Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T20:17:30.193Z Has data issue: false hasContentIssue false

The path functor and faithful representability of banach lie algebras

Published online by Cambridge University Press:  09 April 2009

W. T. van Est
Affiliation:
Rijksuniversiteit, Leiden, Holland.
S. Świerczkowski
Affiliation:
Queen's University, Kingston, Ont., Canada.
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 note “vector space” will mean “Banach space” unless otherwise specified. Accordingly “Lie algebra” will stand for “Banach Lie algebra”. Morphisms between Lie algebras will be assumed continuous. A Banach algebra B will be always assumed associative, and it will be also viewed as a Lie algebra with product [X, YXYYX. In particular, the Lie algebra gl(V) of endomorphisms of a vector space V will be equipped with the uniform norm. A morphism of Lie algebras L → gl(V) will b called a representation of L in gl(V). Also, if B is a Banach algebra, a morphism of Lie algebras L → B will be called a representation of L in B. From such one evidently obtains a representation of L in gl(B). A representation will be called faithful if it is injective.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Ado, I.D., ‘The representation of Lie algebras by matrices’, Amer. Math. Soc. Transi. (1) 9 (1962),308327.Google Scholar
[2]Banach, S., Théorie des opérations linéaries (Warszawa, 1932).Google Scholar
[3]van Est, W.T. and Korthagen, Th.J., ‘Non-enlargible Lie algebras’, Indag. Math. 26 (1964), 1531.CrossRefGoogle Scholar
[4]van Est, W.T., ‘On Ado's theorem’, Indag. Math. 28 (1966), 176191.CrossRefGoogle Scholar
[5]Lang, Serge, Introduction to differentiable manifolds (Interscience [John Wiley & Sons], New York-London, 1962).Google Scholar
[6]Shiga, Kôji, ‘Representations of a compact group on a Banach space’, J. Math. Soc. Japan 7 (1955), 224248.CrossRefGoogle Scholar
[7]Šwierczkowski, S., ‘The path functor on Banach Lie algebras’, Indag. Math. 33 (1971), 235239.CrossRefGoogle Scholar
[8]Wielandt, Helmut, ‘Über die Unbeschränktheit der Operatoren der Quantenmechanik’, Math. Ann. 121 (1949), 21.CrossRefGoogle Scholar
[9]Wintner, Aurel, ‘The unboundedness of quantum-mechanical matrices’, Phys. Rev. 71 (1947), 738739.CrossRefGoogle Scholar