Some non-trivial cocycles

AbstractFor each locally compact non-compact σ-compact Abelian group G whose dual is an I-group and each compact Abelian group Σ, such that there is a one-to-one continous homomorphism from G onto a dense subgroup of Σ, we show the existence of cocycles on Σ whose associated unitary representations of G have purely singular continuous spectrum, as well as the existence of cocycles on Σ whose associated unitary representations of G have purely Haar spectrum