# Three-space problems for the approximation property

Journal of the European Mathematical Society (2009)

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

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topSzankowski, 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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