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arXiv · 2210.09225

Free dual spaces and free Banach lattices

Enrique García-Sánchez·Pedro Tradacete
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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 $\ell_1$.

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Enrique García-Sánchez, Pedro Tradacete. 2022-10-17. Free dual spaces and free Banach lattices. https://doi.org/10.1016/j.jmaa.2023.127931

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