Extremal structure in ultrapowers of Banach spaces

TL;DR

This paper investigates the extremal structure of ultrapowers of Banach spaces, characterizing how extremal points behave under ultrafilter constructions and exploring specific cases like super weakly compact and uniformly convex sets.

Contribution

It provides new conditions under which extremal points in ultrapowers correspond to extremal points in the original space, and analyzes extremal structures for special classes of sets.

Findings

Extreme points in ultrapowers are characterized by uniformity conditions.

Every extreme point of an ultrapower is strongly extreme.

Exposed points in ultrapowers are strongly exposed with countably incomplete ultrafilters.

Abstract

Given a bounded convex subset $C$ of a Banach space $X$ and a free ultrafilter $\mathcal U$, we study which points $(x_i)_\mathcal U$ are extreme points of the ultrapower $C_\mathcal U$ in $X_\mathcal U$. In general, we obtain that when $\{x_i\}$ is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then $(x_i)_\mathcal U$ is an extreme point (respectively denting point, strongly exposed point) of $C_\mathcal U$. We also show that every extreme point of $C_{\mathcal U}$ is strongly extreme, and that every point exposed by a functional in $(X^*)_{\mathcal U}$ is strongly exposed, provided that $\mathcal U$ is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of $C_\mathcal U$ in the case that $C$ is a super weakly compact or uniformly convex set.