Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures. (English) Zbl 0855.46006

Emmanuele, G.

Commentat. Math. Univ. Carol. 37, No. 2, 217-228 (1996).

Summary: In the paper [Stud. Math. 104. No. 2, 111-123 (1993; Zbl 0811.46038)] 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 \(\text{cabv} (\mu, X)\) 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 [“Linear projections from a space of measures onto its Bochner integrable functions subspace”, preprint] 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 \(\text{cabv} (\mu, X)\). Here, we show that the complementability of \(L_1 (\mu, X)\) in \(\text{cabv} (\mu, X)\) together with that one of \(X\) in the bidual \(X^{**}\) 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, thanks to the existence of general results stating that a space in one of those families is complemented in the bidual.

MSC:

46B20	Geometry and structure of normed linear spaces
46E27	Spaces of measures
46E40	Spaces of vector- and operator-valued functions
46L10	General theory of von Neumann algebras
46B42	Banach lattices

Keywords:

spaces of vector measures and vector functions; preduals of \(W^*\)-algebras; quotient spaces; Radon-Nikodym property; Banach lattice; complementability; bidual

Citations:

Zbl 0811.46038