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Characterizations of amenability for noncommutative dynamical systems and Fell bundles

Alcides Buss, Damián Ferraro

TL;DR

The paper develops a unified framework to characterize amenability for C*-dynamical systems and Fell bundles over locally compact groups via diagonal maximal tensor products, linking amenability to the weak containment property and to Exel–Ng approximation properties (AP/BCAP). It proves that AP, BCAP, and wcp are equivalent in this setting and provides permanence results for amenability and nuclearity under subgroup restrictions and quotient constructions. By establishing diagonal tensor criteria and kernel-action amenability, the authors connect dynamical amenability to structural properties of cross-sectional algebras and extend prior results of several authors to a cohesive, general theory. The work thus furnishes new tools for identifying amenability and analyzing nuclearity in a broad class of noncommutative dynamical systems and Fell bundles, with clear implications for permanence under standard C*-algebraic constructions.

Abstract

We prove that for a locally compact group $G$, a $C^*$-dynamical system $(A,G,α)$ is amenable if and only if, for every other system $(B,G,β)$, the diagonal system $(A \otimes_{\max} B, G, α\otimes^d_{\max} β)$ has the weak containment property (wcp). For Fell bundles over $G$, we construct a diagonal tensor product $\otimes^d_{\max}$ and show that a Fell bundle $\mathcal{A}$ has the positive approximation property of Exel and Ng (AP) precisely when $\mathcal{A} \otimes^d_{\max} \mathcal{B}$ has the wcp for every Fell bundle $\mathcal{B}$ over $G$. Equivalently, $\mathcal{A}$ has the AP if and only if the natural action of $G$ on the $C^*$-algebra of kernels of $\mathcal{A}$ is amenable. We show that the approximation properties introduced by Abadie and by Bédos-Conti are equivalent to the AP. We also study the permanence of the wcp, the AP, and the nuclearity of cross-sectional $C^*$-algebras under restrictions, quotients, and other constructions. Our results extend and unify previous characterizations of amenability for $C^*$-dynamical systems and Fell bundles, and provide new tools to analyze structural properties of associated $C^*$-algebras.

Characterizations of amenability for noncommutative dynamical systems and Fell bundles

TL;DR

The paper develops a unified framework to characterize amenability for C*-dynamical systems and Fell bundles over locally compact groups via diagonal maximal tensor products, linking amenability to the weak containment property and to Exel–Ng approximation properties (AP/BCAP). It proves that AP, BCAP, and wcp are equivalent in this setting and provides permanence results for amenability and nuclearity under subgroup restrictions and quotient constructions. By establishing diagonal tensor criteria and kernel-action amenability, the authors connect dynamical amenability to structural properties of cross-sectional algebras and extend prior results of several authors to a cohesive, general theory. The work thus furnishes new tools for identifying amenability and analyzing nuclearity in a broad class of noncommutative dynamical systems and Fell bundles, with clear implications for permanence under standard C*-algebraic constructions.

Abstract

We prove that for a locally compact group , a -dynamical system is amenable if and only if, for every other system , the diagonal system has the weak containment property (wcp). For Fell bundles over , we construct a diagonal tensor product and show that a Fell bundle has the positive approximation property of Exel and Ng (AP) precisely when has the wcp for every Fell bundle over . Equivalently, has the AP if and only if the natural action of on the -algebra of kernels of is amenable. We show that the approximation properties introduced by Abadie and by Bédos-Conti are equivalent to the AP. We also study the permanence of the wcp, the AP, and the nuclearity of cross-sectional -algebras under restrictions, quotients, and other constructions. Our results extend and unify previous characterizations of amenability for -dynamical systems and Fell bundles, and provide new tools to analyze structural properties of associated -algebras.
Paper Structure (18 sections, 49 theorems, 128 equations)

This paper contains 18 sections, 49 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Amenability of dynamical systems
  3. Preliminaries
  4. SIN groups and amenability
  5. Amenability of Fell bundles and C*-partial actions
  6. Amenability and nuclearity
  7. An approximation property of Bédos and Conti
  8. Completely positive maps between Fell bundles
  9. Characterization of completely positive maps between semidirect product bundles
  10. Ccp maps and amenability
  11. Reductions and quotient bundles
  12. Reductions to open subgroups and full cross-sectional C*-algebras
  13. Reductions to open subgroups and reduced cross-sectional C*-algebras
  14. The universal completion of the partial cross-sectional bundle
  15. The reduced completion of the partial cross-sectional bundle
  16. ...and 3 more sections

Key Result

Theorem 1

Let $G$ be a locally compact group and let $\alpha$ be a C*-action of $G$ on a C*-algebra $A$. Using the universal unitary implementation $(A\subset \mathbb{B}(X_\alpha),U^{\alpha})$, denote by $A'_\alpha$ the commutant of $A$ in $\mathbb{B}(X_\alpha)$ and let $\beta$ be the continuous part of $\ope

Theorems & Definitions (112)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5: Corollary 6.6 of mckee2020amenable
  • ...and 102 more