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A note on reduced and von Neumann algebraic free wreath products. (English) Zbl 1355.46056
Summary: 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 F. Lemeux and P. Tarrago [J. Funct. Anal. 270, No. 10, 3828–3883 (2016; Zbl 1356.46057)], 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 \(\Gamma\).

MSC:
46L55 Noncommutative dynamical systems
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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