Groupoids associated to semigroup actions

TL;DR

This paper constructs a semi-direct product groupoid from topological semigroup actions on compact spaces, showing its equivalence to a transformation groupoid and applying it to Wiener-Hopf C*-algebras.

Contribution

It introduces a new construction of groupoids associated with semigroup actions and demonstrates their equivalence to transformation groupoids under mild conditions.

Findings

Constructed a semi-direct product groupoid with a Haar system

Proved the groupoid is equivalent to a transformation groupoid

Applied the construction to Wiener-Hopf C*-algebras

Abstract

In this paper, we consider topological semigroup actions on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf $C^{*}$-algebras.