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

Abstract

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.