Lattice isomorphic Banach lattices of polynomials
Christopher Boyd, Vinícius Miranda
TL;DR
The paper extends Diaz–Dineen-type questions to the lattice setting for regular homogeneous vector-valued polynomials on Banach lattices, showing that if $E^*$ and $F^*$ are lattice isomorphic with at least one having order continuous norm, then for every $n$ and Banach lattice $G$, the spaces $\mathcal{P}^r(^n E; G^*)$ and $\mathcal{P}^r(^n F; G^*)$ are lattice isomorphic. It develops a transfer mechanism via Arens extensions and lattice-isomorphisms, enabling lattice isomorphisms to pass from $E^*$ and $F^*$ to the corresponding polynomial spaces, and extends the results to regular compact, regular weakly compact, orthogonally additive, and regular nuclear polynomials. In the nuclear setting, the authors introduce $\mathcal{P}_{\mathcal N}^r(^n E; F)$ and prove a lattice isomorphism under LAP assumptions for $E^*$, $F^*$ and $G$, using finite-dimensional approximation lemmas to realize the isomorphism. Overall, the work unifies and broadens Diaz–Dineen-type phenomena for Banach lattices, highlighting the central role of order-continuity of dual norms and lattice structure in transferring isomorphisms across polynomial spaces.
Abstract
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.