Three versions of categorical crossed-product duality

In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be formulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomorphisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed-product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an "inversion" of the process of forming a crossed product. In some cases, we obtain what we call "good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Pedersen to obtain a fixed-point functor that is quasi-inverse to the reduced-crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's theorem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open.