A note on the quasi-diagonality of inverse semigroup reduced C*-algebras

TL;DR

This paper investigates the conditions under which the reduced C*-algebra of an inverse semigroup is quasi-diagonal, linking algebraic structure to operator algebra properties and providing new characterizations.

Contribution

It offers a detailed analysis of the relationship between inverse semigroup subgroups, Green's relation, and the quasi-diagonality of their reduced C*-algebras, including new sufficient conditions.

Findings

Isolated subgroups must be amenable for quasi-diagonality.

Characterization of quasi-diagonality for inverse semigroups with minimal universal groupoid.

Green's $\

Abstract

In this note we start the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal, making explicit use of the inner structure of this class of semigroups in order to produce quasi-diagonal approximations. Given a discrete inverse semigroup, we detail the relationship between its isolated subgroups and the quasi-diagonality of its reduced C*-algebra, and prove that such subgroups must be amenable. Moreover, we give a direct characterization of the quasi-diagonality of inverse semigroup whose universal groupoid is minimal. Lastly, we also study the relevance of Green's $\mathcal{D}$-relation when considering quasi-diagonality questions, and give a sufficient condition for the quasi-diagonality of a general inverse semigroup.