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

摘要

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^*$-algebra as a (full) groupoid algebra. Using it, we characterize simplicty for these algebras. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.