G. Emmanuele
Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures

Comment.Math.Univ.Carolinae 37,2 (1996) 217-228.

Abstract: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 {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. \par We shall also prove that certain quotient spaces inherit that property (Section 3).

Keywords: spaces of vector measures and vector functions, complementability, Banach lattices, preduals of W-algebras, quotient spaces
AMS Subject Classification: 46B20, 46E27, 46E40, 46B30, 46L99

PDF