Amenability and Uniqueness for Groupoids Associated with Inverse Semigroups

TL;DR

This paper explores the conditions under which the universal groupoid of inverse semigroups is amenable and how this relates to the uniqueness of reduced $C^*$-algebras, providing new insights into their structure.

Contribution

It establishes new criteria for amenability of universal groupoids associated with inverse semigroups and links this to the uniqueness theorems for reduced $C^*$-algebras.

Findings

Conditions guaranteeing amenability of the universal groupoid.

Existence of a conditional expectation onto a canonical subalgebra.

Application of results to full $C^*$-algebras.

Abstract

We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff \'{e}tale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.