Maximal compact normal subgroups and pro-Lie groups

Abstract

We are concerned with conditions under which a locally compact group G G has a maximal compact normal subgroup K K and whether or not G / K G/K is a Lie group. If G G has small compact normal subgroups K K such that G / K G/K is a Lie group, then G G is pro-Lie. If in G G there is a collection of closed normal subgroups { H α } \{ {H_\alpha }\} such that ∩ H α = e \cap {H_\alpha } = e and G / H α G/{H_\alpha } is a Lie group for each α \alpha , then G G is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.