Spaces of bochner integrable functions and spaces of representable operators as U-ideals

IN this paper we study the geometry of the space of Bochner integrable functions as a subspace of the space of vector valued countably additive measures of finite variation and that of the space of representable operators as a subspace of the space of bounded linear operators. Let £ be a Banach space and (fi, si, fi) a finite measure space; let cabv(fi, E) denote the space of countably additive £-valued measures of finite variation that are absolutely continuous with respect to fi. Drewnowski and Emmanuele [6] have proved recently that if £ has a copy of c0 then L ^, E), the space of Bochner integrable functions, is not complemented in cabv(fi, £). However, if one is interested in weaker geometric conditions like being locally 1-complemented (recall from [12], that a closed subspace Y of a Banach space X is said to be locally 1-complemented if Y is the kernel of a norm one projection in X*\ such a subspace was called an "ideal" in [9]) then L(ji, E) is always locally 1-complemented in cabv(/i, £). Similarly 9?(L'(/A), £) the space of representable operators from L{fi) to £ is a locally 1-complemented subspace of if(L(/u.)£), the space of bounded linear operators (see [14]). Here again, if E has a copy of c0 then 9L{L}(JJL), E) is not complemented in 2(V(jt), E) (see [8]). These results suggest that better and more reasonable geometric properties to study in this context are the notions of "L-ideal and approximation property (UKAP) to study similar questions in the context of the space of compact operators,