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

Diego Martínez

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arXiv · 2107.14725

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

Diego Martínez

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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.

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Diego Martínez. 2021-07-30. A note on the quasi-diagonality of inverse semigroup reduced C*-algebras. https://arxiv.org/abs/2107.14725

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