An alternative Daugavet property

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 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 c₀-, l₁- 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.