C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

2019/03/28 by Sergey Bezuglyi, Bezuglyi, Sergey, Zhuang Niu +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1903.12276

openalex publication_date 2019/03/28 · openalex created_date 2019/04/11 · openalex updated_date 2026/07/28

Abstract

We study homeomorphisms of a Cantor set with k (k < +∞) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and their certain orbit-cut sub-C*-algebras. In the case that k≥ 2, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition (X, σ) is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli-Vershik-Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli-Vershik-Kakutani model) must have at least k vertices at each level, and the image of the index map must consist infinitesimals.

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