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On weakly amenable groupoids

Tomás Pacheco

TL;DR

The paper extends weak amenability from discrete groups to groupoids via Renault's Fourier algebra, establishing a tight link between groupoid-level approximation properties and the CBAP for quasi Cartan pairs. It proves a fundamental inequality $\Lambda_{cb}(C_r^*(\mathcal{G}),C_0(\mathcal{G}^{(0)})) \leq \Lambda_{cb}(\mathcal{G})$ and identifies key classes (discrete groupoids and partial action groupoids) where equality holds, using Fell absorption and diagonal coactions as central tools. By developing a Fell absorption principle for discrete groupoids and constructing explicit $C_0(X)$-rank-one multipliers from groupoid data, the work bridges operator-algebraic and groupoid-analytic viewpoints. It also delineates three variants of weak amenability (measured, measurewise, and topological) and proves inner exactness for topologically weakly amenable groupoids, while highlighting open questions for the general étale case. Overall, the results illuminate how groupoid approximation properties reflect algebraic and dynamical structures, with concrete equality results for important subclasses and a clear path for future explorations.

Abstract

In this work, we study groupoids and their approximation properties, generalizing both the definitions and some known results for the group case. More precisely, we introduce weak amenability for groupoids using the definition of the Fourier algebra given by Renault. We prove that weakly amenable groupoids are inner exact. We also generalize its algebraic counterpart, the CBAP. To do this we introduce the notion of a quasi Cartan pair $(B,A)$ and see that $(C_r^*(G),C_0(G^0))$ can be viewed as such. We then define what it means for a pair $(B,A)$ to have the CBAP. We introduce the Cowling-Haagerup constants associated to these approximation properties and prove that $Λ_{\text{cb}}(C_r^*(G),C_0(G^0)) \leq Λ_{\text{cb}}(G)$. We then study some classes of groupoids where we could achieve equality, that is, $Λ_{\text{cb}}(G) = Λ_{\text{cb}}(C_r^*(G),C_0(G^0))$. They are discrete groupoids and groupoids arising from partial actions of a discrete group $Γ$ on a locally compact Hausdorff space $X$.

On weakly amenable groupoids

TL;DR

The paper extends weak amenability from discrete groups to groupoids via Renault's Fourier algebra, establishing a tight link between groupoid-level approximation properties and the CBAP for quasi Cartan pairs. It proves a fundamental inequality and identifies key classes (discrete groupoids and partial action groupoids) where equality holds, using Fell absorption and diagonal coactions as central tools. By developing a Fell absorption principle for discrete groupoids and constructing explicit -rank-one multipliers from groupoid data, the work bridges operator-algebraic and groupoid-analytic viewpoints. It also delineates three variants of weak amenability (measured, measurewise, and topological) and proves inner exactness for topologically weakly amenable groupoids, while highlighting open questions for the general étale case. Overall, the results illuminate how groupoid approximation properties reflect algebraic and dynamical structures, with concrete equality results for important subclasses and a clear path for future explorations.

Abstract

In this work, we study groupoids and their approximation properties, generalizing both the definitions and some known results for the group case. More precisely, we introduce weak amenability for groupoids using the definition of the Fourier algebra given by Renault. We prove that weakly amenable groupoids are inner exact. We also generalize its algebraic counterpart, the CBAP. To do this we introduce the notion of a quasi Cartan pair and see that can be viewed as such. We then define what it means for a pair to have the CBAP. We introduce the Cowling-Haagerup constants associated to these approximation properties and prove that . We then study some classes of groupoids where we could achieve equality, that is, . They are discrete groupoids and groupoids arising from partial actions of a discrete group on a locally compact Hausdorff space .

Paper Structure

This paper contains 14 sections, 42 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Measured Groupoids and their Operator Algebras
  3. Fourier-Stieltjes and Fourier Algebras
  4. Duality and Multipliers
  5. Weak Amenability
  6. Completely Bounded Approximation Property of Cartan Pairs
  7. Weak Amenability of Partial Actions
  8. Partial Actions
  9. Partial Action Groupoids
  10. Weak Amenability
  11. Weak Amenability of Discrete Groupoids
  12. Fell Absorption for Discrete Groupoids
  13. Weak Amenability
  14. Open Questions and Further Work

Key Result

Theorem 1.1

A $C^*$-algebra $A$ is nuclear if and only if there exists a net of finite rank completely positive operators of norm one $T_i : A \rightarrow A$ such that $T_i \rightarrow \text{id}_A$ in the strong operator topology.

Theorems & Definitions (94)

  • Theorem 1.1: lance1973nuclear
  • Proposition 1.2
  • Definition 1.3
  • Theorem 1.4: brown2008textrm
  • Theorem 1.5: haagerup2016group
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 84 more