Free wreath products by finite group duals

Abstract

Let $\mathbb{G}$ be a compact quantum group and let $\Lambda$ be a finite group. We study $\mathbb{W}(\mathbb{G},\Lambda ):=\mathbb{G}{\wr }_{\ast ,{\beta }_{\Lambda }}\hat{\Lambda }$, a natural family parallel to Bichon's free wreath products $\mathbb{G}{\wr }_{\ast }{S}_N^{+}$. We obtain an explicit formula for the Haar state of $\mathbb{W}(\mathbb{G},\Lambda )$. Under natural Kac-type factoriality assumptions, we also derive central decompositions of the associated von Neumann and reduced C*-algebras, with full ${\mathrm{II}}_1$-factor and simple unique-trace summands.

20 pages. Revised and substantially extended version