Actions of Fell bundles

We introduce and study actions of Fell bundles over discrete groups on Hilbert bundles. Many examples of such actions are presented. We discuss the connection with positive definite bundle maps between Fell bundles, culminating in the unital case in a Gelfand-Raikov type theorem. We also use these actions to construct C*-correspondences over cross-sectional C*-algebras of Fell bundles.