The approximation property does not imply the bounded approximation property

Figiel, T.; Johnson, W. B.

doi:10.1090/S0002-9939-1973-0341032-5

The approximation property does not imply the bounded approximation property

Authors: T. Figiel and W. B. Johnson
Journal: Proc. Amer. Math. Soc. 41 (1973), 197-200
MSC: Primary 46B05; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-1973-0341032-5
MathSciNet review: 0341032
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: There is a Banach space which has the approximation property but fails the bounded approximation property. The space can be chosen to have separable conjugate, hence there is a nonnuclear operator on the space which has nuclear adjoint. This latter result solves a problem of Grothendieck [2],

[1] Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 0402468, https://doi.org/10.1007/BF02392270
[2] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). MR 0075539
[3] 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, https://doi.org/10.1007/BF02762804
[4] 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, https://doi.org/10.1007/BF02771464
[5] Joram Lindenstrauss, On James’s paper “Separable conjugate spaces”, Israel J. Math. 9 (1971), 279–284. MR 0279567, https://doi.org/10.1007/BF02771677
[6] A. Pełczyński, Collected letters (unpublished).