On characters of subgroups

Extension of positive definite functions from subgroups of nilpotent locally compact groups

“…This latter result can also directly be deduced from [11,Theorem 2]. On the other hand, in [10] an example is given of a 2-step solvable compactly generated locally compact group that has the extension property and nevertheless fails to be a SIN-group.…”

“…However, the two properties are not equivalent in general. For instance, the semidirect product group Z R, where n ∈ Z acts on R by multiplication with 2 n , has the extension property [10], but not the separation property.…”

“…We say that G has the extension property when each closed subgroup of G is extending. These properties have been studied by several authors [2], [3], [6], [9], [10], [16], [17]. Fundamental to all this has been Douady's observation (see [5, p. 204] and [12, (34.28)]) that if A is an abelian closed normal subgroup of a locally compact group G and χ is a character of A, then χ extends to some continuous positive definite function on G only if the stabiliser G χ = {x ∈ G : χ(x −1 ax) = χ(a) for all a ∈ A} is open in G.…”

On one-sided harmonic analysis

“…Since the extension property as stated in Theorem 1 is inherited by closed subgroups we can use Theorem 1 in [3] to see that in groups satisfying the onesided Wiener property (L) every connected closed subgroup is the direct product of a compact group and a vector group. By analyzing the proof of Theorem 1, however, we can show that the same holds under some weaker conditions and we shall give a proof which is considerably shorter than those given in [3] and [4]. If 77 is abelian and not contained in the center of G, then there is a character y of 77 such that F(y) =/= G; hence G cannot be connected.…”

“…The arguments used in the proof are essentially the same as those used in [3] to prove Theorem 1. Theorem 2.…”