Applications of Non-Commutative Duality to Crossed Product C^*-Algebras Determined by an Action or Coaction

Abstract

For a separable amenable group G and a separable C*-algebra A, let α denote an action of G on A, δ a coaction of G on A, and G×αA (respectively G×δA) the corresponding crossed product C*-algebras. We employ non-commutative duality theory to develop a notion of induced representation in the coaction case, and for both actions and coactions, to develop a duality between induction and restriction. We characterize ideals of G ×α A invariant under the dual coaction α ^ , as well as ideals of G ×δ A invariant under the dual action δ, and show that, in the coaction case, both the notions of induced ideal and of quasi-orbit in the primitive ideal space PR(A) of A are well-defined. For both actions and coactions, the ‘quasi-orbit map’ which maps a primitive ideal of the crossed product algebra to the quasi-orbit in PR(A) ‘over which it lives’ is continuous, open and surjective. As a consequence, if in addition G is compact and A is Type I AF, the crossed product algebra G ×α A is also AF.