The tight groupoid of the inverse semigroups of left cancellative small categories

Abstract

We fix a path model for the space of filters of the inverse semigroup ${\mathcal{S}}_{\Lambda }$ associated to a left cancellative small category $\Lambda$. Then, we compute its tight groupoid, thus giving a representation of its $C^{\ast }$-algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.

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