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 $VN(S)$ generated by $C^*_\delta(S)$.