Stability results of octahedrality in tensor product spaces

TL;DR

This paper demonstrates that certain geometric properties like octahedrality and the strong diameter two property are not necessarily preserved in tensor product spaces, providing a negative answer to a previously open question.

Contribution

It constructs specific finite-dimensional Banach spaces showing that octahedrality and the strong diameter two property are not inherited in tensor products, resolving an open problem.

Findings

Existence of a finite-dimensional Banach space X with specific tensor product properties.

Failure of strong diameter two property in injective tensor products.

Failure of octahedral norm in projective tensor products.

Abstract

We prove that there exists a finite-dimensional Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_\pi X^*$ fails to have octahedral norm. This proves that the octahedrality of the norm (respectively the strong diameter two property) is not automatically inherited from one factor by taking projective tensor product (respectively injective tensor product), which answers [16,Question 4.4].