Polish spaces of separable Banach lattices
Abstract
We compare two Polish presentations of separable real Banach lattices: closed sublattices of a universal lattice and closed ideals of FBL[ℓ1]. We construct translations preserving lattice isometry in both directions, continuous and injective from sublattices to ideals and Borel in the reverse direction. Natural generation and Fremlin tensor operations are Σ02-measurable with Gδ graphs. Isometric lattice isomorphism in either universal coding is Borel bireducible with a complete orbit equivalence relation of Polish-group actions. Uniform monotonicity and order uniform smoothness are Π03-complete in both codings, whereas strict monotonicity, order smoothness, and order continuity are coanalytic. Abstract Lp and AM-spaces form closed classes of sublattice codes. We also give explicit Borel bounds for atom counts, compact order intervals, and the lattice-isometry classes of Lp[0,1], ℓp, and c0.
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BibTeX
@misc{niwinski2026polish,
title = {{Polish spaces of separable Banach lattices}},
author = {Mariusz Niwiński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08064},
eprint = {2607.08064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08064},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polish spaces of separable Banach lattices
AU - Mariusz Niwiński
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2607.08064
UR - https://arxiv.org/abs/2607.08064
AB - We compare two Polish presentations of separable real Banach lattices: closed sublattices of a universal lattice and closed ideals of FBL[ℓ_1]. We construct translations preserving lattice isometry in both directions, continuous and injective from sublattices to ideals and Borel in the reverse direction. Natural generation and Fremlin tensor operations are Σ^0_2-measurable with G_δ graphs. Isometric lattice isomorphism in either universal coding is Borel bireducible with a complete orbit equivalence relation of Polish-group actions. Uniform monotonicity and order uniform smoothness are Π^0_3-complete in both codings, whereas strict monotonicity, order smoothness, and order continuity are coanalytic. Abstract L_p and AM-spaces form closed classes of sublattice codes. We also give explicit Borel bounds for atom counts, compact order intervals, and the lattice-isometry classes of L_p[0,1], ℓ_p, and c_0.
ER -