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PreprintCC BY 4.0arXiv

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 FBL(p)[X] and the Radon-Nikodym property of X. In addition, we give an example of X ⊂ Y such that FBL(p)[X] is not a regular sublattice of FBL(p)[Y].

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Cite

BibTeXRIS
BibTeX
@misc{ianina2026upper,
  title = {{Upper bound properties of free and related Banach lattices via operators}},
  author = {Anastasiia Ianina and Timur Oikhberg},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11928},
  eprint = {2609.11928},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11928},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Upper bound properties of free and related Banach lattices via operators
AU  - Anastasiia Ianina
AU  - Timur Oikhberg
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11928
UR  - https://arxiv.org/abs/2609.11928
AB  - 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 FBL^(p)[X] and the Radon-Nikodym property of X. In addition, we give an example of X ⊂ Y such that FBL^(p)[X] is not a regular sublattice of FBL^(p)[Y].
ER  -