Three-space problems for the approximation property

A. Szankowski

Journal of the European Mathematical Society (2009)

  • Volume: 011, Issue: 2, page 273-282
  • ISSN: 1435-9855

Abstract

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It is shown that there is a subspace Z q of ℓ q for 1 < q < 2 which is isomorphic to ℓ q such that ℓ q / Z q does not have the approximation property. On the other hand, for 2 < p < ∞ there is a subspace Y p of ℓ p such that Y p does not have the approximation property (AP) but the quotient space ℓ p / Y p is isomorphic to ℓ p . The result is obtained by defining random “Enflo-Davie spaces” Y p which with full probability fail AP for all 2 < p ≤ ∞ and have AP for all 1 ≤ p ≤ 2 . For 1 < p ≤ 2 , Y p are isomorphic to ℓ p .

How to cite

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Szankowski, A.. "Three-space problems for the approximation property." Journal of the European Mathematical Society 011.2 (2009): 273-282. <http://eudml.org/doc/277626>.

@article{Szankowski2009,
abstract = {It is shown that there is a subspace $Z_q$ of $\ell _q$ for $1<q<2$ which is isomorphic to $\ell _q$ such that $\ell _q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty $ there is a subspace $Y_p$ of $\ell _p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space $\ell _p/Y_p$ is isomorphic to $\ell _p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty $ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to $\ell _p$.},
author = {Szankowski, A.},
journal = {Journal of the European Mathematical Society},
keywords = {quotients of Banach spaces; approximation property; approximation property; reflexive spaces; -spaces; Kashin decomposition; Enflo-Davie construction},
language = {eng},
number = {2},
pages = {273-282},
publisher = {European Mathematical Society Publishing House},
title = {Three-space problems for the approximation property},
url = {http://eudml.org/doc/277626},
volume = {011},
year = {2009},
}

TY - JOUR
AU - Szankowski, A.
TI - Three-space problems for the approximation property
JO - Journal of the European Mathematical Society
PY - 2009
PB - European Mathematical Society Publishing House
VL - 011
IS - 2
SP - 273
EP - 282
AB - It is shown that there is a subspace $Z_q$ of $\ell _q$ for $1<q<2$ which is isomorphic to $\ell _q$ such that $\ell _q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty $ there is a subspace $Y_p$ of $\ell _p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space $\ell _p/Y_p$ is isomorphic to $\ell _p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty $ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to $\ell _p$.
LA - eng
KW - quotients of Banach spaces; approximation property; approximation property; reflexive spaces; -spaces; Kashin decomposition; Enflo-Davie construction
UR - http://eudml.org/doc/277626
ER -

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