On non-archimedean Gurarii spaces

TL;DR

This paper constructs a non-archimedean Gurarii Banach space with almost universal disposition for finite-dimensional spaces and characterizes when such spaces are isometrically isomorphic.

Contribution

It introduces the construction of non-archimedean Gurarii spaces and characterizes their isometric uniqueness based on field properties.

Findings

All such spaces are $ ext{ extasciitilde}$-isometric.

They are isometrically isomorphic iff the field is spherically complete with a specific value set.

Abstract

Let $U_{FNA}$ be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurarii Banach space $G$ over a non-archimedean valued field $K$ is constructed, i.e. a non-archimedean Banach space $G$ of countable type which is of almost universal disposition for the class $U_{FNA}$. This means: for every isometry $g : X \to Y$ , where $Y \in U_{FNA}$ and $X$ is a subspace of $G$, and every $\varepsilon \in (0,1)$ there exists an $\varepsilon$-isometry $f : Y \to G$ such that $f(g(x)) = x$ for all $x \in X$. We show that all non-archimedean Banach spaces of almost universal disposition for the class $U_{FNA}$ are $\varepsilon$-isometric. Furthermore, all non-archimedean Banach spaces of almost universal disposition for the class $U_{FNA}$ are isometrically isomorphic if and only if $K$ is spherically complete and $\{|\lambda| : \lambda \in K \setminus \{0\} \} = (0,\infty)$.