Lattice isomorphic Banach lattices of polynomials

We study Díaz-Dineen’s problem for regular homogeneous vector-valued polynomials. In particular, we prove that if $E^$ and $F^$ are lattice isomorphic with at least one having order continuous norm, then $\mathcal{P}^r(^n E; G^)$ and $\mathcal{P}^r(^n F; G^)$ are lattice isomorphic for every $n\in \N$ and every Banach lattice $G$. We also study the analogous problem for the classes of regular compact, regular weakly compact, orthogonally additive and regular nuclear polynomials.

💡 Research Summary

The paper addresses a lattice‑theoretic version of the Díaz‑Dineen problem, which asks whether dual isomorphism of Banach spaces implies isomorphism of spaces of homogeneous polynomials. The authors focus on Banach lattices and regular (i.e., difference of positive) homogeneous vector‑valued polynomials. Their main theorem states that if the duals (E^{}) and (F^{}) are lattice‑isomorphic and at least one of them has an order‑continuous norm, then for every degree (n\in\mathbb N) and every Banach lattice (G) the spaces of regular (n)-homogeneous polynomials (\mathcal P^{r}({}^nE;G^{})) and (\mathcal P^{r}({}^nF;G^{})) are lattice‑isomorphic.

A crucial preliminary result (Proposition 2.1) shows that, for a Banach lattice (E), the following are equivalent: (i) (E^{}) has order‑continuous norm; (ii) (E^{}) contains no sublattice isomorphic to (\ell_{1}); (iii) every positive bilinear form on (E\times E) is Arens‑regular; (iv) every positive operator (E\to E^{*}) is weakly compact; (v) every positive symmetric bilinear form is Arens‑regular; (vi) (E) itself is Arens‑regular. Thus, in the lattice setting, positive Arens‑regularity coincides with the classical notion, a fact that distinguishes Banach lattices from general Banach spaces.

The authors exploit the well‑known identification between regular multilinear operators and linear operators on the positive projective tensor product (E_{1},\widehat\otimes_{|\pi|}\cdots\widehat\otimes_{|\pi|}E_{n}). For a regular (n)-homogeneous polynomial (P) the associated symmetric multilinear map (T_{P}) linearizes to an operator (P^{\otimes}) on the symmetric positive projective tensor power (\widehat\otimes^{n,s}_{|\pi|}E). This linearization is an isometric lattice isomorphism (see