Groups of quasi-invariance and the Pontryagin duality

A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that 1) there exists a continuous monomorphism of G to X, and 2) for each g ∈ X either g ∈ G and the shift μg is equivalent to μ or g 6∈ G and μg is orthogonal to μ. It is proved that G is a σ-compact subset of X. We show that there exists a quotient group T2 of l 2 modulo a discrete subgroup which is a Polish monothetic non locally quasi-convex (and hence nonreflexive) pathwise connected QI-group, and such that the bidual of T2 is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of X.