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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.

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Abstract. We prove that if X is a locally compact σ-compact space then on its quotient, γ(X) say, determined by the algebra of all real valued bounded continuous functions on X, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if X is second countable locally compact then γ(X) is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in [1] and [7] where the primitive ideal space of a C * -algebra was considered.

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