Lifting of the approximation property from Banach spaces to their dual spaces

Oja, Eve

doi:10.1090/S0002-9939-07-08996-4

Lifting of the approximation property from Banach spaces to their dual spaces
HTML articles powered by AMS MathViewer

by Eve Oja PDF

Proc. Amer. Math. Soc. 135 (2007), 3581-3587 Request permission

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.