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1993 | 104 | 2 | 111-123
Tytuł artykułu

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

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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.
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Bibliografia
  • [1] J. Bourgain, Dunford-Pettis operators on $L^1$ and the Radon-Nikodym property, Israel J. Math. 37 (1980), 34-47.
  • [2] S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21-41.
  • [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.
  • [6] L. Drewnowski, Un théorème sur les opérateurs de $l_∞(Γ)$, C. R. Acad. Sci. Paris 281 (1976), 967-969.
  • [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] L. Drewnowski and G. Emmanuele, On Banach spaces with the Gelfand-Phillips property. II, Rend. Circ. Mat. Palermo (2) 38 (1989), 377-391.
  • [9] G. Emmanuele, On complemented copies of $c_0$ in $L_X^p$, 1 ≤ p < ∞, Proc. Amer. Math. Soc. 104 (1988), 785-786.
  • [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.
  • [12] N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278.
  • [13] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, New York 1977.
  • [14] J. Mendoza, Copies of $l_∞$ in $L^p(μ; X)$, 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.
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bwmeta1.element.bwnjournal-article-smv104i2p111bwm
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