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G. Emmanuele 4A=HI  JDA ?FAAJ=>EEJO B IF=?AI B *?DAH E JACH=>A BK?JEI E IF=?AI B LA?JH A=IKHAI

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G. Emmanuele

Remarks on the complementability of spaces of Bochner in- tegrable 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 c0 then L1(µ, X) is not complemented in cabv(µ, X) and conjectured that the same result is true ifX is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez- Piazza ([7]) disproved this conjecture, by showing that ifµis a finite measure and X is a Banach lattice not containing copies ofc0, thenL1(µ, X) is complemented incabv(µ, X). Here, we show that the complementability ofL1(µ, X) incabv(µ, X) together with that one ofXin the bidualX∗∗is equivalent to the complementability of L1(µ, X) in its bidual, so obtaining that for certain families of Banach spaces not containingc0 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).

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

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