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

Joan Bosa, Nathanial P. Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White, Wilhelm Winter

Research output: Book/ReportBook

50 Citations (Scopus)
11 Downloads (Pure)


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.
Original languageEnglish
PublisherAmerican Mathematical Society
Number of pages97
ISBN (Electronic)978-1-4704-4949-0
ISBN (Print)978-1-4704-3470-0
Publication statusPublished - 30 Jan 2019

Publication series

NameMemoirs of the American Mathematical Society
PublisherAmerican 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).


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