Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T03:16:06.472Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Bourbaki, N., Groupes et algèbres de Lie, Eléments de Mathématiques (Hermann, Paris, 1960).Google Scholar
2. Halmos, P. R., Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933.Google Scholar
3. Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
4. Putnam, C. R., Commutation properties of Hilbert space operators and related topics (Springer-Verlag, New York, 1967).Google Scholar
5. Putnam, C. R., Séminaire “Sophys Lie”, 1954-1955. Théorie des Algèbres de Lie; Topologie des groupes de Lie (Secrétariat Mathématique, Paris, 1955).Google Scholar
6. Serre, J. P., Lie algebras and Lie groups, Lectures given at Harvard University (Benjamin, New York, Amsterdam, 1965).Google Scholar
7. Stampfli, J. G., The norm of a derivation, Pacific J. Math. 38 (1970), 737–747.Google Scholar
8. Sabac, M., Une géneralisation du théorème de Lie, Bull. Sci. Math. 95 (1971), 53–57.Google Scholar
9. Vasilescu, F.-H., On Lie1 s theorem in operator algebras, Trans. Amer. Math. Soc. (to appear).Google Scholar