### Impacto

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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Official URL: http://www.jstor.org/stable/2048371?seq=3

## Abstract

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 |
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Uncontrolled Keywords: | Bochner-integrable functions |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 16888 |

References: | 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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