A note on reduced and von Neumann algebraic free wreath products

TL;DR

This paper investigates operator algebraic properties of free wreath products involving compact quantum groups, establishing conditions for simplicity, uniqueness of trace, and absence of property Gamma in their von Neumann algebras.

Contribution

It proves that the free wreath product is of Kac type if the original quantum group is, and that the reduced algebra is simple with a unique trace for sufficiently large N.

Findings

The free wreath product is of Kac type if the original quantum group is.

The reduced algebra is simple with a unique trace for N ≥ 8.

The reduced von Neumann algebra does not have property Gamma.

Abstract

In this paper, we study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $\mathbb G \wr_* S_N^+$, where $\mathbb G$ is a compact matrix quantum group. Based on recent result on their corepresentation theory by Lemeux and Tarrago, we prove that $\mathbb G \wr_* S_N^+$ is of Kac type whenever $\mathbb G$ is, and that the reduced version of $\mathbb G \wr_* S_N^+$ is simple with unique trace state whenever $N \geq 8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb G \wr_* S_N^+$ does not have property $\Gamma$.