L- Orthogonality, octahedrality and daugavet property in banach spaces

Abstract

We prove that the abundance of almost L-orthogonal vectors in a Banach space X (almost Daugavet property) implies the abundance of nonzero vectors in X** being L-orthogonal to X. In fact, we get that a Banach space X verfies the Daugavet property if, and only if, the set of vectors in X** being L-orthogonal to X is weak-star dense in X**. In contrast with the separable case, we prove that the existence of almost L-orthogonal vectors in a nonseparable Banach space X (octahedrality) does not imply the existence of nonzero vectors in X** being L-orthogonal to X, which shows that the answer to an environment question is negative (see [7, Section 9] and [13, Section 4]). Also, in contrast with the separable case, we obtain that the existence of almost L-orthogonal vectors in a nonseparable Banach space X (octahedrality) does not imply the abundance of almost L-orthogonal vectors in Banach space X (almost Daugavet property), which solves an open question in [20]. Some consequences on Daugavet property in the setting of L-embedded spaces are also obtained.