Covering Dimension of C*-Algebras and 2-Coloured Classification
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-3470-0 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-3470-0 (ISBN)
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Introduces the concept of finitely coloured equivalence for unital $^*$-homomorphisms between $/mathrm C^*$-algebras, for which unitary equivalence is the $1$-coloured case.
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.
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.
Joan Bosa, University of Glasgow, Scotland, United Kingdom. Nathanial P. Brown, The Pennsylvania State University, University Park, Pennsylvania. Yasuhiko Sato, Kyoto University, Japan. Aaron Tikuisis, University of Aberdeen, Scotland, United Kingdom. Stuart White, University of Glasgow, Scotland, United Kingdom, and University of Munster, Germany. Wilhelm Winter, University of Munster, Germany.
Introduction
Preliminaries
A $2/times 2$ matrix trick
Ultrapowers of trivial $/mathrm W^*$-bundles
Property (SI) and its consequences
Unitary equivalence of totally full positive elements
$2$-coloured equivalence
Nuclear dimension and decomposition rank
Quasidiagonal traces
Kirchberg algebras
Addendum
Bibliography.
| Erscheinungsdatum | 04.02.2019 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 210 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-3470-9 / 1470434709 |
| ISBN-13 | 978-1-4704-3470-0 / 9781470434700 |
| Zustand | Neuware |
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