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Pro-Lie groups. (English) Zbl 0575.22006

This paper studies the structure of locally compact groups which are determined by Lie groups in one form or another. The best understood class if that of pro-Lie groups G (which are locally compact groups with arbitrarily small compact normal subgroups N such that G/N is a Lie group. Theorem 1 says that these are precisely those groups G such that for each neighborhood U of the identity there is a neighborhood W such that for every subgroup H of G inside W and every g in G one has \(gHg^{-1}\subseteq U.\)
The authors call a locally compact group an L-group, if for every neighborhood U of the identity and every compact subset C of G there is a neighborhood W such that for every subgroup H of G inside U and every g in G one has \(gHg^{-1}\cap C\subseteq U\). This condition is weaker than being pro-Lie, but the authors show that both conditions agree for compactly generated locally compact groups (and this means that they should be generated by a compact identity neighborhood).
A locally compact group is called a residual Lie group, if the Lie group quotients separate the points. Every such is an L-group; the reverse seems to be unsettled. But every compactly generated residual Lie group is pro-Lie (Theorem 1.4). The paper contains numerous useful conditions for locally compact groups to be pro-Lie for which we refer to the paper. It also contains a helpful diagram of the various implications and nonimplications known to exist between various properties of locally compact groups being determined by Lie groups.
Reviewer: K.H.Hofmann

MSC:

22D05 General properties and structure of locally compact groups
22E99 Lie groups
20E18 Limits, profinite groups
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