Isometric rigidity and Fra\"iss\'e properties of Orlicz sequence spaces

TL;DR

This paper investigates the approximate rigidity and Fra"iss"e properties of Orlicz sequence spaces, showing that almost isometric embeddings nearly preserve disjointness and basic vectors, with implications for their structural classification.

Contribution

It introduces an approximate rigidity result for Orlicz sequence spaces and characterizes their Fra"iss"e properties, answering a specific open question.

Findings

Almost isometric embeddings nearly preserve disjointness in large classes of Orlicz spaces.

In certain cases, embeddings nearly preserve basic vectors.

Some Orlicz spaces are guarded Fra"iss"e but not $\,\omega$-categorical, and do not contain $\,\ell_2$.

Abstract

We provide an approximate version of a rigidity result by Randrianantoanina: for a large class of Orlicz sequence spaces, almost isometric embeddings almost preserve disjointness. In specific cases, we can even prove that such embeddings almost preserve basic vectors. As a consequence, we prove that some Orlicz sequences spaces are guarded Fra\"iss\'e but not $\omega$-categorical; moreover, they do not contain copies of $\ell_2$ and their age is not closed. This answers a question of C\'uth-de Rancourt-Doucha.