In the paper [5] L. Drewnowski and the author proved that if $X$ is a Banach space containing a copy of $c_0$ then $L_1({\mu },X)$ is {\it not} complemented in $cabv({\mu },X)$ and conjectured that the same result is true if $X$ is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez-Piazza ([7]) disproved this conjecture, by showing that if $\mu$ is a finite measure and $X$ is a Banach lattice not containing copies of $c_0$, then $L_1({\mu },X)$ is complemented in $cabv({\mu },X)$. Here, we show that the complementability of $L_1({\mu },X)$ in $cabv({\mu },X)$ together with that one of $X$ in the bidual $X^{\ast\ast}$ is equivalent to the complementability of $L_1({\mu },X)$ in its bidual, so obtaining that for certain families of Banach spaces not containing $c_0$ complementability occurs (Section 2), thanks to the existence of general results stating that a space in one of those families is complemented in the bidual. We shall also prove that certain quotient spaces inherit that property (Section 3).
@article{118827,
author = {Giovanni Emmanuele},
title = {Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {37},
year = {1996},
pages = {217-228},
zbl = {0855.46006},
mrnumber = {1398997},
language = {en},
url = {http://dml.mathdoc.fr/item/118827}
}
Emmanuele, Giovanni. Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 217-228. http://gdmltest.u-ga.fr/item/118827/
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