A correspondence between inverse subsemigroups, open wide subgroupoids and cartan intermediate C*-subalgebras

Abstract For a given inverse semigroup action on topological space, one can associate an étale groupoid. We prove that there exists correspondence between the certain subsemigroups and open wide subgroupoids in case is strongly tight. Combining with recent result of Brown et al., we obtain Cartan intermediate subalgebras groupoid C*-algebra.