The universality of $\ell_1$ as a dual space

TL;DR

This paper demonstrates that any Banach space with a separable dual can be embedded into an $ ext{L}____$ space whose dual is isomorphic to $ ext{l}_1$, with additional properties related to reflexivity and incomparability.

Contribution

It establishes the universality of $ ext{l}_1$ as a dual space for Banach spaces with separable duals, including constructions preserving reflexivity and incomparability.

Findings

Any Banach space with a separable dual embeds into an $ ext{L}____$ space with dual $ ext{l}_1$.

Constructs $ ext{L}____$ spaces that are totally incomparable to given spaces.

Shows that for separable reflexive spaces, the constructed $ ext{L}____$ space can be made reflexive.

Abstract

Let $X$ be a Banach space with a separable dual. We prove that $X$ embeds isomorphically into a $\cL_\infty$ space $Z$ whose dual is isomorphic to $\ell_1$. If, moreover, $U$ is a space so that $U$ and $X$ are totally incomparable, then we construct such a $Z$, so that $Z$ and $U$ are totally incomparable. If $X$ is separable and reflexive, we show that $Z$ can be made to be somewhat reflexive.