Let $\mathcal{ G}$ be a locally compact Hausdorff \'{ e}tale groupoid. We call a tracial state $\tau$ on a general groupoid $C^*$-algebra $C_\nu^*(\mathcal{ G})$ canonical if $\tau=\tau|_{ C_0(\mathcal{ G}^{ (0)})} \circ E$, where $E:C^*_\nu(\mathcal{ G}) \to C_0(\mathcal{ G}^{ (0)})$ is the canonical conditional expectation. In this paper, we consider so-called fixed point traces on $C_c(\mathcal{ G})$, and prove that $\mathcal{ G}$ is essentially free if and only if any tracial state on $C_\nu^*(\mathcal{ G})$ is canonical and any fixed point trace is extendable to $C_\nu^*(\mathcal{ G})$. As applications, we obtain the following: 1) a group action is essentially free if every tracial state on the reduced crossed product is canonical and every isotropy group is amenable; 2) if the groupoid $\mathcal{ G}$ is second countable, amenable and essentially free then every (not necessarily faithful) tracial state on the reduced groupoid $C^*$-algebra is quasidiagonal.
Categories: math.OA