Trimming the Johnson bonsai

Abstract

We show that if $p>1$ every subspace of $\ell_p(\Gamma)$ is an $\ell_p$-sum of separable subspaces of $\ell_p$, and we provide examples of subspaces of $\ell_p(\Gamma)$ for $0<p\leq 1$ that are not even isomorphic to any $\ell_p$-sum of separable spaces, notably the kernel of any quotient map $\ell_p(\Gamma)\to L_1(2^{\Gamma})$ with $\Gamma$ uncountable. We involve the separable complementation property (SCP) and the separable extension property (SEP), showing that if $X$ is a Banach space of density character $\aleph_1$ with the SCP then the kernel of any quotient map $\ell_p(\Gamma)\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP.