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

TL;DR

This paper establishes a correspondence between inverse semigroup substructures, open wide subgroupoids, and Cartan intermediate C*-subalgebras in the context of strongly tight actions, linking algebraic and topological structures.

Contribution

It introduces a new correspondence connecting inverse semigroup subsemigroups, subgroupoids, and C*-algebra subalgebras under specific action conditions.

Findings

Established a correspondence between certain subsemigroups and open wide subgroupoids.

Linked inverse semigroup structures to Cartan intermediate subalgebras in groupoid C*-algebras.

Extended previous results to strongly tight actions, broadening the applicability.

Abstract

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