Maximal compact normal subgroups and pro-Lie groups
Abstract: We are concerned with conditions under which a locally compact group has a maximal compact normal subgroup and whether or not is a Lie group. If has small compact normal subgroups such that is a Lie group, then is pro-Lie. If in there is a collection of closed normal subgroups such that and is a Lie group for each , then 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.
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-  Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1–40. MR 0284541, https://doi.org/10.1515/crll.1971.246.1
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