Theoretical study reveals conditions for extending invariant measures to traces on étale groupoid C*-algebras, indicating unique tracial states in self-similar groups.
We provide sufficient conditions for the existence of a trace on the essential ‐algebra of a (not necessarily Hausdorff) étale groupoid which extends an invariant measure on the unit space of . In particular, it suffices for the isotropy groups of to be amenable, or for to be essentially free with respect to . We also show that is essentially free with respect to an invariant measure if and only if extends to a unique trace on the full ‐algebra of . 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.
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Miller et al. (2026) studied this question.
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