Extremal structure in ultrapowers of Banach spaces

2021/09/03 by García-Lirola, Luis C., Grelier, Guillaume, Zoca, Abraham Rueda
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2109.01393

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.

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