The dynamics of partial inverse semigroup actions

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Cordeiro, Luiz Gustavo¹  Beuter, Viviane^2,3 

[1] UMPA, UMR 5669, Ecole Normale Super Lyon, CNRS, 46 Allee Italie, F-69364 Lyon 07, France

[2] Univ Fed Santa Catarina, Dept Matemat, BR-88040900 Florianopolis, SC, Brazil

[3] Univ Estado Santa Catarina, Dept Matemat, BR-89219710 Joinville, BR, Brazil

关键词: Partial action of inverse semigroups;    Groupoid;    Steinberg algebras;    Crossed product;    Topologically principal;    Continuous orbit equivalence;   

DOI  :  10.1016/j.jpaa.2019.06.001

来源: Elsevier

【 摘 要 】

Given an inverse semigroup S endowed with a partial action on a topological space X, we construct a groupoid of germs S proportional to X in a manner similar to Exel's groupoid of germs, and similarly a partial action of S on an algebra A induces a crossed product A (sic) S. We then prove, in the setting of partial actions, that if X is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs S proportional to X is isomorphic to the crossed product A(R) (X) (sic) S, where A(R)(X) is the Steinberg algebra of X. We also prove that the converse holds, that is, that under natural hypotheses, crossed products of the form A(R) (X) (sic) S are Steinberg algebras of appropriate groupoids of germs of the form S proportional to X. We introduce a new notion of topologically principal partial actions, which correspond to topologically principal groupoids of germs, and study orbit equivalence for these actions in terms of isomorphisms of the corresponding groupoids of germs. This generalizes previous work of the second-named author as well as from others, which dealt mostly with global actions of semigroups or partial actions of groups. We finish the article by comparing our notion of orbit equivalence of actions and orbit equivalence of graphs. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free