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

Alistair Miller, Eduardo Scarparo

Abstract

We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $μ$ 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 $μ$. We also show that $G$ is essentially free with respect to an invariant measure $μ$ if and only if $μ$ 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.

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

Abstract

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.
Paper Structure (6 sections, 32 theorems, 26 equations)

This paper contains 6 sections, 32 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Essentially free measures
  4. Traces on full groupoid $\mathrm{C}^*$-algebras
  5. Existence of traces on the essential C*-algebra
  6. Finite-state self-similar groups

Key Result

Theorem A

Let $(G,\mathcal{L})$ be a twisted étale groupoid and $\mu$ an invariant Radon measure on $G^0$. Suppose that at least one of the following conditions holds: Then there exists a trace on $\mathrm{C}^*_\mathrm{ess}(G,\mathcal{L})$ which extends $\mu$. In case ess free intro, the canonical extension of $\mu$ descends to $\mathrm{C}^*_\mathrm{ess}(G,\mathcal{L})$.

Theorems & Definitions (73)

  • Theorem A
  • Theorem B: Theorems \ref{['thm:u']} and \ref{['thm:nu']}
  • Theorem C: Corollary \ref{['cor:trace space']}
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • ...and 63 more