An alternative Daugavet property

Abstract

We introduce a strictly weaker version of the Daugavet property as follows: a Banach space X has this alternative Daugavet property (ADP in short) if the norm identity (aDE) max |ω|=1 ∥ Id+ωT∥=1+∥T∥ holds for all rank-one operators T :X→X . In such a case, all weakly compact operators on X also satisfy ( aDE). We give some geometric characterizations of the alternative Daugavet property in terms of the space and its successive duals. We prove that the ADP is stable for c 0-, l 1- and l ∞-sums and characterize when some vector-valued function spaces have the property. Finally, we show that a C ∗ -algebra (or the predual of a von Neumann algebra) has the ADP if and only if its atomic projection (respectively, the atomic projection of the algebra) are central. We also establish some geometric properties of JB ∗ -triples, and characterize JB ∗ -triples possessing the ADP and the Daugavet property.