The uniform Roe algebra of an inverse semigroup

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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.
Inverse semigroup, Schützenberger graph, Uniform Roe algebra, Property a
Bibliographic citation
Lledó, F., & Martínez, D. (2021). The uniform roe algebra of an inverse semigroup. Journal of Mathematical Analysis and Applications, 499(1), 124996.