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Almost-Lp-projections and Lp isomorphisms

Published online by Cambridge University Press:  14 November 2011

Michael Cambern ,
Krzysztof Jarosz  and
Georg Wodinski
Michael Cambern
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.
Krzysztof Jarosz
Affiliation:
Department of Mathematics, Southern Illinois University at Edwardsville, Edwardsville, IL 62026, U.S.A.
Georg Wodinski
Affiliation:
Institut für Mathematik I, Freie Universität Berlin, Arnimalle 3, D-1000 Berlin 33, Germany
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Synopsis

Lp -summands and Lp -projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value of p, l ≦ p ≦ ∞, p ≠ 2, any two Lp -projections on a given Banach space E commute. Here we introduce the notion of almost-Lp -projections, and we establish a result which generalises Behrends' theorem, while also simplifying its proof. Almost-Lp-projections are then applied to the study of small-bound isomorphisms of Bochner LP -spaces. It is shown that if the Banach space E satisfies a geometric condition which, in the finite-dimensional case, reduces to the absence of non-trivial Lp-summands, then for separable measure spaces, the existence of a small-bound isomorphism between Lp (λ1, E) and LP(λ2, E) implies that these Bochner spaces are, in fact, isometric.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 13 - 25
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Alspach, D.. Small into isomorphisms on L p spaces. Illinois J. Math. 27 (1983), 300–314.CrossRefGoogle Scholar
2Banach, S.. Théorie des opérations linéaires (Monografje Matematyczne, Warszawa, 1932).Google Scholar
3Behrends, E.. L p-Struktur in Banachräumen. Studio Math. 55 (1976), 71–85.CrossRefGoogle Scholar
4Behrends, E. et al. Lp-structure in real Banach Spaces, Lecture Notes in Mathematics 613 (Berlin: Springer, 1977).CrossRefGoogle Scholar
5Benyamini, Y.. Near isometries in the class of L 1-preduals. Israeli. Math. 20 (1975), 275–281.CrossRefGoogle Scholar
6Cambern, M.. The isometries of L p(X, K). Pacific J. Math. 55 (1974), 9–17.CrossRefGoogle Scholar
7Fleming, R. J. and Jamison, J. E.. Classes of operators on vector-valued integration spaces. J. Austral. Math. Soc. Ser. A 24 (1977), 129–138.CrossRefGoogle Scholar
8Greim, P.. Hilbert spaces have the Banach-Stone property for Bochner spaces. Bull. Austral. Math. Soc. 27 (1983), 121–128.CrossRefGoogle Scholar
9Lamperti, J.. On the isometries of certain function spaces. Pacific J. Math. 8 (1958), 459–466.CrossRefGoogle Scholar
10Jarosz, K.. Perturbations of Banach Algebras, Lecture Notes in Mathematics 1120 (Berlin: Springer, 1985).CrossRefGoogle Scholar
11Jarosz, K.. A generalization of the Banach–Stone theorem. Studia Math. 73 (1982), 33–39.CrossRefGoogle Scholar
12Sourour, A. R.. The isometries of L p(Ω, X). J. Fund. Anal. 30 (1978), 276–285.CrossRefGoogle Scholar