Dimensions And C Ast Algebras
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Dimensions and $C^\ast $-Algebras
Author: Edward G. Effros
language: en
Publisher: American Mathematical Soc.
Release Date: 1981
Discusses elementary algebras and $C DEGREES*$-algebras, namely those which are direct limits of complex semi simple al
Classification of Ring and $C^\ast $-Algebra Direct Limits of Finite-Dimensional Semisimple Real Algebras
Author: K. R. Goodearl
language: en
Publisher: American Mathematical Soc.
Release Date: 1987
Motivated by (i) Elliott's classification of direct limits of countable sequences of finite-dimensional semisimple complex algebras and complex AF C*-algebras, (ii) classical results classifying involutions on finite-dimensional semisimple complex algebras, and (iii) the classification by Handelman and Rossmann of automorphisms of period two on the algebras appearing in (i) we study the real algebras described above and completely classify them, up to isomorphism, Morita equivalence, or stable isomorphism. We also show how our classification easily distinguishes various types of algebras within the given classes, and we partially solve the problem of determining exactly which values are attained by the invariants used in classifying these algebras.
Covering Dimension of C*-Algebras and 2-Coloured Classification
Author: Joan Bosa
language: en
Publisher: American Mathematical Soc.
Release Date: 2019-02-21
The authors introduce the concept of finitely coloured equivalence for unital -homomorphisms between -algebras, for which unitary equivalence is the -coloured case. They use this notion to classify -homomorphisms from separable, unital, nuclear -algebras into ultrapowers of simple, unital, nuclear, -stable -algebras with compact extremal trace space up to -coloured equivalence by their behaviour on traces; this is based on a -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, -stable -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 -algebras with finite nuclear dimension.