Extremal structure in ultrapowers of Banach spaces

Abstract

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