Tracial states on groupoid $C^{*}$-algebras and essential freeness

Let \mathcal{G} be a locally compact Hausdorff étale 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.