Extremal structure in ultrapowers of Banach spaces

Abstract

Given 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.

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

“…The hypothesis yields that the elementary molecule δ(x)−δ(y) d(x,y) is an extreme point of B F (M) which is not strongly extreme [2]. However, every extreme point of the unit ball of an ultrapower is strongly extreme [10]. Example 6.5.…”

Almost squareness and strong diameter two property in tensor product spaces

“…[15]) or the weak compactness in ultrapowers, which is intimately related to the study of super weakly compact sets (see [13,30] and references therein). From the geometric point of view, the expression of the norm in a ultrapower space has also motivated the study of properties of Banach spaces of isometric nature: the property of being a Lindenstrauss space [17], the study of extreme points [10,29], the study of strongly exposed points [10] or ASQ Banach spaces [16].…”