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

Abstract

Let $\mathcal{G}$ be a locally compact Hausdorff étale groupoid. We call a tracial state $\tau$ on a general groupoid $C^{\ast }$-algebra $C_{\nu }^{\ast }(\mathcal{G})$ canonical if $\tau =\tau {|}_{C_0({\mathcal{G}}^{(0)})}\circ E$, where $E:C_{\nu }^{\ast }(\mathcal{G})\to C_0({\mathcal{G}}^{(0)})$ 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 $C_{\nu }^{\ast }(\mathcal{G})$ is canonical and any fixed point trace is extendable to $C_{\nu }^{\ast }(\mathcal{G})$.
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 $C^{\ast }$-algebra is quasidiagonal.