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

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

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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.
Original languageEnglish
Number of pages97
JournalMemoirs of the American Mathematical Society
Volume257
Issue number1233
Early online date10 Jan 2019
DOIs
Publication statusPublished - 30 Jan 2019

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Covering Dimension
Algebra
C*-algebra
Unital
Trace
Equivalence
Homomorphisms
Ultrapower
Homotopy Equivalence
Theorem
Rigidity
Homotopy
Isomorphism
Classify
Topology
Imply
Calculate
Geometry
Zero

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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

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

In: Memoirs of the American Mathematical Society, Vol. 257, No. 1233, 30.01.2019.

Research output: Contribution to journalArticle

Bosa, J, Brown, NP, Sato, Y, Tikuisis, A, White, S & Winter, W 2019, 'Covering Dimension of C*-Algebras and 2-Coloured Classification' Memoirs of the American Mathematical Society, vol. 257, no. 1233. https://doi.org/10.1090/memo/1233
Bosa, Joan ; Brown, Nathanial P. ; Sato, Yasuhiko ; Tikuisis, Aaron ; White, Stuart ; Winter, Wilhelm. / Covering Dimension of C*-Algebras and 2-Coloured Classification. In: Memoirs of the American Mathematical Society. 2019 ; Vol. 257, No. 1233.
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