Tracial states and $\mathbb G$-invariant states of discrete quantum groups
Volume 279 / 2024
Abstract
We investigate the tracial states and $\mathbb G$-invariant states on the reduced $C^*$-algebra $C_r(\widehat {\mathbb G})$ of a discrete quantum group $\mathbb G$. Here, we denote by $\widehat {\mathbb G}$ the dual compact quantum group. Our main result is that a state on $C_r(\widehat {\mathbb G})$ is tracial if and only if it is $\mathbb G$-invariant. This generalizes a fact known for unimodular discrete quantum groups and builds upon the work of Kalantar, Kasprzak, Skalski, and Vergnioux. As one consequence of this, we find that $C_r(\widehat {\mathbb G})$ is nuclear and admits a tracial state if and only if $\mathbb G$ is amenable. This resolves an open problem due to C.-K. Ng and Viselter, and Crann, in the discrete case. As another consequence, we prove that tracial states on $C_r(\widehat {\mathbb G})$ “concentrate” on $\widehat {\mathbb G}_F$, where $\mathbb G_F$ is the cokernel of the Furstenberg boundary. Furthermore, under certain assumptions, we characterize the existence of traces on $C_r(\widehat {\mathbb G})$ in terms of whether or not $\widehat {\mathbb G}_F$ is Kac type. We also characterize the uniqueness of (idempotent) traces in terms of whether or not $\widehat {\mathbb G}_F$ is equal to the canonical Kac quotient of $\widehat {\mathbb G}$. These results rely on the following, of which we give proofs: Sołtan’s canonical Kac quotient construction, whether it is applied to the universal or the reduced CQG $C^*$-algebra of $\widehat {\mathbb G}$ (when the latter admits a trace), yields the maximal Kac type closed quantum subgroup of $\widehat {\mathbb G}$.
