Free dual spaces and free Banach lattices

TL;DR

This paper explores the relationship between free Banach lattices and dual spaces, showing how free $p$-convex Banach lattices relate to their double duals and characterizing conditions for their duality properties.

Contribution

It establishes isometric lattice embeddings between free $p$-convex Banach lattices generated by a space and its double dual, and characterizes when the double dual is a free dual lattice.

Findings

$FBL^p[E^{**}]$ embeds into $FBL^p[E]^{**}$ isometrically.

$FBL^p[E]^{**}$ is the free dual $p$-convex lattice for $p>1$.

For $p=1$, this holds iff $E$ lacks complemented $oldsymbol{ ext{l}}_1$ copies.

Abstract

The relation between the free Banach lattice generated by a Banach space and free dual spaces is clarified. In particular, it is shown that for every Banach space $E$ the free $p$-convex Banach lattice generated by $E^{**}$, denoted $FBL^p[E^{**}]$, admits a canonical isometric lattice embedding into $FBL^p[E]^{**}$ and $FBL^p[E^{**}]$ is lattice finitely representable in $FBL^p[E]$. Moreover, we also show that for $p>1$, $FBL^p[E]^{**}$ can actually be considered as the free dual $p$-convex Banach lattice generated by $E$, whereas for $p=1$ this happens precisely when $E$ does not contain complemented copies of $\ell_1$.