Trimming the Johnson bonsai

We show that if \(p>1\) $p>1$ every subspace of \(\ell _p(\Gamma )\) ${\ell }_p(\mathrm{\Gamma })$ is an \(\ell _p\) ${\ell }_p$ -sum of separable subspaces of \(\ell _p\) ${\ell }_p$ , and we provide examples of subspaces of \(\ell _p(\Gamma )\) ${\ell }_p(\mathrm{\Gamma })$ for \(0<p\le 1\) $0<p\le 1$ that are not even isomorphic to any \(\ell _p\) ${\ell }_p$ -sum of separable spaces, notably the kernel of any quotient map \(\ell _p(\Gamma )\rightarrow L_1(2^{\Gamma })\) ${\ell }_p(\mathrm{\Gamma })\to L_1(2^{\mathrm{\Gamma }})$ with \(\Gamma \) $\mathrm{\Gamma }$ 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 \(\aleph _1\) ${\aleph }_1$ with the SCP then the kernel of any quotient map \(\ell _p(\Gamma )\rightarrow X\) ${\ell }_p(\mathrm{\Gamma })\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP.