Nonfree almost finite actions for locally finite-by-virtually Z ${\mathbb {Z}}$ groups

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Kang Li, Xin Ma
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引用次数: 0

Abstract

In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let α : G X $\alpha:G\curvearrowright X$ be a minimal action of a locally finite-by-virtually Z ${\mathbb {Z}}$ group G $G$ on the Cantor set X $X$ . We prove that under certain assumptions, the action α $\alpha$ is almost finite in measure if and only if α $\alpha$ is essentially free. As an application, we obtain that any minimal topologically free action of a virtually Z ${\mathbb {Z}}$ group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property Γ $\Gamma$ and Z $\mathcal {Z}$ -stability for their crossed product C $C^*$ -algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.

局部有限虚Z ${mathbb {Z}}$ 群的无自由几乎有限作用
在本文中,我们将研究非自由作用的几乎有限性和几乎有限性度量。设 α : G ↷ X $\alpha:G\curvearrowright X$ 是局部有限虚数 Z ${\mathbb {Z}}$ 群 G $G$ 在康托集 X $X$ 上的最小作用。我们证明,在某些假设条件下,当且仅当 α $\alpha$ 本质上是自由的,作用 α $\alpha$ 在度量上几乎是有限的。作为一个应用,我们得到,在具有小边界性质的无限紧凑可元空间上,虚Z ${mathbb {Z}}$ 群的任何最小自由拓扑作用都是几乎有限的。这是这个方向上第一个只假定拓扑自由性的一般性结果,这些结果引出了关于其交叉积 C∗ $C^*$ -代数的均匀性质Γ $\Gamma$ 和 Z $\mathcal {Z}$ -稳定性的新结果。本文提供了一些最小拓扑自由(但非自由)子移动的具体例子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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