On neutral subgroups of topological groups

On a separation property of positive definite functions on locally compact groups

“…For (ii) ⇒ (iii) see [13]. Finally, when G is almost connected or locally connected, (iii) ⇒ (iv) by Theorem 1.7 of [13].…”

“…The preceding corollary applies to groups G such as Example 2.6 of [13] to provide subgroups H such that G has the H-separation property.…”

“…Also, slightly generalizing the result of [5] mentioned above, it was shown in [10] that G has the H-separation property whenever H is a neutral subgroup of G. Here a subgroup H is called neutral if given any neighbourhood U of the identity, there exists a neighbourhood V of the identity such that V H ⊆ HU . Such subgroups were first considered in [16] in the context of invariant measures on homogeneous spaces and have later been studied extensively in [13] (see also [17]). …”

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

“…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]). Very recently, Losert [25] has given an intricate characterization of neutral subgroups of locally compact groups that possess an almost connected open normal subgroup.…”

Invariant projections and convolution operators

“…According to Leischner and Roelcke [9] H is said to be locally neutral in G if there is a compact neighborhood U 0 of e such that for every neighborhood U of e there is a neighborhood V of e with (HV H) ∩ U 0 ⊂ U H. Proposition 2. Let G be a locally compact group and H a closed subgroup locally neutral in G. We assume that G/H admits an invariant measure.…”