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Isometric rigidity and Fraïssé properties of Orlicz sequence spaces

Noé de Rancourt, Micheline Fakhoury

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 Fraïssé but not $ω$-categorical; moreover, they do not contain copies of $\ell_2$ and their age is not closed. This answers a question of Cúth-de Rancourt-Doucha.

Isometric rigidity and Fraïssé properties of Orlicz sequence spaces

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 Fraïssé but not -categorical; moreover, they do not contain copies of and their age is not closed. This answers a question of Cúth-de Rancourt-Doucha.

Paper Structure

This paper contains 8 sections, 10 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Fraïssé theory for Banach spaces
  3. Oligomorphy and related notions
  4. Orlicz spaces
  5. Results
  6. Disjointness preservation results
  7. Basis preservation results
  8. Cofinally Fraïssé Orlicz sequence spaces

Key Result

Theorem 1.5

Assume that $M$ is $\mathcal{C}^2$ on $(0, +\infty)$, satisfies the $\Delta_{2+}$-condition at infinity, that $M'(0) = 0$, $M"(t) > 0$ whenever $t > 0$, and $M"(t)$ tends to either $0$ or $\infty$ when $t \to 0$. Then all isometric embeddings $L_M([0, 1]) \to L_M([0, 1])$ preserve disjointness. $\bl

Theorems & Definitions (28)

  • Definition 1.1: Ferenczi--López-Abad--Mbombo--Todorcevic 2020
  • Definition 1.2: Cúth--de Rancourt--Doucha 2024+
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Randrianantoanina 1998
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 18 more