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On Lie groups in varieties of topological groups. (English) Zbl 0917.22003

A variety of topological groups is a class of topological groups closed with respect to the operations of forming arbitrary products (C), proceeding to subgroups (S), and quotient groups (Q). Forming products over finite families is denoted by P, and passage to closed subgroups by \(\overline{\text{S}}\). The first author has shown [Colloq. Math. 46, 147-165 (1982; Zbl 0501.22002)] that \(V(\Omega) :=\overline{\text{SCQSP}}(\Omega)\) is the smallest variety of topological groups containing \(\Omega\). The paper under review shows (by means of a counterexample) that Proposition 5.3 in the paper by K. H. Hofmann, S. A. Morris and the reviewer [Colloq. Math. 70, 151-163 (1996; Zbl 0853.22001)] is false. Generalizing the notion of minimal group topologies, the paper under review introduces locally minimal group topologies: a Hausdorff topological group \((G,\tau)\) is called locally minimal if there exists a neighborhood \(V\) of the identity with the property that every group topology \(\sigma\subseteq\tau\) such that the \(\sigma\)-interior of \(V\) is nonempty satisfies \(\sigma=\tau\). Every Banach-Lie group is uniformly free from small subgroups, in the sense of P. Enflo [Isr. J. Math. 8, 230-252 (1970; Zbl 0214.28402)]. If \(G\) is uniformly free from small subgroups then \(G\) is locally minimal. In the paper under review, the flawed proposition is replaced by Theorem 3.10: All members of \(V(\Omega)\) that are locally minimal and have no small subgroups are contained in \(\text{SPQ}\overline{\text{S}}\text{P}(\Omega)\subseteq \text{QSP}(\Omega)\). As a consequence, Theorem 5.5 of the paper by Hofmann et al. [loc.cit.] is extended in Theorem 1.1 of the paper under review: every Banach–Lie group in \(V(\Omega)\) is contained in \(\text{Q}\overline{\text{S}} \text{P}(\Omega)\).

MSC:

22A05 Structure of general topological groups
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