Quantum semigroups generated by locally compact semigroups

TL;DR

This paper constructs quantum semigroups from certain subsemigroups of locally compact groups, showing that their associated $C^*$-algebras and von Neumann algebras have natural quantum semigroup structures.

Contribution

It introduces a method to generate quantum semigroups from subsemigroups of locally compact groups, expanding the theory of quantum group-like structures.

Findings

$C^*_ abla(S)$ admits a comultiplication making it a compact quantum semigroup

The von Neumann algebra $VN(S)$ also admits a compatible quantum semigroup structure

The results apply to subsemigroups with $S^{-1}S=G$ in second countable locally compact groups

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 $C^*_\delta(S)$.