Abstract
We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation property and is 1-complemented in its bidual, then it has the weak metric approximation property. We also study the lifting of the weak metric approximation property from Banach spaces to their dual spaces. This enables us, in particular, to show that the subspace of c0, constructed by Johnson and Schechtman, does not have the weak metric approximation property.
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References
Casazza, P.G.: Approximation properties. In: W.B. Johnson and J. Lindenstrauss (eds.) Handbook of the Geometry of Banach Spaces. Volume 1, Elsevier, pp. 271–316 (2001)
Davis, W.J., Figiel, T., Johnson, W.B., Pelczyński, A.: Factoring weakly compact operators. J. Funct. Anal. 17, 311–327 (1974)
Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies 176 (1993)
Diestel, J., Uhl, J.J.Jr.:Vector Measures. Mathematical Surveys 15, Amer. Math. Soc., Providence, Rhode Island, 1977
Feder, M., Saphar, P.D.: Spaces of compact operators and their dual spaces. Israel J. Math. 21, 38–49 (1975)
Godefroy, G.: The Banach space c0. Extracta Math. 16, 1–25 (2001)
Godefroy, G., Saphar, P.D.: Duality in spaces of operators and smooth norms on Banach spaces. Illinois J. Math. 32, 672–695 (1988)
Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955)
Harmand, P., Werner, D., Werner, W.: M-ideals in Banach Spaces and Banach Algebras. Lecture Notes in Math. 1547, Springer, Berlin Heidelberg, 1993
Johnson, W.B.: A complementary universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13, 301–310 (1972)
Johnson, W.B., Oikhberg, T.: Separable lifting property and extensions of local reflexivity. Illinois J. Math. 45, 123–137 (2001)
Lima, V., Lima, Å.: Ideals of operators and the metric approximation property. J. Funct. Anal. 210, 148–170 (2004)
Lima, Å., Nygaard, O., Oja, E.: Isometric factorization of weakly compact operators and the approximation property. Israel J. Math. 119, 325–348 (2000)
Lindenstrauss, J., Tzafriri, L. Classical Banach Spaces I. Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer, Berlin Heidelberg, 1977
Oja, E.: Géométrie des espaces de Banach ayant des approximations de l'identité contractantes. C. R. Acad. Sci. Paris, Sér. I, 328, 1167–1170 (1999)
Oja, E.: Geometry of Banach spaces having shrinking approximations of the identity. Trans. Amer. Math. Soc. 352, 2801–2823 (2000)
Oja, E., Pelander, A.: The approximation property in terms of the approximability of weak*-weak continuous operators. J. Math. Anal. Appl. 286, 713–723 (2003)
Phelps, R.: Convex Functions, Monotone Operators and Differentiability. Second edition. Lecture Notes in Math. 1364, Springer, Berlin Heidelberg, 1993
Reinov, O.I.: Un contre-exemple à une conjecture de A. Grothendieck. C. R. Acad. Sc. Paris, Sér. I, 296, 597–599 (1983)
Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics, Springer, Berlin Heidelberg, 2002
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The research of the second-named author was partially supported by Estonian Science Foundation Grant 5704 and the Norwegian Academy of Science and Letters.