Trimming the Johnson bonsai

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