Extension problems and non-abelian duality for C*-algebras

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Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation U of a crossed product C*(G) (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-abelian duality for crossed products of C*-algebras.