Free Banach-Lie algebras, couniversal Banach-Lie groups, and more
Here we apply the concept of a free complete normed Lie algebra to the theory of topological and Lie groups. Our main result is that there exists a couniversal separable connected Banach-Lie group, that is, such a separable connected Banach-Lie group that any other such Banach-Lie group is its quotient Lie group. This follows from the observation that any free Banach-Lie algebra is enlargeable, that is, comes from an appropriate Banach-Lie group. Also we give entirely new and rather transparent proofs of two earlier known results. The first was obtained by W. T. van Est and independently by \textit{S. Świerczkowski} [see Indag. Math. 33, 235-239 (1971; Zbl 0213.137)]: any Banach-Lie algebra is a quotient algebra of an enlargeable Banach-Lie algebra. As to the second one, some two decades ago \textit{I. Kaplansky} [Lie algebras and locally compact groups (Chicago, 1971; Zbl 0223.17001)] asked whether a quotient group of a topological group with no small subgroups (NSS group) is again an NSS group. It was proved by successive efforts of \textit{S. A. Morris} and \textit{H. B. Thompson} [Proc. Am. Math. Soc. 46, 431-437 (1974; Zbl 0294.22001)], \textit{H. B. Thompson} [J. Aust. Math. Soc. 18, 482-484 (1974; Zbl 0297.22001)], \textit{O. V. Sipacheva} and \textit{V. V. Uspenskij} [Vestn. Mosk. Univ., Ser. I 1987, No. 4, 21-24 (1987; Zbl 0623.22002)], and the author [``Topological groups and algebraic envelopes of topological spaces'', Ph.D. thesis, Moscow State University, Moscow, 1983, 78 pp. (in Russian)] that each topological group is a quotient group of an NSS group. The proof is ``hard'' -- it relies on combinatorial technique of words in free groups. The concept of free Banach-Lie algebra enables us to provide a purely Lie-theoretic and certainly ``soft'' proof.
couniversal separable connected Banach-Lie group
topological group with no small subgroups
scientific article; zbMATH DE number 508856