The uniform Roe algebra of an inverse semigroup

摘要

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 Lambda(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(infinity) (S), with the uniform Roe algebra over the metric space, namely C-u* (Lambda(S)). This yields a characterization of when R-S = C-u*(Lambda(S)), which generalizes finite generation of S. We have termed this by admitting a finite labeling(or being FL), since it holds when Lambda(S) can be labeled in a finitary manner.