Extension Problems For

When A = C, H = N is normal and K = G, this is a classical problem which has been studied using a variety of methods (see [3, 8, 1, 12], for example), and its solution for irreducible representations is a crucial ingredient in the Mackey machine [17]. In [10], we considered the problem for C-dynamical systems with H = N normal and K = G, and found a condition on the induced representation IndGN(π×U) of the crossed product A×αG which is equivalent to the existence of (π, V ) [10, Theorem 4]. For fixed K, we can apply this theorem to obtain a criterion involving the induced representation IndKN (π × U) of A×α K. This is not a very satisfactory solution to Problem 1, though, since it requires that we consider all the induced representations IndKN (π × U) as K varies. In our first main theorem, we describe a criterion which uses the same induced representation IndGN(π × U) of A×α G for every subgroup K (see Theorem 3.1). Our proof of Theorem 3.1, like that of [10, Theorem 4], uses ideas from non-abelian duality for crossed products of C-algebras, and hence it is natural to consider also the analogue of Problem 1 for crossed products by coactions. In stating Problem 1, we made implicit use of our ability to restrict α to actions of the subgroups H and K. Coactions of G restrict to coactions of quotients of G, and hence the most natural dual analogue of Problem 1 involves a pair of closed normal subgroups of G. A precise statement of this dual analogue is given in Problem 2 in §4. Our solutions to Problems 1 and 2 follow the same general pattern. Each proof has two main ingredients: a theorem describing one aspect of the duality between induction and restriction of representations, of the sort proved in [16, 6, 5], and an imprimitivity theorem which allows us to recognise induced representations. When dealing with duality for crossed products of C-algebras, we have to make choices: we can use full crossed