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

TL;DR

This paper characterizes various geometric properties of Banach spaces through their ultrapowers, establishing ultrafilter-independent dualities that connect properties like MLUR, LUR, and URED with ultrapower elements.

Contribution

It introduces ultrafilter-independent characterizations of Banach space geometric properties via ultrapowers, linking classical and ultrapower geometric features.

Findings

MLUR points correspond to extreme points in ultrapowers

LUR points relate to the absence of line segments in ultrapower spheres

UREd spaces have no distinct points in ultrapower with difference in the canonical image

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 $x-y \in j(X)$.