The dynamics of partial inverse semigroup actions

Abstract

Given an inverse semigroup $S$ endowed with a partial action on a topological space $X$, we construct a groupoid of germs $S⋉X$ 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 $A⋊S$. 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 $S⋉X$ is isomorphic to the crossed product $A_R(X)⋊S$, 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 $A_R(X)⋊S$ are Steinberg algebras of appropriate groupoids of germs of the form $S⋉X$. 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 first-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.

34 pages

References in corpus (15)

• Etale groupoids and their quantales
• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
• Homology and topological full groups of etale groupoids on totally disconnected spaces
• Graph inverse semigroups, groupoids and their C*-algebras
• The Path Space of a Directed Graph
• Graph algebras and orbit equivalence
• Partial actions and KMS states on relative graph $C^{\ast }$-algebras
• Diagonal-preserving graded isomorphisms of Steinberg algebras
• Inverse semigroup actions as groupoid actions
• General Non-Commutative Locally Compact Locally Hausdorff Stone Duality
• C*-algebras Associated do Stationary Ordered Bratteli Diagrams
• C*-crossed products by partial actions and actions of inverse semigroups
• A general Banach-Stone type theorem and applications
• Semigroupoid C*-Algebras
• C*-algebras of Boolean inverse monoids - traces and invariant means

Cited by in corpus (3)

• Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
• Étale inverse semigroupoids - the fundamentals
• Reccurence Sets for Partial Inverse, Semigroup Actions and Related Structures