Morita enveloping Fell bundles

Abstract: We introduce notions of weak and strong equivalence for non-saturated 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.