A note on the orbit equivalence of injective actions

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.