C*-algebras associated to topological Ore semigroups

Abstract: Let $G$ be a locally compact group and $P \subset G$ be a closed Ore semigroup containing the identity element. Let $V: P \to B(\clh)$ be a representation such that for every $a \in P$, $V_{a}$ is an isometry and the final projections of ${V_{a}: a \in P}$ commute. In this article, we study the $C^{*}$-algebra $\mathcal{W}{V}(P,G)$, generated by ${\int f(a)V{a} da: f \in L^{1}(P)}$. We show that there exists a universal $C^{*}$-algebra, which admits a groupoid description, of which $\mathcal{W}_{V}(P,G)$ is a quotient. If $P=G$, then this universal algebra is just $C^{*}(G)$.