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.

Approximation Properties for Crossed Products by Actions and Coactions

“…Curiously, it does not seem possible to prove this without first considering the special case of Theorem 4.1. A similar situation arose in the fourth section of [7].…”

“…where π is a faithful representation of A on H, and M f denotes multiplication by f on L 2 (G) (Definition 1.4 of [7]). The coaction identity (2.5) and non-degeneracy (2.6) ensure that this is a C * -algebra, [27].…”

Full C*-Crossed product duality

“…denotes the C*-algebra of compact operators on the Hilbert space L (G). He leaves open the question whether this result [2] Full C*-crossed product duality 35 extends to twisted actions, which we answer in the affirmative (Theorem 3.6) for the case of twisted actions in Green's sense. He also shows that if A carries a coaction of G, then there is natural way to define a dual action of G on the crossed product Ax G, and asks whether the dual crossed product AxGxG is isomorphic to A ®X.…”

The ideal property in crossed products

“…But, by [GL,Proposition 3.11 (ii)] it follows that I = I α G for every α-invariant ideal I of A. Again using the fact that G is amenable, by a result of E.C.…”