Slice diameter two property in ultrapowers

TL;DR

This paper investigates when ultrapower spaces inherit the slice diameter two property, providing characterizations and examples that show the property is not always preserved in ultrapowers.

Contribution

It offers a characterization of the inheritance of the slice diameter two property in ultrapowers and presents counterexamples showing it is not always inherited.

Findings

Ultrapowers of many Banach spaces retain the slice diameter two property.

Counterexamples show ultrapowers can lack the property even if the original space has it.

The slice diameter two property is not necessarily inherited by ultrapower spaces.

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 $\eta>0$, 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 $\eta$ 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.