Fixed-point algebras for weakly proper Fell bundles

We define weakly proper Fell bundles and construct exotic fixed-point algebras for such bundles. Three alternative constructions of such algebras are given. Under a kind of freeness condition, one of our constructions implies that every exotic cross-sectional C*-algebra of a weakly proper Fell bundle is Morita equivalent to an exotic fixed-point algebra. The other constructions are used to show that ours generalizes that of Buss and Echterhoff on weakly proper actions on C*-algebras. We also generalize to Fell bundles the fact that every C*-action which is proper in Kasparov's sense is amenable.