Quantum hypergroups arising from ergodic coactions

Abstract

Given a compact quantum group $\mathbb{G}$ and an ergodic action $L^{\infty }(\mathbb{X})\stackrel{\hat{I}}{\pm }\curvearrowleft \mathbb{G}$ with algebraic core $\mathcal{O}(\mathbb{X})$, we show that the unital $\ast$-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 $\ast$-algebra admits two (generally distinct) $C^{\ast }$-algebra completions (`reduced' and `universal'), both carrying the structure of a $C^{\ast }$-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.

30 pages + references. Comments are welcome! v2: New material on the universal version in Section 3, some other material removed. Paper content reorganized