Free dual spaces and free Banach lattices
Enrique Garc\'ia-S\'anchez, Pedro Tradacete

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 the free -convex Banach lattice generated by , denoted , admits a canonical isometric lattice embedding into and is lattice finitely representable in . Moreover, we also show that for , can actually be considered as the free dual -convex Banach lattice generated by , whereas for this happens precisely when does not contain complemented copies of .
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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Topology and Set Theory
