Categorical Landstad duality for actions

TL;DR

This paper extends Landstad duality to a categorical framework, showing equivalences between actions of a group on C*-algebras and certain comma categories of coactions, thereby characterizing crossed products categorically.

Contribution

It establishes a categorical equivalence between actions of a locally compact group on C*-algebras and comma categories of coactions, extending classical Landstad duality.

Findings

Categorical equivalence between actions and comma categories of coactions

Identification of crossed products via categorical objects

Extension of classical Landstad duality to a broader categorical context

Abstract

We show that the category A(G) of actions of a locally compact group 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),delta_G); 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.