Trimming the Johnson bonsai

TL;DR

This paper investigates the structure of subspaces of l_p(\u03b3) for various p, showing they are sums of separable subspaces when p>1, and provides examples of more complex subspaces for 0<pnd discusses properties like SCP and SEP.

Contribution

It establishes the structure of subspaces of l_p(3) for p>1 and constructs examples for 0<pemonstrating more intricate subspace behavior, linking to properties like SCP and SEP.

Findings

Subspaces of l_p(3) are l_p-sums for p>1.

Existence of non-l_p-sum subspaces for 0<pemonstrated.

Kernel of quotient maps relate to properties like SCP and SEP.

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.