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Copies of l∞ in Lp(μ;X).


Mendoza Casas, José (1990) Copies of l∞ in Lp(μ;X). Proceedings of the American Mathematical Society, 109 (1). pp. 125-127. ISSN 0002-9939

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Let (Ω,Σ,μ) be any measure space, X a Banach space and for 1≤p<+∞ let Lp(μ,X) be the Banach space of all X-valued Bochner "pth power integrable'' functions on Ω, with the usual "Lp''-norm. A natural question is: do properties enjoyed by Lp(μ,X) "descend'' to X? In the present paper it is proved that Lp(μ,X) contains l∞ isomorphically (if and) only if X does.
In a sense, the author's result completes earlier ones [e.g., S. Kwapien, Studia Math. 52 (1974), 187–188; G. Pisier, C. R. Acad. Sci. Paris Sér. A 286 (1978), no. 17, 747–749; ; L. Drewnowski, "Copies of l∞ in the operator spaces Kω∗(X∗,Y)'', to appear].
The proof of the theorem is achieved by applying three earlier results; one is from the paper of Drewnowski [op. cit.], and the other two from a paper by H. P. Rosenthal [Studia Math. 37 (1970), 13–36].
Another recent paper by Drewnowski ["When does ca(Σ,X) contain a copy of l∞ or c0?'', Proc. Amer. Math. Soc., to appear] is also relevant to the present paper.

Item Type:Article
Uncontrolled Keywords:Bochner-integrable functions
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16888

J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys No. 15, Amer. Math. Soc., Providence, RI, 1977.

L. Drewnowski, Copies of l∞ in the operator space Kw∗(X∗,Y), (to appear).

N. Dunford and J. T. Schwartz, Linear operators, vol. I. New York, Interscience, 1955.

N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278.

S. Kwapien, Sur les espaces de Banach contenant c0, Studia Math. 52 (1974), 187-188.

S. Lang, Analysis II, Addison-Wesley, Reading, MA, 1969.

G. Pisier, Une propriété de stabilité de la classe des espaces ne contenant pas l1, C. R. Acad. Sci. Paris Sér. A 286 (1978), 747-749.

H. P. Rosenthal, On relatively disjoint families of measures with some applications to Banach space theory, Studia Math. 37 (1970), 13-16

Deposited On:26 Oct 2012 08:28
Last Modified:13 Nov 2013 16:40

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