Amalgamation and Ramsey properties of $L_p$ spaces

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 $p\ne 4,6,8,\dots$ they are $\epsilon$-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 Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces $L_p(0,1)$, $p\ne 4,6,8,\dots$ and the Gurarij space. We also give a proof of the Ramsey property of the classes $\{{\ell }_p^n{\}}_n$, $p\ne 2,\infty$, 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 Matoušek and Rödl. 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 $X$.

56 pages, 1 figure. To appear in Advances in Math