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. We give two applications of our results: a “noncommutative” version of Ellis joint continuity theorem for semitopological groups, a corollary to which is a new C∗-algebraic proof of the theorem for classical semitopological semigroup; 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.