On embedding l 1 as a complemented subspace of Orlicz vector valued function spaces.

Abstract

A Banach space E is said to have property A [property B] if E contains a copy isomorphic with l 1 [a complemented subspace isomorphic with l 1 ]. G. Pisier proved that the Lebesgue-Bochner space L p (μ,E) , 1<p<∞ , has property A if E has the same property. The author of the paper under review extended Pisier's result to the case of Orlicz-Bochner function spaces L Φ (μ,E) , where Φ is a Young function [the author, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 1, 107–112;]. In the same paper the author further proved that if E is a Banach lattice, and Φ satisfies Δ 2 -condition, and if the measure μ is nonpurely atomic then L Φ (μ,E) has property B if and only if L Φ or E has property B. In the present paper the author continues to study the problem of characterizing spaces L Φ (μ,E) with property B, dropping the assumption that E is a Banach lattice. One such result asserts that if E has the weak (V ∗ ) property and Φ satisfies the Δ 2 -condition, then L Φ (μ,E) has property B if and only if L Φ or E has property B. The author states several results related to the open problem of characterizing completely the spaces L Φ (μ,E) with property B in terms of properties of E or L Φ .