# Three-space problems for the approximation property

### A. Szankowski

The Hebrew University of Jerusalem, Israel

## 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 $2 < p\leq\infty$ and have AP for all $1 \leq p \leq 2$. For $1 < p \leq$ 2, $Y_p$ are isomorphic to $\ell_p$.