Appears in collection : Arbre de Noël du GDR « Géométrie non-commutative »

For a discrete group G, the tracial states on its reduced group $C^_$-algebra $C^∗_r (G)$ are exactly the conjugation invariant states. This makes the traces on $C^∗_r (G)$ amenable to group dynamical techniques. In the setting of a discrete quantum group ${\mathbb G}$, there is a quantum analog of the conjugation action of $G$ on $C^∗_r (G)$. Recent work of Kalantar, Kasprzak, Skalski, and Vergnioux shows that ${\mathbb G}$-invariant states on the quantum group reduced $C^_$-algebra $C_r( \widehat{\mathbb G})$ are in one-to-one correspondence with certain KMS-states, exhibiting a disparity between tracial states and ${\mathbb G}$-invariant states unless ${\mathbb G}$ is unimodular. We will show there is still enough of a connection between traceability and G-invariance to say interesting things about the tracial states of $C_r( \widehat{\mathbb G})$.