Maximality of dual coactions on sectional $C^*$-algebras of Fell bundles and applications
We give a simple proof of the maximality of dual coactions on full cross-sectional $C^*$-algebras of Fell bundles over locally compact groups. This result was only known for discrete groups or for saturated (separable) Fell bundles over locally compact groups. Our proof, which is derived as an application of the theory of universal generalised fixed-point algebras for weakly proper actions, is different from these previously known cases and works for general Fell bundles over locally compact groups. As applications, we extend certain exotic crossed-product functors in the sense of Baum, Guentner and Willett to the category of Fell bundles and the category of partial actions, and we obtain results about the $K$-theory of (exotic) cross-sectional algebras of Fell bundles over $K$-amenable groups. As a bonus, we give a characterisation of maximal coactions of discrete groups in terms of maximal tensor products.