Positive definiteness and Fell bundles over discrete groups

TL;DR

This paper develops a new concept of positive definiteness for bundle maps between Fell bundles over discrete groups, leading to functorial completely positive maps between their cross-sectional $C^*$-algebras and a generalized approximation property.

Contribution

It introduces a novel notion of positive definiteness for bundle maps, extending the theory of Fell bundles and their $C^*$-algebras with new approximation properties.

Findings

Defines positive definiteness for bundle maps between Fell bundles.

Establishes functorial completely positive maps between cross-sectional $C^*$-algebras.

Introduces a generalized approximation property for Fell bundles.

Abstract

We introduce a natural concept of positive definiteness for bundle maps between Fell bundles over (possibly different) discrete groups and describe several examples. Such maps induce completely positive maps between the associated full cross-sectional $C^*$-algebras in a functorial way. Under the assumption that the kernel of the homomorphism connecting the groups under consideration is amenable, they also induce completely positive maps between the associated reduced cross-sectional $C^*$-algebras. As an application, we define an approximation property for a Fell bundle over a discrete group which generalizes Exel's approximation property and still implies the weak containment property. Both approximation properties coincide when the unit fibre is nuclear.