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

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.

11 pages; minor revision, accepted for publication in J. Op. Th

References in corpus (3)

• Invariant means and finite representation theory of C*-algebras
• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• The uniform Roe algebra of an inverse semigroup