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Non-commutative LP-spaces

Published online by Cambridge University Press:  24 October 2008

F. J. Yeadon
F. J. Yeadon
Affiliation:
The University of Hull
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Extract

1. Introduction. The spaces L1 and L2 of unbounded operators associated with a regular gauge space (von Neumann algebra equipped with a faithful normal semi-finite trace) are defined by Segal(5) definitions 3.3, 3.7. The spaces Lp (1 < p < ∞, p ± 2) are defined by Dixmier(2) as the abstract completions of their bounded parts. Dixmier makes use of the Riesz convexity theorem to prove the Hölder inequality, and the uniform convexity, and hence reflexivity, of LLp (2 < p < ∞).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 1 , January 1975 , pp. 91 - 102
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Dixmier, J.Les algèbres d'opérateurs dans l'espace Hilbertien. Gauthier-Villars (Paris, 1969).Google Scholar
(2)Dixmier, J.Formes linéaires sur un anneau d'operateurs. Bull. Soc. Math. France 81 (1953), 9–39.CrossRefGoogle Scholar
(3)Garling, D. J. H.On ideals of operators in Hilbert space. Proc. London Math. Soc. 17 (1967), 115–138.CrossRefGoogle Scholar
(4)Grotthendieck, A.Rearrangements de fonctions et inégalités de convexité dans les algébres de von Neumann munies d'une trace. Séminaire Bourbaki (1955).Google Scholar
(5)Segal, I. E.A non-commutative extension of abstract integration. Ann. of Math. 57 (1952), 401–457.CrossRefGoogle Scholar
(6)Stinespring, W. F.Integration theorems for gages and duality for unimodular groups. Trans. Amer. Math. Soc. 90 (1959), 15–56.CrossRefGoogle Scholar
(7)Yeadon, F. J.Convergence of measurable operators. Proc. Cambridge Philos. Soc. 74 (1973), 257–268.CrossRefGoogle Scholar