Almost square and octahedral norms in tensor products of Banach spaces

TL;DR

This paper investigates geometric properties such as octahedrality and almost squareness in tensor product spaces of Banach spaces, establishing how these properties are preserved or transferred through tensor products.

Contribution

It provides new results on the preservation of octahedrality and almost squareness in various tensor product constructions of Banach spaces.

Findings

Injective tensor product of two octahedral Banach spaces is octahedral

Injective tensor product of an almost square Banach space with any Banach space is almost square

Injective symmetric tensor product of an octahedral Banach space is octahedral

Abstract

The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.