The action of an inverse semigroup on its Stone-\v{C}ech compactification

TL;DR

This paper studies the properties of the Stone-eh compactification groupoid of an inverse semigroup, establishing conditions for Hausdorffness, effectiveness, and their implications for associated C*-algebras.

Contribution

It introduces the study of the Stone-eh transformation groupoid for inverse semigroups and links its topological properties to algebraic and operator algebraic conditions.

Findings

Hausdorff, principal, and effective properties are equivalent for the groupoid.

Algebraic conditions on the semigroup characterize the groupoid's Hausdorffness.

Hausdorffness of the tight groupoid is necessary for the Hausdorffness of the main groupoid.

Abstract

We initiate the study of the Stone-\v{C}ech transformation groupoid $\mathcal{G} = \mathcal{S}\ltimes\beta\mathcal{S}$ of an inverse semigroup $\mathcal{S}$. We prove that the properties of being Hausdorff, principal, and effective are all equivalent for $\mathcal{G}$, and give an algebraic condition on $\mathcal{S}$ equivalent to the Hausdorffness of $\mathcal{G}$. We show that the Hausdorffness of Exel's tight groupoid $\mathcal{G}_{\text{tight}}(\mathcal{S})$ is necessary for the Hausdorffness of $\mathcal{G}$. Finally, we clarify the connection between several crossed product constructions involving this groupoid, and show that when it is Hausdorff and $\mathcal{S}$ has the Property (FL) of Lled\'o and Mart\'inez, then $\mathcal{G}$ is amenable if and only if the reduced C$^*$-algebra $\text{C}^*_r(\mathcal{S})$ is exact.