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Lifting of the approximation property from Banach spaces to their dual spaces
Author:
Eve Oja
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3581-3587
MSC (2000):
Primary 46B20, 46B28, 47L05.
Posted:
June 22, 2007
MathSciNet review:
2336573
Full-text PDF Free Access
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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.
- 1.
Peter
G. Casazza, Approximation properties, Handbook of the geometry
of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001,
pp. 271–316. MR 1863695
(2003f:46012), http://dx.doi.org/10.1016/S1874-5849(01)80009-7
- 2.
W.
J. Davis, T.
Figiel, W.
B. Johnson, and A.
Pełczyński, Factoring weakly compact operators,
J. Functional Analysis 17 (1974), 311–327. MR 0355536
(50 #8010)
- 3.
J.
Diestel and J.
J. Uhl Jr., Vector measures, American Mathematical Society,
Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical
Surveys, No. 15. MR 0453964
(56 #12216)
- 4.
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 0341032
(49 #5782), http://dx.doi.org/10.1090/S0002-9939-1973-0341032-5
- 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
(89j:47026)
- 6.
Alexandre
Grothendieck, Produits tensoriels topologiques et espaces
nucléaires, Mem. Amer. Math. Soc. 1955 (1955),
no. 16, 140 (French). MR 0075539
(17,763c)
- 7.
William
B. Johnson, A complementary universal conjugate Banach space and
its relation to the approximation problem, Proceedings of the
International Symposium on Partial Differential Equations and the Geometry
of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 301–310
(1973). MR
0326356 (48 #4700)
- 8.
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
(2002j:46014)
- 9.
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 0280983
(43 #6702)
- 10.
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
(96m:46022), http://dx.doi.org/10.1515/crll.1995.461.137
- 11.
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
(2002b:46031), http://dx.doi.org/10.1007/BF02810673
- 12.
Åsvald
Lima and Eve
Oja, The weak metric approximation property, Math. Ann.
333 (2005), no. 3, 471–484. MR 2198796
(2006i:46025), http://dx.doi.org/10.1007/s00208-005-0656-0
- 13.
Joram
Lindenstrauss, Extension of compact operators, Mem. Amer.
Math. Soc. No. 48 (1964), 112. MR 0179580
(31 #3828)
- 14.
Joram
Lindenstrauss and Lior
Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin,
1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete,
Vol. 92. MR
0500056 (58 #17766)
- 15.
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
(2002b:47138), http://dx.doi.org/10.1006/jfan.2000.3674
- 16.
Eve
Oja, Geometry of Banach spaces having
shrinking approximations of the identity, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2801–2823. MR 1675226
(2000j:46034), http://dx.doi.org/10.1090/S0002-9947-00-02521-6
- 17.
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.
- 1.
- P.G. CASAZZA.
Approximation properties.
In: W.B. Johnson and J. Lindenstrauss (eds.) Handbook of the Geometry of Banach Spaces. Volume 1, Elsevier (2001) 271-316. MR 1863695 (2003f:46012)
- 2.
- W.J. DAVIS, T. FIGIEL, W.B. JOHNSON, AND A. PECZY´NSKI.
Factoring weakly compact operators.
J. Funct. Anal. 17 (1974) 311-327. MR 0355536 (50:8010)
- 3.
- J. DIESTEL AND J.J. UHL.
Vector Measures.
Mathematical Surveys 15. American Mathematical Society, Providence, Rhode Island, 1977. MR 0453964 (56:12216)
- 4.
- 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 0341032 (49:5782)
- 5.
- G. GODEFROY AND P.D. SAPHAR.
Duality in spaces of operators and smooth norms on Banach spaces.
Illinois J. Math. 32 (1988) 672-695. MR 955384 (89j:47026)
- 6.
- A. GROTHENDIECK.
Produits tensoriels topologiques et espaces nucléaires.
Mem. Amer. Math. Soc. 16 (1955). MR 0075539 (17:763c)
- 7.
- W.B. JOHNSON.
A complementary universal conjugate Banach space and its relation to the approximation problem.
Israel J. Math. 13 (1972) 301-310. MR 0326356 (48:4700)
- 8.
- W.B. JOHNSON AND T. OIKHBERG.
Separable lifting property and extensions of local reflexivity.
Illinois J. Math. 45 (2001) 123-137. MR 1849989 (2002j:46014)
- 9.
- 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 0280983 (43:6702)
- 10.
- N.J. KALTON AND D. WERNER.
Property (M), M-ideals, and almost isometric structure of Banach spaces.
J. Reine Angew. Math. 461 (1995) 137-178. MR 1324212 (96m:46022)
- 11.
- Å. LIMA, O. NYGAARD, AND E. OJA.
Isometric factorization of weakly compact operators and the approximation property.
Israel J. Math. 119 (2000) 325-348. MR 1802659 (2002b:46031)
- 12.
- Å. LIMA AND E. OJA.
The weak metric approximation property.
Math. Ann. 333 (2005) 471-484. MR 2198796 (2006i:46025)
- 13.
- J. LINDENSTRAUSS.
Extension of compact operators.
Mem. Amer. Math. Soc. 48 (1964). MR 0179580 (31:3828)
- 14.
- J. LINDENSTRAUSS AND L. TZAFRIRI.
Classical Banach Spaces I.
Springer, Berlin-Heidelberg-New York, 1977. MR 0500056 (58:17766)
- 15.
- T. OIKHBERG AND H.P. ROSENTHAL.
Extension properties for the space of compact operators.
J. Funct. Anal. 179 (2001) 251-308. MR 1809112 (2002b:47138)
- 16.
- E. OJA.
Geometry of Banach spaces having shrinking approximations of the identity.
Trans. Amer. Math. Soc. 352 (2000) 2801-2823. MR 1675226 (2000j:46034)
- 17.
- 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.
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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
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08996-4
PII:
S 0002-9939(07)08996-4
Keywords:
Approximation properties,
extendable local reflexivity,
projective tensor product of Banach spaces.
Received by editor(s):
August 10, 2006
Posted:
June 22, 2007
Additional Notes:
This research was partially supported by Estonian Science Foundation Grant 5704
Communicated by:
Jonathan M. Borwein
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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