$C^*$-simplicity and boundary actions of discrete quantum groups
Benjamin Anderson-Sackaney, Roland Vergnioux
TL;DR
The paper develops quantum analogues of Powers' averaging property and boundary dynamics to study $C^*$-simplicity for discrete quantum groups. It defines PAP and PAP$_h$, links them to boundary envelopes and equivariant ucp maps to the quantum Furstenberg boundary, and introduces strong $C^*$-faithfulness to connect boundary actions with simplicity. In particular, it shows that a strongly $C^*$-faithful boundary action implies PAP and $C^*$-simplicity, and applies these results to the free unitary quantum groups $ f U_F$, proving they satisfy quantum PAP and admit a strongly $C^*$-faithful boundary action on their quantum Gromov boundary. The boundary of $ f U_F$ is established as a genuine $ f U_F$-boundary through a unique stationary harmonic state, providing a dynamical route to $C^*$-simplicity for $N\ge 3$. Overall, the work extends boundary-action technology to the quantum group setting, yielding new criteria and verifiable instances of $C^*$-simplicity in the noncommutative realm.
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 $σ$-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.