Duality for Actions

Let G be a locally compact group. We show that the category A(G) of actions of G on C∗-algebras (with equivariant nondegenerate ∗-homomorphisms into multiplier algebras) is equivalent, via a full-crossed-product functor, to a comma category of maximal coactions of G under the comultiplication (C∗(G), δG); and also that A(G) is equivalent, via a reduced-crossed-product functor, to a comma category of normal coactions under the comultiplication. This extends classical Landstad duality to a category equivalence, and allows us to identify those C∗-algebras which are isomorphic to crossed products by G as precisely those which form part of an object in the appropriate comma category. Introduction Landstad duality (a term coined by the second author in [8]) refers to a particular method of characterizing crossed product C-algebras. The first appearance of this method is in [6], where Landstad characterized reduced crossed products by actions of locally compact groups in terms of the existence of suitably equivariant reduced coactions. The second author proved a dual version in [8], giving a characterization of crossed products by coactions in terms of the existence of suitably equivariant actions. In [5], the authors applied the recently developed theory of maximal coactions (see [1]) to give a version of Landstad’s characterization for full rather than reduced crossed products by actions. In the present paper we analyze the method of Landstad duality more deeply, shifting the focus from characterizing crossed products to developing a process which recovers the action from its crossed product. It is useful to compare this with the more well-established crossedproduct duality, which uses the crossed product by the dual coaction to recover the action up to Morita-Rieffel equivalence (see [2, Appendix] Date: April 19, 2008. 2000 Mathematics Subject Classification. Primary 46L55; Secondary 46M15, 18A25.