On characters of subgroups

φ{symbol} is a continuous positive definite function of a locally compact group G. Considering subgroups of G for which the restriction of φ{symbol} is a character a short proof of Douady's Theorem and its converse can be given. We show that in connected groups G which are not the direct product of a vector group and a compact group there always exist a closed subgroup H and a character of H which cannot be extended to G as a continuous positive definite function. Moreover, if G is almost connected the restriction map B(G) → B(H) of the Fourier-Stieltjes algebras is surjective for every closed subgroup H if and only if G has arbitrarily small invariant neighbourhoods at e.