The uniform Roe algebra of an inverse semigroup

Abstract 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 Schutzenberger graphs. This graph, which we denote by Λ S , inherits much of the structure of S. In this article we compare the C*-algebra R S , generated by the left regular representation of S on l 2 ( S ) and l ∞ ( S ) , with the uniform Roe algebra over the metric space, namely C u ⁎ ( Λ S ) . This yields a characterization of when R S = C u ⁎ ( Λ S ) , which generalizes finite generation of S. We have termed this by admitting a finite labeling (or being FL), since it holds when Λ S can be labeled in a finitary manner. The graph Λ S , and the FL condition, also allow to analyze large scale properties of Λ 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., [6] ) is a quasi-isometric invariant of Λ S . Moreover, we characterize property A of Λ 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.