On bi-exactness of discrete quantum groups

TL;DR

This paper extends the concept of bi-exactness to discrete quantum groups and investigates the structural properties of their von Neumann algebras, revealing the absence of Cartan subalgebras in certain subfactors.

Contribution

It introduces a new notion of bi-exactness for discrete quantum groups and proves structural results about their von Neumann algebras, including the non-existence of Cartan subalgebras in specific cases.

Findings

Non-amenable subfactors of free quantum group von Neumann algebras lack Cartan subalgebras.

Established structural properties of von Neumann algebras associated with discrete quantum groups.

Extended Ozawa's bi-exactness concept to the setting of discrete quantum groups.

Abstract

We define Ozawa's notion of bi-exactness to discrete quantum groups, and then prove some structural properties of associated von Neumann algebras. In particular, we prove that any non amenable subfactor of free quantum group von Neumann algebras, which is an image of a faithful normal conditional expectation, has no Cartan subalgebras.