Amalgamation and Ramsey properties of $L_p$ spaces

TL;DR

This paper investigates the symmetry and combinatorial properties of $L_p$ spaces, extending classical principles and introducing the concept of Fra"issé Banach spaces, with implications for isometry groups and Ramsey theory.

Contribution

It introduces the notion of Fra"issé Banach spaces, extends the equimeasurability principle, and links isometry group dynamics with Ramsey properties of finite-dimensional subspaces.

Findings

For $p eq 4,6,8,\

Established the Ramsey property for classes $\\{\\ell_p^n\\

Linked dynamics of isometry groups with Ramsey properties via a new correspondence.

Abstract

We study the dynamics of the group of isometries of $L_p$-spaces. In particular, we study the canonical actions of these groups on the space of $\delta$-isometric embeddings of finite dimensional subspaces of $L_p(0,1)$ into itself, and we show that for $p \neq 4,6,8,\ldots$ they are $\varepsilon$-transitive provided that $\delta$ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fra\"iss\'e Banach space which underlies these results and of which the known separable examples are the spaces $L_p(0,1)$, $p \neq 4,6,8,\ldots$ and the Gurarij space. We also give a proof of the Ramsey property of the classes $\{\ell_p^n\}_n$, $p\neq 2,\infty$, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matou\v{s}ek and R\"{o}dl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space $X$ with a Ramsey property of the collection of finite dimensional subspaces of $X$.