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 $$(X)_{\mathcal {U}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>U</mml:mi> </mml:msub> </mml:math> , the ultrapower of X through a free ultrafilter $${\mathcal {U}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> , 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 $$\eta &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , an example of a Banach space X with the Daugavet property such that the unit ball of $$(X)_{\mathcal {U}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>U</mml:mi> </mml:msub> </mml:math> contains a slice of diameter smaller than $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> for every free ultrafilter $${\mathcal {U}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> over $${\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> . This proves, in particular, that the slice diameter two property is not in general inherited by taking ultrapower spaces.