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

Giovanni Emmanuele

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 2, page 217-228
  • ISSN: 0010-2628

Abstract

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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 ( μ , X ) is not complemented in c a b v ( μ , 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 μ is a finite measure and X is a Banach lattice not containing copies of c 0 , then L 1 ( μ , X ) is complemented in c a b v ( μ , X ) . Here, we show that the complementability of L 1 ( μ , X ) in c a b v ( μ , X ) together with that one of X in the bidual X * * is equivalent to the complementability of L 1 ( μ , 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).

How to cite

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Emmanuele, Giovanni. "Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures." Commentationes Mathematicae Universitatis Carolinae 37.2 (1996): 217-228. <http://eudml.org/doc/247943>.

@article{Emmanuele1996,
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. We shall also prove that certain quotient spaces inherit that property (Section 3).},
author = {Emmanuele, Giovanni},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {spaces of vector measures and vector functions; complementability; Banach lattices; preduals of W$^\ast $-algebras; quotient spaces; spaces of vector measures and vector functions; preduals of -algebras; quotient spaces; Radon-Nikodym property; Banach lattice; complementability; bidual},
language = {eng},
number = {2},
pages = {217-228},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures},
url = {http://eudml.org/doc/247943},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Emmanuele, Giovanni
TI - Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 2
SP - 217
EP - 228
AB - 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. We shall also prove that certain quotient spaces inherit that property (Section 3).
LA - eng
KW - spaces of vector measures and vector functions; complementability; Banach lattices; preduals of W$^\ast $-algebras; quotient spaces; spaces of vector measures and vector functions; preduals of -algebras; quotient spaces; Radon-Nikodym property; Banach lattice; complementability; bidual
UR - http://eudml.org/doc/247943
ER -

References

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