Hereditary approximation property

W. B. Johnson; A. Szankowski

doi:http://doi.org/10.4007/annals.2012.176.3.10

Abstract

If $X$ is a Banach space such that the isomorphism constant to $\ell_2^n$ from $n$-dimensional subspaces grows sufficiently slowly as $n\to \infty$, then $X$ has the approximation property. A consequence of this is that there is a Banach space $X$ with a symmetric basis but not isomorphic to $\ell_2$ so that all subspaces of $X$ have the approximation property. This answers a problem raised in 1980. An application of the main result is that there is a separable Banach space $X$ that is not isomorphic to a Hilbert space, yet every subspace of $X$ is isomorphic to a complemented subspace of $X$. This contrasts with the classical result of Lindenstrauss and Tzafriri that a Banach space in which every closed subspace is complemented must be isomorphic to a Hilbert space.

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@article {cgj, MRKEY = {1840107}, AUTHOR = {Casazza, P. G. and Garc{\'ı}a, C. L. and Johnson, W. B.}, TITLE = {An example of an asymptotically {H}ilbertian space which fails the approximation property}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {129}, YEAR = {2001}, NUMBER = {10}, PAGES = {3017--3023}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {46B20 (46B07)}, MRNUMBER = {1840107}, MRREVIEWER = {Deba Prasad Sinha}, DOI = {10.1090/S0002-9939-01-06142-1}, ZBLNUMBER = {0983.46022}, }
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Hereditary approximation property

Abstract

Authors