Banach spaces which always produce octahedral spaces of operators

Abstract

AbstractWe 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, $$\ell _\infty $$ ℓ ∞ can be finitely-representable in a part of X kind of $$\ell _1$$ ℓ 1 -orthogonal to Z. We also prove that L(Y, X) is octahedral for every Y if, and only if, $$L(\ell _p^n,X)$$ L ( ℓ p n , X ) is octahedral for every $$n\in {\mathbb {N}}$$ n ∈ N and $$1<p<\infty $$ 1 < p < ∞ . Finally, we find examples of Banach spaces satisfying the above conditions like $${\textrm{Lip}}_0(M)$$ Lip 0 ( M ) spaces with octahedral norms or $$L_1$$ L 1 -preduals with the Daugavet property.