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spaces of vector measures and vector functions; complementability; Banach lattices; preduals of W$^\ast$-algebras; quotient spaces
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).
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