Translations in quantum groups

Abstract

Let $H$ be the Hopf $C^*$-algebra of continuous functions on a (locally) compact quantum group of either reduced or full type. We show that endomorphisms of $H$ that respect its right regular comodule structure are translations by elements of largest classical subgroup of $G$. Furthermore, we show that for compact $G$ such an endomorphism is automatically an automorphism regardless of the quantum group norm on the $C^*$-algebra $H$; this answers a question of Piotr M. Hajac.