Copies of l∞ in Lp(μ;X).

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.