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

The uniform Roe algebra of an inverse semigroup Articles