Persistence of Banach lattices under nonlinear order isomorphisms

TL;DR

This paper explores when nonlinear order isomorphisms between Banach lattices imply the preservation of their linear lattice structures, showing that certain isomorphisms ensure linear isomorphism.

Contribution

It demonstrates that order isomorphisms between specific Banach lattices and classical spaces imply linear lattice isomorphisms, extending understanding of their structural persistence.

Findings

Order isomorphism to C(K) implies linear isomorphism to C(K)

Order isomorphism to c_0 implies linear isomorphism to c_0

Containment of a closed sublattice order isomorphic to c_0 ensures linear structure preservation

Abstract

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection between them that preserves order. We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to c_0, and for Banach lattices that contain a closed sublattice order isomorphic to c_0.