Slice diameter two property in 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 $\eta>0$, 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.