New axiomatizable classes of Banach spaces via disjointness-preserving isometries

TL;DR

This paper introduces new axiomatizable classes of Banach spaces derived from Banach lattices via disjointness-preserving isometries, expanding the understanding of their structural properties and providing examples like Musielak-Orlicz and Nakano spaces.

Contribution

It establishes conditions under which classes of Banach lattices lead to axiomatizable classes of Banach spaces, including new examples such as Musielak-Orlicz and Nakano spaces.

Findings

Axiomatizability preserved under disjointness-preserving isometries

Extension of axiomatizable classes to Banach spaces without lattice structure

Identification of new classes like Musielak-Orlicz and Nakano spaces

Abstract

Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class C^B of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its "script class" SC, so does C^B with SC^B. This allows us to give new examples of axiomatizable classes of Banach spaces, namely certain Musielak-Orlicz spaces, Nakano spaces, and mixed norm spaces.