Quantum hypergroups arising from ergodic coactions

Abstract

Given a compact quantum group $\mathbb{G}$ and an ergodic action $L^\infty(\mathbb{X})\stackrel{\alpha}\curvearrowleft \mathbb{G}$ with algebraic core $\mathcal{O}(\mathbb{X})$, we show that the unital $*$-algebra $\mathcal{O}(\mathbb{X}\times_{\mathbb{G}} \bar{\mathbb{X}}):= \mathcal{O}(\mathbb{X})\square\overline{\mathcal{O}(\bar{\mathbb{X}})}$ carries the structure of an algebraic compact quantum hypergroup. This $*$-algebra admits two (generally distinct) $C^*$-algebra completions (`reduced' and `universal'), both carrying the structure of a $C^*$-algebraic compact quantum hypergroup. This provides a large class of new examples of (analytical) compact quantum hypergroups. We provide characterizations of coamenability for these compact quantum hypergroups, making use of the theory of equivariant correspondences.