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

By J. Bosa, N. Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White, W. Winter. Memoirs of the American Mathematical Society, 2015

131 citations recorded by the source corpus.

We introduce the concept of finitely 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 classification 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 finite topological covering dimension, this confirms 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 finite nuclear dimension.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b5670f82ed138a0c9dc4c0186bad083e2d268a04

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Citations in the corpus

Subalgebras of simple AF-algebras (2018, background): “The trace-kernel extension has been extensively studied in connection with the TomsWinter conjecture (with a modified definition when B has more than one trace); see [5, 44, 54, 55, 75, 76, 83] for example.”
A random matrix approach to the Peterson-Thom conjecture (2020, background): “351-352]), the special case of convex subsets of the trace space appeared recently in the study of classification of nuclear C-algebras and their homomorphisms (see [5, 13, 12]).”
Nuclear dimension of simple $$\mathrm {C}^*$$-algebras (2019, methodology): “In the main results of our paper, unitality enters through Theorem H, as the methods from [3] are only available in the presence of a unit.”
Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, background): “Given the increasingly clear parallels between concepts for vonNeumann algebras and for C∗-algebras, which has driven the C∗-algebraic theory to its recent breakthroughs [5,14,21,78], one would expect that there is much more to discover about amenable group actions on C∗-algebras.”
Nuclear dimension of simple stably projectionless C∗-algebras (2019, methodology): “2), which allows us to assume the domains of certain maps are unital in places where simplicity and Z-stability are only really needed on the codomain side in [8].”