On one-sided harmonic analysis

Abstract

Abstract. It is shown that, in locally compact groups satisfying the onesided version of the Wiener property introduced by Leptin, characters of closed subgroups have always continuous positive definite extensions onto the whole group. We give a quick proof that for such groups the connected component of the identity is the direct product of a compact group and a vector group.

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

“…Conversely, a connected group G has the extension property only if it is a SIN-group (equivalently, being connected, G is the direct product of a vector group and a compact group) [3,Corollary 2]. 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.…”

Extension and separation properties of positive definite functions on locally compact groups

“…Consequently, every group with small conjugation invariant neighbourhoods (SIN-group) satisfies both the extension and the separation property. Conversely, if G is either almost connected or a compactly generated nilpotent group, then either of these properties forces G to be a SIN-group (see [7], [15], [19] and [20]). Also, for G almost connected, the separating subgroups of G have been identified as precisely the so-called neutral subgroups [21] (for the notion and basic theory of neutral subgroups see [23] and [32]).…”