# The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of ${c}_{0}$

Studia Mathematica (1993)

- Volume: 104, Issue: 2, page 111-123
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topDrewnowski, L., and Emmanuele, G.. "The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of $c_{0}$." Studia Mathematica 104.2 (1993): 111-123. <http://eudml.org/doc/215963>.

@article{Drewnowski1993,

abstract = {Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of $c_0$. Then the Bochner space $L^1(m;X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.},

author = {Drewnowski, L., Emmanuele, G.},

journal = {Studia Mathematica},

keywords = {Banach space; isomorphic copy of $c_0$; spaces of vector measures; Bochner integrable functions; Radon-Nikodym property; uncomplemented subspace; Bochner space; Banach space of all -continuous vector measures; bounded variation; relatively compact range},

language = {eng},

number = {2},

pages = {111-123},

title = {The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of $c_\{0\}$},

url = {http://eudml.org/doc/215963},

volume = {104},

year = {1993},

}

TY - JOUR

AU - Drewnowski, L.

AU - Emmanuele, G.

TI - The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of $c_{0}$

JO - Studia Mathematica

PY - 1993

VL - 104

IS - 2

SP - 111

EP - 123

AB - Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of $c_0$. Then the Bochner space $L^1(m;X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.

LA - eng

KW - Banach space; isomorphic copy of $c_0$; spaces of vector measures; Bochner integrable functions; Radon-Nikodym property; uncomplemented subspace; Bochner space; Banach space of all -continuous vector measures; bounded variation; relatively compact range

UR - http://eudml.org/doc/215963

ER -

## References

top- [1] J. Bourgain, Dunford-Pettis operators on ${L}^{1}$ and the Radon-Nikodym property, Israel J. Math. 37 (1980), 34-47. Zbl0457.46017
- [2] S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21-41. Zbl0175.14503
- [3] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer, New York 1984.
- [4] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
- [5] P. Domański and L. Drewnowski, Uncomplementability of the spaces of norm continuous functions in some spaces of "weakly" continuous functions, Studia Math. 97 (1991), 245-251. Zbl0736.46026
- [6] L. Drewnowski, Un théorème sur les opérateurs de ${l}_{\infty}\left(\Gamma \right)$, C. R. Acad. Sci. Paris 281 (1976), 967-969. Zbl0323.46014
- [7] L. Drewnowski, Another note on copies of ${l}_{\infty}$ and ${c}_{0}$ in ca(Σ, X), and the equality ca(Σ, X) = cca(Σ, X), preprint, 1990.
- [8] L. Drewnowski and G. Emmanuele, On Banach spaces with the Gelfand-Phillips property. II, Rend. Circ. Mat. Palermo (2) 38 (1989), 377-391. Zbl0689.46004
- [9] G. Emmanuele, On complemented copies of ${c}_{0}$ in ${L}_{X}^{p}$, 1 ≤ p < ∞, Proc. Amer. Math. Soc. 104 (1988), 785-786. Zbl0692.46016
- [10] G. Emmanuele, About the position of ${K}_{w}*(E*,F)$ inside ${L}_{w}*(E*,F)$, Atti Sem. Mat. Fis. Univ. Modena, to appear.
- [11] M. Feder, On the non-existence of a projection onto the space of compact operators, Canad. Math. Bull. 25 (1982), 78-81. Zbl0432.46008
- [12] N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278. Zbl0266.47038
- [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, New York 1977. Zbl0362.46013
- [14] J. Mendoza, Copies of ${l}_{\infty}$ in ${L}^{p}(\mu ;X)$, Proc. Amer. Math. Soc. 109 (1990), 125-127.
- [15] H. P. Rosenthal, On relatively disjoint families of measures, with some application to Banach space theory, Studia Math. 37 (1970), 13-36. Zbl0227.46027

## Citations in EuDML Documents

top- Lech Drewnowski, Nonseparability of the quotient space cabv(∑,m;X)/L¹(m;X) for Banach spaces X without the Radon-Nikodym property
- María J. Rivera, On vector valued measure spaces of bounded $\Phi $-variation containing copies of ${\ell}_{\infty}$
- Giovanni Emmanuele, Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures
- Giovanni Emmanuele, On the position of the space of representable operators in the space of linear operators${}^{1}$
- J. Bonet, Paweł Domański, M. Lindström, Cotype and complemented copies of ${c}_{0}$ in spaces of operators

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.