Partial Representations and Amenable Fell Bundles over Free Groups

TL;DR

This paper proves that certain structured Fell bundles over free groups are always amenable if they satisfy orthogonality and semi-saturation conditions, expanding understanding of their algebraic properties.

Contribution

It establishes the amenability of orthogonal, semi-saturated Fell bundles over free groups, a novel result in the theory of operator algebras.

Findings

Fell bundles over free groups are amenable under specified conditions.

Orthogonality and semi-saturation imply amenability.

Extends the class of known amenable Fell bundles.

Abstract

We show that a Fell bundle B = {B_t}_{t \in F}, over an arbitrary free group F, is amenable, whenever it is orthogonal (in the sense that B_x^* B_y = 0, if x and y are distinct generators of F) and semi-saturated (in the sense that B_{ts} coincides with the closed linear span of B_t B_s, when the multiplication ``ts'' involves no cancelation).