Octahedral norms in spaces of operators

TL;DR

This paper investigates conditions under which spaces of bounded linear operators between Banach spaces have octahedral norms, linking properties of the domain and codomain spaces and exploring implications for tensor products.

Contribution

It proves that $L(X,Y)$ has octahedral norm if $X^*$ and $Y$ do, and characterizes when $L( ext{l}_1,X)$ has octahedral norm, also analyzing stability of diameter 2 properties in tensor products.

Findings

$L(X,Y)$ has octahedral norm if $X^*$ and $Y$ do.

$L( ext{l}_1,X)$ has octahedral norm iff $X$ has octahedral norm.

Stability results for strong diameter 2 property in tensor products.

Abstract

We study octahedral norms in the space of bounded linear operators between Banach spaces. In fact, we prove that $L(X,Y)$ has octahedral norm whenever $X^*$ and $Y$ have octahedral norm. As a consequence the space of operators $L(\ell_1 ,X)$ has octahedral norm if, and only if, $X$ has octahedral norm. These results also allows us to get the stability of strong diameter 2 property for projective tensor products of Banach spaces, which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of Banach spaces.