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W*-Amenability for Fell bundles over discrete groups

Alcides Buss, Damián Ferraro

TL;DR

The paper develops a comprehensive W*-amenability theory for Fell bundles over discrete groups by introducing a canonical enlargement to ll^∞(G,bla) and linking amenability to equivariant conditional expectations at the bundle and kernel levels. It establishes that W*-amenability is equivalent to the existence of conditional expectations from the enlarged bundle to the original one and from the kernel algebra to the reduced crossed product, and it derives permanence results under subgroup restrictions and normal-quotient decompositions. It further connects Fell bundles to group coactions on von Neumann algebras, showing that amenability properties translate into coaction amenability and co-amenability via dual actions and the W*-AP, and applies the framework to Green twisted actions. Overall, the work unifies C*- and W*-amenability theories for noncommutative dynamical systems and provides structural tools for analyzing amenability across reductions, quotients, and coactions.

Abstract

We investigate amenability for $W^*$-Fell bundles over a discrete group $G$, with a focus on its characterization via approximation properties and conditional expectations. Building on the notion of $W^*$-amenability, we construct an enlarged $W^*$-Fell bundle analogous to $\ell^\infty(G, M)$ for a group action $G$ on a von Neumann algebra $M$, and relate amenability to the existence of suitable conditional expectations at both the bundle and crossed-product levels. Our results unify and extend several approaches to amenability for noncommutative dynamical systems. As applications of our methods, we prove that amenability of Fell bundles passes to restrictions to subgroups and that a Fell bundle over a group $G$ is amenable if and only if both its restriction to a normal subgroup $H \trianglelefteq G$ and the associated quotient Fell bundle over $G/H$ are amenable. This provides a powerful structural tool that extends classical permanence results for group amenability to the setting of \Wstar Fell bundles and also \cstar algebraic Fell bundles. We also discuss how Fell bundles and their amenability interact with group coactions on $C^*$-algebras and von Neumann algebras.

W*-Amenability for Fell bundles over discrete groups

TL;DR

The paper develops a comprehensive W*-amenability theory for Fell bundles over discrete groups by introducing a canonical enlargement to ll^∞(G,bla) and linking amenability to equivariant conditional expectations at the bundle and kernel levels. It establishes that W*-amenability is equivalent to the existence of conditional expectations from the enlarged bundle to the original one and from the kernel algebra to the reduced crossed product, and it derives permanence results under subgroup restrictions and normal-quotient decompositions. It further connects Fell bundles to group coactions on von Neumann algebras, showing that amenability properties translate into coaction amenability and co-amenability via dual actions and the W*-AP, and applies the framework to Green twisted actions. Overall, the work unifies C*- and W*-amenability theories for noncommutative dynamical systems and provides structural tools for analyzing amenability across reductions, quotients, and coactions.

Abstract

We investigate amenability for -Fell bundles over a discrete group , with a focus on its characterization via approximation properties and conditional expectations. Building on the notion of -amenability, we construct an enlarged -Fell bundle analogous to for a group action on a von Neumann algebra , and relate amenability to the existence of suitable conditional expectations at both the bundle and crossed-product levels. Our results unify and extend several approaches to amenability for noncommutative dynamical systems. As applications of our methods, we prove that amenability of Fell bundles passes to restrictions to subgroups and that a Fell bundle over a group is amenable if and only if both its restriction to a normal subgroup and the associated quotient Fell bundle over are amenable. This provides a powerful structural tool that extends classical permanence results for group amenability to the setting of \Wstar Fell bundles and also \cstar algebraic Fell bundles. We also discuss how Fell bundles and their amenability interact with group coactions on -algebras and von Neumann algebras.

Paper Structure

This paper contains 13 sections, 26 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. W*-Fell bundles over discrete groups
  3. A concrete description of the W*-algebra of kernels
  4. The reduced cross-sectional W*-algebra
  5. The W*-algebra of kernels as a reduced cross-sectional W*-algebra
  6. The W*-partial cross-sectional Fell bundle
  7. An example: semidirect product bundles
  8. Amenability, fixed points and conditional expectations
  9. Fell bundles and group coactions
  10. Coactions on von Neumann algebras and W*-Fell bundles
  11. Aproximation property in terms of coactions
  12. Reductions and quotients of Fell bundles
  13. Amenability for Green twisted actions

Key Result

Theorem 1

For a W*-Fell bundle $\mathcal{M}=\{M_t\}_{t\in G},$ the following statements are equivalent. Here $\ell^\infty(G,\mathcal{M}):=\{\ell^\infty(G, M_t)\}_{t \in G}$ denotes the canonical enlargement of a $W^*$-Fell bundle, to be introduced in Section sec:enlarged-Fell-bundles, $W^*_{\mathop{\rm r}}(\mathcal{M})$ denotes the W*-crossed section algebra and $\mathbb{k}_{\operatorname{\rm w}^*}(\m

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.6
  • Remark 2.7
  • ...and 49 more