The action of an inverse semigroup on its Stone-Čech compactification

We initiate the study of the Stone-Čech transformation groupoid $\mathcal{G}=\mathcal{S}⋉\beta \mathcal{S}$ 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 Lledó and Martínez, then $\mathcal{G}$ is amenable if and only if the reduced C${}^{\ast }$-algebra ${\text{C}}_r^{\ast }(\mathcal{S})$ is exact.