## Illinois Journal of Mathematics

### A note on reduced and von Neumann algebraic free wreath products

Jonas Wahl

#### 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 $\Gamma$.

#### Article information

Source
Illinois J. Math., Volume 59, Number 3 (2015), 801-817.

Dates
Revised: 22 March 2016
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.ijm/1475266409

Digital Object Identifier
doi:10.1215/ijm/1475266409

Mathematical Reviews number (MathSciNet)
MR3554234

Zentralblatt MATH identifier
1355.46056

Subjects
Primary: 46L54: Free probability and free operator algebras

#### Citation

Wahl, Jonas. A note on reduced and von Neumann algebraic free wreath products. Illinois J. Math. 59 (2015), no. 3, 801--817. doi:10.1215/ijm/1475266409. https://projecteuclid.org/euclid.ijm/1475266409

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