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