Abstract
We show that ifl p(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceL ϕ (which does not containc 0) then the Banach spaceX is isomorphic to a subspace ofL p. The same remains true forp = 2 whenL ϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.
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References
D. Aldous,Subspaces of L 1 via random measures, Trans. Am. Math. Soc.267 (1981), 445–463.
I. Aharoni, B. Maurey and B. S. Mityagin,Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces, Isr. J. Math.52 (1985), 251–265.
J. Bretagnolle and D. Dacunha-Castelle,Applications de l’étude de certaines formes linéaires aléatories au plongement d’espaces de Banach dans les espaces L p, Ann. Sci. Ec. Norm. Sup. 4° série,2 (1969), 437–480.
J. Bretagnolle, D. Dacunha-Castelle and J. J. Krivine,Lois stables et espaces L p, Ann. Inst. Henri Poincaré, Sect. B (1966), 231–259.
I. Berkes and H. P. Rosenthal,Almost exchangeable sequences of random variables, Z. Wahrscheinlichkeitstheor. Verw. Geb.70 (1985), 473–507.
E. Dubinski, A. Pelczynski and H. P. Rosenthal,On Banach spaces X for which ∏ 2(L ∞,X)=B(L ∞,X, Studia Math.44 (1972), 617–648.
S. Guerre,Sur les sous-espaces de L p, Séminaire d’Analyse Fonctionnelle, Université Paris 7, 1983–84.
D. J. H. Garling,Stable Banach spaces, random measures and Orlicz function spaces, inProbability Measures on Groups, Lecture Notes in Math. 928, Springer-Verlag, Berlin, 1982.
S. Guerre and M. Lévy,Espaces l p dans les sous-espaces de L1, Trans. Am. Math. Soc.279 (1983), 611–616.
R. G. Haydon, M. Lévy and Y. Raynaud,Randomly normed spaces, preprint.
W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri,Symmetric structures in Banach spaces, Am. Math. Soc. Memoirs19 (1979), 217.
N. Kalton,Banach spaces embedding into L 0, Isr. J. Math.52 (1985), 305–319.
J. L. Krivine,Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973–74, exposé 22, Ecole Polytechnique, Paris.
J. L. Krivine and B. Maurey,Espaces de Banach stables, Isr. J. Math.39 (1981), 273–295.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I: Sequence Spaces, Springer-Verlag, Berlin, 1977.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II: Function Spaces, Springer-Verlag, Berlin, 1979.
B. Maurey, Oral communication.
V. Milman and G. Schechtman,Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. No. 1200, Springer-Verlag, Berlin, 1986.
J. Musielak,Orlicz spaces and modular spaces, Lecture Notes in Math. No. 1034, Springer-Verlag, Berlin, 1983.
Y. Raynaud,Sur les sous-espaces de L p(Lq), Séminaire de Géométrie des Espaces de Banach, Universités Paris 6–7, 1984–85.
Y. Raynaud,Finie repésentabilité de l p(X) dans les espaces d’Orlicz, C.R. Acad. Sci. Paris 304, SérieI (1987), 331–334.
Y. Raynaud,Almost isometric methods in some isomorphic embedding problems, Proceedings of the 1987 Iowa Workshop on Banach Space Theory, Contemporary Math. 85, Am. Math. Soc., 1989.
W. Wnuk,Representations of Orlicz lattices, Diss. Math.235 (1984).
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Raynaud, Y. Finite representability ofl p(X) in Orlicz function spaces. Israel J. Math. 65, 197–213 (1989). https://doi.org/10.1007/BF02764860
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DOI: https://doi.org/10.1007/BF02764860