KS-Groupoids and KS-Crossed Products associated with Inverse Semigroup Actions

For an action of an inverse semigroup $S$ on a locally compact Hausdorff space $X$, we construct an étale groupoid $S⋉X^{\mathrm{K}\mathrm{S}}$, called the KS-groupoid associated with the action. This construction may be viewed as a dynamical analogue of the construction of universal groupoids for inverse semigroups. We provide a dynamical version of Paterson’s theorem which states that the full and reduced crossed products (in the sense of Khoshkam and Skandalis) of $C_0(X)$ are canonically isomorphic to the full and reduced groupoid C*-algebras of $S⋉X^{\mathrm{K}\mathrm{S}}$, respectively. We further investigate some topological properties, functoriality and universality of KS-groupoids.