Almost Fra\"iss\'e Banach spaces

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.