On a quantum version of Ellis joint continuity theorem

TL;DR

This paper generalizes Ellis's joint continuity theorem to the quantum setting, providing conditions under which a compact semitopological quantum semigroup becomes a quantum group and exploring the existence of Haar states.

Contribution

It introduces a necessary and sufficient condition for a compact semitopological quantum semigroup to be a quantum group, extending classical results to the noncommutative context.

Findings

Characterization of quantum groups via semitopological quantum semigroups

Generalization of Ellis's joint continuity theorem

Existence criteria for Haar states in quantum semigroups

Abstract

We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. In particular, we obtain a generalisation of Ellis's joint continuity theorem. We also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a "noncommutative" version of the converse Haar's theorem.