Invariant measures and traces on groupoid $\mathrm{C}^\ast$-algebras

TL;DR

This paper establishes conditions under which invariant measures on groupoid unit spaces extend to traces on their C*-algebras, highlighting the role of amenability and essential freeness, with applications to self-similar groups.

Contribution

It provides new criteria for the existence and uniqueness of traces on groupoid C*-algebras, including cases with infinite measures and twisted groupoids.

Findings

Invariant measures extend to traces when isotropy groups are amenable.

Essential freeness of the groupoid corresponds to unique trace extension.

Finite-state self-similar groups have a unique tracial state.

Abstract

We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) \'etale groupoid $G$ which extends an invariant measure $\mu$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $\mu$. We also show that $G$ is essentially free with respect to an invariant measure $\mu$ if and only if $\mu$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$. We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids. As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state.