Quantum semigroups generated by locally compact semigroups

Abstract

Let $S$ be a subsemigroup of a second countable locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^{*}$-algebra $C^{*}_{\delta }(S)$ generated by the operators of translation by all elements of $S$ in $L^{2}(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. The same is proved for the von Neumann algebra $\operatorname{VN}(S)$ generated by $C^{*}_{\delta }(S)$.