Covering Dimension of C*-Algebras and 2-Coloured Classification
Covering Dimension of C*-Algebras and 2-Coloured Classification
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DOI：
10.1090/memo/1233
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发表时间：
2015-06
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影响因子：
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通讯作者：
J. Bosa;N. Brown;Yasuhiko Sato;A. Tikuisis;Stuart White;W. Winter
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作者：
J. Bosa;N. Brown;Yasuhiko Sato;A. Tikuisis;Stuart White;W. Winter
我们为 C* 代数之间的酉 *-同态引入有限颜色等价的概念，其中酉等价是 1 色的情况。我们使用这个概念将来自可分离、单位、核 C* 代数的 *-同态分类为简单、单位、核、Z 稳定 C* 代数的超幂，通过它们在迹上的行为，具有紧致极值迹空间直至 2 色等价；这是基于某些零阶图的 1 色分类定理，也是在轨迹数据方面。作为一个应用，我们计算具有紧极值迹空间的非 AF、简单、可分离、单位、核、Z 稳定 C* 代数的核维数：为 1。在极值迹空间也具有有限拓扑覆盖维数的情况下，这证实了 Toms-Winter 猜想的剩余开蕴涵。受几何和拓扑中同伦刚性定理的启发，我们针对具有有限核维数的大类 C* 代数得出了“同伦等价意味着同构”的结果。
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.