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 $\ast$-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.

arXiv comment30 pages + references. Comments are welcome! v2: New material on the universal version in Section 3, some other material removed. Paper content reorganized