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 (aDE) 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.  2004 Elsevier Inc. All rights reserved.