Asymptotically Hilbertian Modular Banach Spaces: Examples of Uncountable Categoricity

TL;DR

This paper establishes a criterion for a class of modular Banach spaces to be uncountably categorical, providing new examples and insights into the model theory of Banach spaces.

Contribution

It introduces a criterion for elementary classes of modular Banach spaces to include all direct sums with Hilbert spaces, and provides new examples of uncountably categorical Banach spaces.

Findings

Criterion for elementary class characterization of modular Banach spaces

Construction of new uncountably categorical Banach space examples

Identification of Nakano direct sums satisfying the criterion

Abstract

We give a criterion ensuring that the elementary class of a modular Banach space E (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of E) consists of all direct sums E\oplus_m H, where H is an arbitrary Hilbert space and \oplus_m denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.