Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups

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 its dual compact quantum group by $\widehat{\mathbb{G}}$. 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 known fact 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, given 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 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: So\l tan'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 $\widehat{\mathbb{G}}$.