Quantum hypergroups arising from ergodic coactions

TL;DR

This paper introduces a new class of compact quantum hypergroups derived from ergodic coactions of locally compact quantum groups, with characterizations of their coamenability.

Contribution

It constructs quantum hypergroups from ergodic coactions and characterizes their coamenability using equivariant correspondences.

Findings

Established a natural coassociative map on the von Neumann algebra of the quantum hypergroup.

Provided a large class of examples of analytical compact quantum hypergroups.

Characterized coamenability for these hypergroups using equivariant correspondence theory.

Abstract

Given a locally compact quantum group $\mathbb{G}$ and an ergodic, integrable action $L^\infty(\mathbb{X})\stackrel{\alpha}\curvearrowleft \mathbb{G}$, the von Neumann algebra $L^\infty(\mathbb{X}\times_{\mathbb{G}}\bar{\mathbb{X}}):= L^\infty(\mathbb{X})\square \overline{L^\infty(\mathbb{X})}$ is shown to carry a natural normal ucp coassociative map $\Delta_{\mathbb{X}\times_{\mathbb{G}}\bar{\mathbb{X}}}: L^\infty(\mathbb{X}\times_{\mathbb{G}}\bar{\mathbb{X}})\to L^\infty(\mathbb{X}\times_{\mathbb{G}}\bar{\mathbb{X}})\bar{\otimes} L^\infty(\mathbb{X}\times_{\mathbb{G}}\bar{\mathbb{X}})$. Restricting to the class of compact quantum groups, 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.