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 $C^{\ast }$-algebra $C_{\delta }^{\ast }(S)$ 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 $C_{\delta }^{\ast }(S)$.

Revised and cleaned version

References in corpus (1)

• Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras