# Continuous projections on ℓ1\ell_1 with norm >1>1

I was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $S=\mathbb{N}$.

I’ve found one quite general condition for the closed linear subspace to be image of a continuous projection. Such a subspace must be the closure of the linear span of so-called relatively disjoint vectors. This subspaces gives us explicit examples of projections with norm greater than 1. For details see the paper of H. P. Rosenthal On relatively disjoint families of measures, with some applications to Banach space theory.

As a special case we can get subspaces that are the closure of the linear span of disjointly supported vectors. These subspaces give us examples of norm one projections. Moreover, only such subspaces are give rise to norm one projections. For details see the survey by Beata Randrianantoanina Norm one projections in Banach spaces.

Thus, for projections of norm 1 we have a complete description. For the rest quite a big source of examples. In the first mentioned paper the author states that he doesn’t know any other examples of continuous projections on $\ell_1(S)$ that are not generated by some relatively disjoint family of vectors.

So, could someone give me a reference where I can read about other examples of projections on $\ell_1(S)$, or may be their complete characterization?

Also I will be grateful if you give me some explicit examples of discontinuous projections on $\ell_1(S)$.

The same question on mathoverflow.net.

Without loss of generality, regard $\ell^1(S)$ for $S=\{0,1\}$.

Let $e_0=(R,0)$ and $e_1=(R,r)$ for $R$ and $r$ both nonzero.

Then $\|e_0\|_1=R$ while $\|e_1-e_0\|_1=r$.

From this we obtain a projection $P:\ell^1(S)\to\ell^1(S)$ given by

whose norm is bounded from below by $R/r$.

In particular, for $R>r$ we obtain $\|P\|>1$.