We introduce notions of weak and strong equivalence for nonsaturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional C * -algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.

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We study partial actions of exact discrete groups on C * -algebras. We show that the partial crossed product of a commutative C * -algebra by an exact discrete group is nuclear whenever the full and reduced partial crossed products coincide. This generalises a result by Matsumura in the context of global actions. In general, we prove that a partial action of an exact discrete group on a C * -algebra A has Exel's approximation property if and only if the full and reduced partial crossed products associated to the diagonal partial action on A ⊗ max A op coincide. We apply our results to show that the reduced semigroup C * -algebra C * λ (P ) of a submonoid of an exact discrete group is nuclear if the left regular representation on 2 (P ) is an isomorphism between the full and reduced C * -algebras. We also show that nuclearity is equivalent to the weak containment property in the case of C * -algebras associated to separated graphs.

We give a notion of equivalence for Fell bundles over groups, not necessarily saturated nor separable. The equivalence between two Fell bundles is implemented by a bundle of Hilbert bimodules with some extra structure. Suitable cross-sectional spaces of such a bundle turn out to be imprimitivity bimodules for the cross-sectional C∗-algebras of the involved Fell bundles. We show that amenability is preserved under this equivalence.