On a quantum version of Ellis joint continuity theorem

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.