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

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

Research output: Contribution to journalArticle

3 Citations (Scopus)
4 Downloads (Pure)

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

Fingerprint

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

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

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.
@article{b8a49ab69baa4c26ba974f9c65d0a662,
title = "Covering Dimension of C*-Algebras and 2-Coloured Classification",
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.",
author = "Joan Bosa and Brown, {Nathanial P.} and Yasuhiko Sato and Aaron Tikuisis and Stuart White and Wilhelm Winter",
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). Print ISBN: 978-1-4704-3470-0 Electronic ISBN: 978-1-4704-4949-0",
year = "2019",
month = "1",
day = "30",
doi = "10.1090/memo/1233",
language = "English",
volume = "257",
journal = "Memoirs of the American Mathematical Society",
issn = "0065-9266",
publisher = "American Mathematical Society",
number = "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

UR - http://www.scopus.com/inward/record.url?scp=85065421446&partnerID=8YFLogxK

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 -