Deformations of Fell bundles and twisted graph algebras

TL;DR

This paper introduces a method to deform Fell bundles over discrete groups using two-cocycles, leading to new twisted graph algebras and a C*-bundle structure that unifies these deformations.

Contribution

It defines deformations of Fell bundles via two-cocycles and demonstrates their application to twisted graph algebras, establishing a C*-bundle framework.

Findings

Deformations produce new Fell bundles with deformed multiplication.

Twisted graph algebras can be realized as deformations of standard graph algebras.

Different cocycle deformations form the fibers of a C*-bundle.

Abstract

We consider Fell bundles over discrete groups, and the C*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles with the same underlying Banach bundle but with the multiplication deformed by a two-cocycle on the group. Every graph algebra can be viewed as the C*-algebra of a Fell bundle, and there are are many cocycles of interest with which to deform them. We thus obtain many of the twisted graph algebras of Kumjian, Pask and Sims. We demonstate the utility of our approach to these twisted graph algebras by proving that the deformations associated to different cocycles can be assembled as the fibres of a C*-bundle.