Ebook: Covering Dimension of C*-Algebras and 2-Coloured Classification
Author: Joan Bosa, Nathanial P. Brown, Yasuhiko Sato
- Tags: C*-algebras., Homomorphisms (Mathematics), Extremal problems (Mathematics)
- Series: Memoirs of the American Mathematical Society Ser.
- Year: 2018
- Publisher: American Mathematical Society
- City: Providence, United States
- Edition: 1
- Language: English
- pdf
The authors introduce the concept of finitely coloured equivalence for unital $^*$-homomorphisms between $mathrm C^*$-algebras, for which unitary equivalence is the $1$-coloured case. They use this notion to classify $^*$-homomorphisms from separable, unital, nuclear $mathrm C^*$-algebras into ultrapowers of simple, unital, nuclear, $mathcal Z$-stable $mathrm C^*$-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 the authors calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, $mathcal Z$-stable $mathrm C^*$-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, the authors derive a "homotopy equivalence implies isomorphism" result for large classes of $mathrm C^*$-algebras with finite nuclear dimension.
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