Slice diameter two property in ultrapowers

TL;DR

The paper investigates whether the slice diameter two property (slice-D2P) is preserved by ultrapowers of Banach spaces. It introduces the uniform slice-D2P and proves a sharp equivalence: $(X)_\mathcal{U}$ has the slice-D2P for all free ultrafilters $\mathcal{U}$ iff $\lim_{n\to\infty} C_n^\alpha(X)=0$ for every $0<\alpha<2$, where $C_n^\alpha(X)$ encodes how well the unit ball can be generated by convex combinations of far-apart pairs. It shows that the uniform version holds for many classical spaces (including LASQ, $L_1$-preduals, Lipschitz spaces, and uniform algebras) and is preserved under standard sum constructions, thereby ensuring ultrapower slice-D2P in those cases. Crucially, it also constructs a Daugavet-space $X$ such that $(X)_\mathcal{U}$ fails the slice-D2P for every ultrafilter, or even has unit-ball slices of arbitrarily small diameter, establishing that slice-D2P is not generally inherited by ultrapowers.

Abstract

In this note we study the inheritance of the slice diameter two property by ultrapower spaces. Given a Banach space $X$, we give a characterisation of when $(X)_\mathcal U$, the ultrapower of $X$ through a free ultrafilter $\mathcal U$, has the slice diameter two property obtaining that this is the case for many Banach spaces which are known to enjoy the slice diameter two property. We also provide, for every $η>0$, an example of a Banach space $X$ with the Daugavet property such that the unit ball of $(X)_\mathcal U$ contains a slice of diameter smaller than $η$ for every free ultrafilter $\mathcal U$ over $\mathbb N$. This proves, in particular, that the slice diameter two property is not in general inherited by taking ultrapower spaces.

Theorems & Definitions (27)

• Proposition 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Definition 3.4
• Example 3.5