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 $\mathbb{G} \wr_{\ast} S_N^+$, 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 $\mathbb{G} \wr_{\ast} S_N^+$ is of Kac type whenever $\mathbb{G}$ is, and that the reduced version of $\mathbb{G} \wr_{\ast} S_N^+$ is simple with unique trace state whenever $N \geq 8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb{G} \wr_{\ast} S_N^+$ does not have property $\Gamma$.
Type
Publication
Illinois Journal of Mathematics (IJM)
Cite
@article{wahlANoteOnReduced2015,
\wr_{\ast} 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 $\mathbb{G} \wr_{\ast} S_N^+$ is of Kac type whenever $\mathbb{G}$ is, and that the reduced version of $\mathbb{G} \wr_{\ast} S_N^+$ is simple with unique trace state whenever $N \geq 8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb{G} \wr_{\ast} S_N^+$ does not have property $\Gamma$.},
author = {Jonas Wahl},
doi = {10.1215/ijm/1475266409},
journal = {Illinois Journal of Mathematics (IJM)},
number = {3},
pages = {801-817},
publisher = {Duke University Press (im Auftrag der University of Illinois)},
title = {A note on reduced and von Neumann algebraic free wreath products},
url = {https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-59/issue-3/A-note-on-reduced-and-von-Neumann-algebraic-free-wreath/10.1215/ijm/1475266409.full},
volume = {59},
year = {2015}
}