilhelm

N1 - 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). Print ISBN: 978-1-4704-3470-0 Electronic ISBN: 978-1-4704-4949-0

PY - 2019/1/30

Y1 - 2019/1/30

N2 - 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.

AB - 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.

UR - http://www.mendeley.com/research/covering-dimension-calgebras-2coloured-classification

U2 - 10.1090/memo/1233

DO - 10.1090/memo/1233

M3 - Article

VL - 257

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 1233

ER -