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Normed Lie Algebras

Published online by Cambridge University Press:  20 November 2018

F.-H. Vasilescu
F.-H. Vasilescu*
Affiliation:
Institute of Mathematics, Bucharest, Rumania; Queen's University, Kingston, Ontario
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In this paper we attempt to define the notion of normed Lie algebra by endowing the corresponding algebraic concept with topologicalmetric properties. More precisely, we define normed Lie algebras as being normed spaces possessing a Lie product, the latter satisfying a compatibility relation. It turns out that any normed algebra, in particular the algebra of continuous linear operators on a normed space, is a normed Lie algebra in the sense denned below, with the usual Lie product given by the additive commutator.

It seems that some algebraic features have within the framework of normed Lie algebras the natural topological extensions. We mention, for instance, the convergence of the well-known Campbell-Hausdorff formula. Let us also mention the occurence of a variant of the Kleinecke-Sirokov theorem, obtained via universal enveloping algebra, which might be unknown in this context.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 580 - 591
Copyright
Copyright © Canadian Mathematical Society 1972

References

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