Positive definiteness and Fell bundles over discrete groups

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.