A note on the orbit equivalence of injective actions

TL;DR

This paper characterizes the groupoid $C^*$-algebras and continuous orbit equivalence for injective actions of discrete Ore semi-groups on compact spaces, linking algebraic and topological properties.

Contribution

It provides a new characterization of groupoid $C^*$-algebras and orbit equivalence for injective semi-group actions, extending the understanding of their structure.

Findings

Characterization of groupoid $C^*$-algebras via reduced crossed products.

Equivalence of continuous orbit equivalence and transformation groupoids.

Description of injective semi-group actions on their compactifications.

Abstract

We characterise the groupoid $C^*$-algebras associated to the transformation groupoids of injective actions of discrete countable Ore semi-groups on compact topological spaces in terms of the reduced crossed product from the dual actions, and characterise the continuous orbit equivalence for injective actions by means of the transformation groupoids, as well as their reduced groupoid $C^*$-algebras. Finally, we characterize the injective action of semi-group on its compactifications.