  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.
