Exotic coactions

If a locally compact group G acts on a C∗-algebra B, we have both full and reduced crossed products and each has a coaction of G. We investigate 'exotic' coactions in between the two, which are determined by certain ideals E of the Fourier-Stieltjes algebra B(G); an approach that is inspired by recent work of Brown and Guentner on new C∗-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C∗-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain 'E-crossed product duality', intermediate between full and reduced duality. We give partial results concerning exotic coactions with the ultimate goal being a classification of which coactions are determined by ideals of B(G).