Trimming the Johnson bonsai

Abstract

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

To appear in BJMA