Thoma type results for discrete quantum groups

Abstract

Thoma's theorem states that a group algebra $C^{\ast }(\Gamma )$ 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 $\pi :C^{\ast }(\Gamma )\to M_F(C(L))$, 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, $\hat{\Gamma }\subset S_N^{+}$, 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 $\le 1$.

arXiv comment20 pages