Thoma type results for discrete quantum groups

Abstract

Thoma's theorem states that a group algebra $C^*(\Gamma)$ is of type I if and only if $\Gamma$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group $\Gamma$, we have a stationary model of type $\pi:C^*(\Gamma)\to M_F(C(L))$, with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\widehat{\Gamma}\subset S_N^+$, by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank $\leq1$.