Full duality for coactions of discrete groups

Abstract

Using the strong relation between coactions of a discrete group G on C*-algebras and Fell bundles over G, we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of G, as opposed to the usual theory which uses the spatially defined reduced crossed products and normal subgroups of G. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.