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Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023  pp. 313-329.

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

Authors:  Diego Martinez
Author institution: Mathematisches Institut, WWU Muenster, Einsteinstr. 62, 48149 Muenster, Germany

Summary:  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 an 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.

Keywords: inverse semigroup, quasi-diagonality, amenability

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