On hereditary properties of quantum group amenability

TL;DR

This paper investigates how the amenability of a locally compact quantum group relates to its closed quantum subgroups and their actions, providing new criteria and insights into quantum group properties.

Contribution

It establishes an equivalence condition for quantum group amenability based on subgroup amenability and actions on quantum homogeneous spaces.

Findings

Amenability of $G$ is equivalent to subgroup amenability and amenable action on $G/H$

Existence of $L^1(Ghat)$-module projections onto quantum subgroup algebras

New characterizations of hereditary properties in quantum group amenability

Abstract

Given a locally compact quantum group $\mathbb{G}$ and a closed quantum subgroup $\mathbb{H}$, we show that $\mathbb{G}$ is amenable if and only if $\mathbb{H}$ is amenable and $\mathbb{G}$ acts amenably on the quantum homogenous space $\mathbb{G}/\mathbb{H}$. We also study the existence of $L^1(\widehat{\mathbb{G}})$-module projections from $L^{\infty}(\widehat{\mathbb{G}})$ onto $L^{\infty}(\widehat{\mathbb{H}})$.