Categorical Landstad duality for actions

Abstract

Let G be a locally compact group. We show that the category A(G) of actions of G on C � -algebras (with equivari- ant 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),�G); and also that A(G) is equivalent, via a reduced-crossed-product functor, to a comma category of normal coactions under the comul- tiplication. 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.