Octahedral norms in tensor products of Banach spaces

TL;DR

This paper explores the conditions under which octahedral norms are preserved in tensor products of Banach spaces, providing solutions to open problems and establishing new preservation results under specific properties.

Contribution

It proves the existence of Banach spaces where certain tensor products lack octahedral norms and shows preservation of octahedrality under the metric approximation property in specific tensor products.

Findings

Existence of Banach spaces with tensor products lacking octahedral norms.

Preservation of octahedrality under the metric approximation property.

Solutions to two open problems in the literature.

Abstract

We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.