Lifting of the approximation property from Banach spaces to their dual spaces
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Abstract:
Inspired by the principle of local reflexivity, due to Lindenstrauss and Rosenthal, a new geometric property of Banach spaces, the extendable local reflexivity, was recently introduced by Rosenthal. Johnson and Oikhberg proved that the extendable local reflexivity permits lifting the bounded approximation property from Banach spaces to their dual spaces. It is not known whether the extendable local reflexivity permits lifting the approximation property. We prove that it does whenever the space is complemented in its bidual. This involves the concept of the weak bounded approximation property, introduced by Lima and Oja.References
- Peter G. Casazza, Approximation properties, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 271–316. MR 1863695, DOI 10.1016/S1874-5849(01)80009-7
- W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 0355536, DOI 10.1016/0022-1236(74)90044-5
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
- T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197–200. MR 341032, DOI 10.1090/S0002-9939-1973-0341032-5
- Gilles Godefroy and Pierre David Saphar, Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32 (1988), no. 4, 672–695. MR 955384
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 (1972), 301–310 (1973). MR 326356, DOI 10.1007/BF02762804
- William B. Johnson and Timur Oikhberg, Separable lifting property and extensions of local reflexivity, Illinois J. Math. 45 (2001), no. 1, 123–137. MR 1849989
- W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
- Nigel J. Kalton and Dirk Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178. MR 1324212, DOI 10.1515/crll.1995.461.137
- Åsvald Lima, Olav Nygaard, and Eve Oja, Isometric factorization of weakly compact operators and the approximation property, Israel J. Math. 119 (2000), 325–348. MR 1802659, DOI 10.1007/BF02810673
- Åsvald Lima and Eve Oja, The weak metric approximation property, Math. Ann. 333 (2005), no. 3, 471–484. MR 2198796, DOI 10.1007/s00208-005-0656-0
- Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Timur Oikhberg and Haskell P. Rosenthal, Extension properties for the space of compact operators, J. Funct. Anal. 179 (2001), no. 2, 251–308. MR 1809112, DOI 10.1006/jfan.2000.3674
- Eve Oja, Geometry of Banach spaces having shrinking approximations of the identity, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2801–2823. MR 1675226, DOI 10.1090/S0002-9947-00-02521-6
- E. Oja. The impact of the Radon-Nikodým property on the weak bounded approximation property. Rev. R. Acad. Cien. Serie A. Mat. 100 (2006) 325–331.
Additional Information
- Eve Oja
- Affiliation: Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia
- Email: eve.oja@ut.ee
- Received by editor(s): August 10, 2006
- Published electronically: June 22, 2007
- Additional Notes: This research was partially supported by Estonian Science Foundation Grant 5704
- Communicated by: Jonathan M. Borwein
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 3581-3587
- MSC (2000): Primary 46B20, 46B28, 47L05
- DOI: https://doi.org/10.1090/S0002-9939-07-08996-4
- MathSciNet review: 2336573