Amenability of locally compact quantum groups and their unitary co-representations

TL;DR

This paper establishes that amenability of a unitary co-representation in locally compact quantum groups is preserved under weak containment, extending classical results and characterizations of group amenability to the quantum setting.

Contribution

It generalizes a known result of Bekka to quantum groups and affirms a question by Bédos, Conti, and Tuset, extending classical group amenability characterizations.

Findings

Amenability passes to weakly contained co-representations.

Extends classical group amenability characterizations to quantum groups.

Provides a quantum analogue of nuclearity-based amenability criteria.

Abstract

We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of B\'edos, Conti and Tuset. As a corollary, we extend to locally compact quantum groups a result of the first-named author, which characterizes amenability of a locally compact group $G$ by nuclearity of the reduced group $C^{*}$-algebra $C_{r}^{*}(G)$ and an additional condition.