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A $𝕋$ -algebras from fiberwise essentially minimal zero-dimensional dynamical systems

Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 8410501, Israel

https://doi.org/10.1142/S0129167X22500355Cited by:1 (Source: Crossref)

Abstract

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product $C^{*}$ -algebra of such a system is an A $𝕋$ -algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product $C^{*}$ -algebra has real rank zero, which tells us that such $C^{*}$ -algebras are classifiable by $K$ -theory. The associated crossed product $C^{*}$ -algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

This research was supported by the Israel Science Foundation grant no. 476/16.

Communicated by Yasuyuki Kawahigashi

Keywords:

AMSC: 46L55, 46L35, 46L80

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