Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers

Abstract

In this note various geometric properties of a Banach space $X$ are characterized by means of weaker corresponding geometric properties involving an ultrapower $X^\mathcal{U}$. The characterizations do not depend on the particular choice of the free ultrafilter $\mathcal{U}$. For example, a point $x\in S_X$ is an MLUR point if and only if $j(x)$ (given by the canonical inclusion $j\colon X \to X^\mathcal{U}$) in $B_{X^\mathcal{U}}$ is an extreme point; a point $x\in S_X$ is LUR if and only if $j(x)$ is not contained in any non-degenerate line segment of $S_{X^\mathcal{U}}$; a Banach space $X$ is URED if and only if there are no $x,y \in S_{X^\mathcal{U}}$, $x\neq y$, with $x-y \in j(X)$.