$\aleph_0$-categorical Banach spaces contain $\ell_p$ or $c_0$

TL;DR

This paper explores the model-theoretic properties of certain Banach spaces, showing that $eth_0$-categorical spaces nearly always embed classical sequence spaces like $c_0$ or $oldsymbol{ extit{ extltilde}}_oldsymbol{p}$, and analyzes the Tsirelson space's properties.

Contribution

It establishes model-theoretic facts about Tsirelson space and proves that $eth_0$-categorical Banach spaces embed classical sequence spaces almost isometrically.

Findings

Tsirelson space has the non independence property (NIP)

$eth_0$-categorical Banach spaces embed $c_0$ or $oldsymbol{ extit{ extltilde}}_oldsymbol{p}$ almost isometrically

The theory of Tsirelson space does not uniquely characterize it up to almost isometry

Abstract

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) $M_{\mathcal{T}}$ has the \emph{non independence property} (NIP); (2) every Banach space that is $\aleph_0$-categorical up to small perturbations embeds $c_0$ or $\ell_p$ ($1\leqslant p<\infty$) almost isometrically; consequently the (continuous) first-order theory of $M_{\mathcal{T}}$ does not characterize $M_{\mathcal{T}}$, up to almost isometric isomorphism.