Polish spaces of separable Banach lattices

Abstract

We study the descriptive complexity of classes of separable Banach lattices. Building on the theory of coding spaces for separable Banach spaces, we introduce two Polish space encodings of separable Banach lattices: one via closed sublattices of the universal lattice $\mathcal{C}=C(Δ;L_1)$, and one via closed order ideals of the free Banach lattice $\operatorname{FBL}[\ell_1]$. We prove that, for every separable Banach lattice $E$, the spaces of closed sublattices and of closed order ideals of $E$ are Polish subspaces of the hyperspace of closed subsets of $E$. We also prove that the Fremlin projective tensor-product operation on ideal codes is $\boldsymbolΣ^0_2$-measurable and has a $G_δ$ graph.