The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of c 0

L. Drewnowski; G. Emmanuele

Studia Mathematica (1993)

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

Abstract

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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.

How to cite

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Drewnowski, 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

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  1. [1] J. Bourgain, Dunford-Pettis operators on L 1 and the Radon-Nikodym property, Israel J. Math. 37 (1980), 34-47. Zbl0457.46017
  2. [2] S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21-41. Zbl0175.14503
  3. [3] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer, New York 1984. 
  4. [4] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977. 
  5. [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. [6] L. Drewnowski, Un théorème sur les opérateurs de l ( Γ ) , C. R. Acad. Sci. Paris 281 (1976), 967-969. Zbl0323.46014
  7. [7] L. Drewnowski, Another note on copies of l and c 0 in ca(Σ, X), and the equality ca(Σ, X) = cca(Σ, X), preprint, 1990. 
  8. [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. [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. [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. [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. [12] N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278. Zbl0266.47038
  13. [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, New York 1977. Zbl0362.46013
  14. [14] J. Mendoza, Copies of l in L p ( μ ; X ) , Proc. Amer. Math. Soc. 109 (1990), 125-127. 
  15. [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

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

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