A note on reduced and von Neumann algebraic free wreath products

Abstract

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 results on their corepresentation theory by Lemeux and Tarrago in [Lemeux and Tarrago (2014)], 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 $N\geq8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb{G}\wr_{*}S_{N}^{+}$ does not have property $\Gamma$.