Tensor Products and the Isomorphic Theory of Daugavet Spaces

Abstract

We construct two separable Banach spaces with the Daugavet property whose injective and projective tensor squares, respectively, fail the Daugavet property. Both examples arise from one sequence of finite-dimensional extensions. The first space is the completion of the increasing union of the extensions, whereas the second is the closed span of their coordinate functionals. We will also investigate the operator Daugavet properties of these spaces later in the paper, also highlighting natural examples of metric spaces \(M\) for which \(\operatorname{Lip}_0(M)\) has the Daugavet property but fails to have the weak operator Daugavet property. Independently of this, we will also prove that every Banach space admits a linear isometric embedding into a Banach space which is not isomorphic to any Banach space with the Daugavet property.