Year: 2026
Type: preprint
Abstract: We compare two Polish presentations of separable real Banach lattices: closed sublattices of a universal lattice and closed ideals of $\operatorname{FBL}[\ell_1]$. 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 $\boldsymbolΣ^0_2$-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 $\boldsymbolΠ^0_3$-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 $L_p[0,1]$, $\ell_p$, and $c_0$.
Source: arXiv (Cornell University)
Author: Niwiński, Mariusz
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167926031