Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}\_p$ which is complementably $\ell\_p$-saturated but admits no unconditional basis. I was wondering if such an example has been proved for the case $p=1$.


I have managed to construct a space which is complementably $\ell\_1$-saturated but admits no symmetric basis. However, it does admit a subsymmetric one---in particular, an unconditional one. I was going to adapt my technique to see if I could also kill off unconditionality. But, it would be more motivating to me to know whether this has been done before ; )


Just to be clear, when I ask for a space $X$ to be "complementably $\ell\_p$-saturated" I mean that for every infinite-dimensional closed subspace $Y$ of $X$ we can find a continuous linear projection $P:X\to X$ such that $PX\subseteq Y$ and $PX\cong\ell\_p$.


Thank you!



example of an $\ell_1$-saturated Banach space without an unconditional basis
Szankowski (Subspaces without the approximation property. Israel J. Math. 30 (1978), no. 1-2, 123–129) for each $1\le p<2$ constructed a subspace $X$ of $\ell\_p$ without the approximation property and thus without an unconditional basis. On the other hand, as a subspace of $\ell\_p$, $1\le p<\infty$, the space $X$ is complementably $\ell\_p$-saturated.


