Almost Fra\"iss\'e Banach spaces

TL;DR

This paper introduces and studies weaker forms of homogeneity in Banach spaces, called Almost ultrahomogeneity and the Almost Fra"iss"e Property, relating them to pseudometrics and ultrapowers.

Contribution

It formally defines weaker homogeneity properties for Banach spaces and explores their relationships, including how ultrapowers of almost Fra"iss"e spaces are ultrahomogeneous.

Findings

Ultrapowers of almost Fra"iss"e Banach spaces are ultrahomogeneous.

Approximately ultrahomogeneous spaces are finitely isometrically extensible.

Relations between various homogeneity properties and pseudometrics on finite-dimensional subspaces.

Abstract

Continuing with the study of Approximately ultrahomogeneous and Fra\"iss\'e Banach spaces introduced by V. Ferenczi, J. L\'opez-Abad, B. Mbombo and S. Todorcevic, we define formally weaker and in some aspects more natural properties of Banach spaces which we call Almost ultrahomogeneity and the Almost Fra\"iss\'e Property. We obtain relations between these different homogeneity properties of a space $E$ and relate them to certain pseudometrics on the class $\mathrm{Age}(E)$ of finite dimensional subspaces of $E$. We prove that ultrapowers of an almost Fra\"iss\'e Banach space are ultrahomogeneous. We also study two properties called finitely isometrically extensible and almost finitely isometrically extensible, respectively, and prove that approximately ultrahomogeneous Banach spaces are finitely isometrically extensible.