Groupoids associated to Ore semigroup actions

Abstract

In this paper, we consider actions of locally compact Ore semigroups on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is Morita-equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf C * -algebras. 1

On the Wiener-Hopf compactification of a symmetric cone

“…Here L ∞ (G) is given the weak * -topology and the action of P is by right translation. For a self-contained proof of the existence and the uniqueness of the Wiener-Hopf compactification, we refer the reader to the article [RS15], in particular to Prop.5.1, [RS15]. The term "Order compactification" is used instead of the Wiener-Hopf compactification in [RS15].…”

“…Remark 2.4 Prop 5.1 of [RS15] requires that the map X × Int(P ) ∋ (x, a) → xa ∈ X is open. However it is proved in Theorem 4.3 of [RS15] that the openness of the map X × Int(P ) ∋ (x, a) → xa ∈ X is equivalent to (C1).…”

“…Let X 0 := {u ⊞ a : a ∈ Ω} = {u ∈ X : u > −1}. By Proposition 6.6 of [RS15], it is enough to show that the K-groups of C 0 (X 0 ) vanish. Note that…”

“…Observe that X 0 = [−∞, 0) × [0, 1). We refer the reader to Section 7 of [RS15] for the proof of the fact that the compact space X, together with the above action of P , is the Wiener-Hopf compactification of the semigroup P .…”

“…We should remark that this could probably be proved directly following the ideas in [AJ07] (Theorem 9). However, we follow a different path by appealing to the axiomatic characterisation of the Wiener-Hopf compactification of an Ore semigroup (Prop 5.1, [RS15]). We believe that this axiomatic approach is elegant and has potential applications to semigroups other than convex cones.…”