Banach spaces which always produce octahedral spaces of operators

TL;DR

This paper characterizes Banach spaces that ensure all operator spaces $L(Y,X)$ are octahedral, linking geometric properties of $X$ with the structure of finite-dimensional subspaces and providing examples like Lipschitz spaces.

Contribution

It provides a characterization of Banach spaces with octahedral operator spaces for all $Y$, connecting finite-dimensional subspace structure to octahedrality, and offers concrete examples.

Findings

Banach spaces with certain finite-dimensional subspace structures produce octahedral $L(Y,X)$.

Octahedrality of $L(Y,X)$ for all $Y$ is equivalent to octahedrality of $L( ext{} ext{ell}_p^n,X)$ for all $n$ and $p$.

Examples include Lipschitz spaces with octahedral norms and $L_1$-preduals with the Daugavet property.

Abstract

We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is octahedral for every non-zero Banach space $Y$. They are those satisfying that, for every finite dimensional subspace $Z$, $\ell_\infty$ can be finitely-representable in a part of $X$ kind of $\ell_1$-orthogonal to $Z$. We also prove that $L(Y,X)$ is octahedral for every $Y$ if, and only if, $L(\ell_p^n,X)$ is octahedral for every $n\in\mathbb N$ and $1<p<\infty$. Finally, we find examples of Banach spaces satisfying the above conditions like $\Lip(M)$ spaces with octahedral norms or $L_1$-preduals with the Daugavet property.