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

Part of: Topological dynamics Selfadjoint operator algebras

Published online by Cambridge University Press:  20 February 2020

SERGEY BEZUGLYI ,
ZHUANG NIU  and
WEI SUN
SERGEY BEZUGLYI
Affiliation:
University of Iowa, Iowa City, IA 52242, USA email sergii-bezuglyi@uiowa.edu
ZHUANG NIU
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, WY82071, USA email zniu@uwyo.edu
WEI SUN
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200062, China email wsun@math.ecnu.edu.cn
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Abstract

We study homeomorphisms of a Cantor set with $k$ ($k<+\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ 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 of infinitesimals.

Keywords

operator algebrastopological dynamicsK-theoryCantor systemsindex map

MSC classification

Primary: 46L35: Classifications of $C^*$-algebras 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1296 - 1341
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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