The uniform Roe algebra of an inverse semigroup

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we equip with natural metric, given by path metric disjoint union of Sch\"{u}tzenberger graphs. This graph, which denote $\Lambda_S$, inherits much structure $S$. this article compare C*-algebra $\mathcal{R}_S$, generated left regular representation on $\ell^2(S)$ $\ell^\infty(S)$, uniform Roe algebra over space, namely $C^*_u(\Lambda_S)$. yields chacterization when $\mathcal{R}_S = C^*_u(\Lambda_S)$, generalizes finite generation We have termed labeability (FL), since it holds $\Lambda_S$ be labeled finitary manner. The graph FL condition above, also allow analyze large scale properties relate them C*-properties algebra. show that domain measurability (a notion generalizing Day's definition amenability semigroup, cf., [5]) is quasi-isometric invariant $\Lambda_S$. Moreover, characterize property A (or components) terms nuclearity exactness corresponding C*-algebras. treat special classes F-inverse E-unitary semigroups from point view.