$C^*$-simplicity and boundary actions of discrete quantum groups

TL;DR

This paper develops quantum analogues of classical boundary action concepts to study $C^*$-simplicity in discrete quantum groups, establishing new criteria and illustrating them with free quantum groups.

Contribution

It introduces quantum versions of Powers' Averaging Property and boundary actions, linking them to $C^*$-simplicity and applying these to free quantum groups.

Findings

Quantum PAP implies $C^*$-simplicity and unique $eta$-KMS states.

Strongly $C^*$-faithful boundary actions imply $C^*$-simplicity.

Unitary free quantum groups satisfy quantum PAP and act strongly $C^*$-faithfully on their boundary.

Abstract

We introduce and investigate several quantum group dynamical notions for the purpose of studying $C^*$-simplicity of discrete quantum groups via the theory of boundary actions. In particular we define a quantum analogue of Powers' Averaging Property (PAP) and a quantum analogue of strongly faithful actions. We show that our quantum PAP implies $C^*$-simplicity and the uniqueness of $\sigma$-KMS states, and that the existence of a strongly $C^*$-faithful quantum boundary action also implies $C^*$-simplicity and, in the unimodular case, the quantum PAP. We illustrate these results in the case of the unitary free quantum groups $\mathbb{F} U_F$ by showing that they satisfy the quantum PAP and that they act strongly $C^*$-faithfully on their quantum Gromov boundary. Moreover we prove that this particular action of $\mathbb{F} U_F$ is a quantum boundary action.