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arXiv · 2609.11928

Upper bound properties of free and related Banach lattices via operators

Anastasiia Ianina·Timur Oikhberg
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Abstract

It is known that the free $p$-convex Banach lattice on a Banach space $X$ can be represented as a space of functions on the unit ball of $X^*$. In this way, it gives rise to certain related (larger) lattices. To investigate such lattices, we introduce a new tool, related to operators into $X$. This tool is then used to (i) determine whether the lattices in question possess, or fail, properties involving upper bounds of upward directed sets -- namely, the Fatou property, and the related property of monotonic boundedness; (ii) investigate the regularity of embeddings between spaces in question. We find connections between the Fatou-like properties of ${\textrm{FBL}}^{(p)}[X]$ and the Radon-Nikodym property of $X$. In addition, we give an example of $X \subset Y$ such that ${\textrm{FBL}}^{(p)}[X]$ is not a regular sublattice of ${\textrm{FBL}}^{(p)}[Y]$.

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Anastasiia Ianina, Timur Oikhberg. 2026-09-10. Upper bound properties of free and related Banach lattices via operators. https://arxiv.org/abs/2609.11928

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