Trimming the Johnson bonsai

Abstract

We show that if $p>1$ every subspace of $\ell_p(Γ)$ is an $\ell_p$-sum of separable subspaces of $\ell_p$, and we provide examples of subspaces of $\ell_p(Γ)$ 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 $\ell_p(Γ)\to L_1(2^Γ)$ with $Γ$ 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 $\ell_p(Γ)\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP.

Theorems & Definitions (14)

• Definition 1.1
• Definition 1.2
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop']}
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Proposition 3.1
• Proposition 3.2