# Complemented subspaces of $\ell_p(I)$ for uncountable $I$

I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one

Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p < \infty$, Atti Sem. Mat. Fis. Univ. Modena, 42, 399–402.

This is just an exercise in the reflexive range once you know the proof in the separable case. Suppose $X$ is a subspace of $\ell_p(I)$ with density character $\aleph$. Since each vector in $X$ has countable support, WLOG $I=\aleph$. If $X$ is complemented, you just need to check that $X$ has a complemented subspace isomorphic to $\ell_p(\aleph)$, for then you can apply the decomposition method. The separable case is known, so assume that $\aleph$ is uncountable. Take a maximal set of disjointly supported unit vectors in $X$. If the cardinality of the set is less than $\aleph$, then the union $A$ of their supports has cardinality less than $\aleph$. Let $P$ be the band projection onto the functions supported on $A$. Since $X$ has density character greater than $|A|$, this projection is not one to one (the adjoint cannot have dense range; this is where reflexivity helps).