Inner amenability and approximation properties of locally compact quantum groups

TL;DR

This paper develops a notion of inner amenability for locally compact quantum groups, explores its properties, and connects it to homological and approximation properties, leading to several generalizations and resolutions of existing conjectures.

Contribution

It introduces a new concept of inner amenability for quantum groups and links it to homological and approximation properties, extending classical results.

Findings

Inner amenability relates to dual quantum group properties.

Approximation properties can be transferred via averaging for inner amenable quantum groups.

Certain operator algebra properties are characterized for specific classes of quantum groups.

Abstract

We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\mathbb{G}$. Similar homological techniques are used to prove that $\ell^1(\mathbb{G})$ is not relatively operator biflat for any non-Kac discrete quantum group $\mathbb{G}$; a discrete Kac algebra $\mathbb{G}$ with Kirchberg's factorization property is weakly amenable if and only if $L^1_{cb}(\widehat{\mathbb{G}})$ is operator amenable, and amenability of a locally compact quantum group $\mathbb{G}$ implies $C_u(\widehat{\mathbb{G}})=L^1(\widehat{\mathbb{G}})\widehat{\otimes}_{L^1(\widehat{\mathbb{G}})}C_0(\widehat{\mathbb{G}})$ completely isometrically. The latter result allows us to partially answer a conjecture of Voiculescu when $\mathbb{G}$ has the approximation property.