TY - JOUR

T1 - An alternative Daugavet property

AU - Martín, Miguel

AU - Oikhberg, Timur

N1 - Funding Information:
* Corresponding author. E-mail addresses: mmartins@ugr.es (M. Martín), toikhber@math.uci.edu (T. Oikhberg). 1 Supported by Spanish MCYT projects BFM2000-1467 and BFM2002-00061. 2 Supported by the NSF grants DMS-0296094 and DMS-0200714.

PY - 2004/6/1

Y1 - 2004/6/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jmaa.2004.02.006

DO - 10.1016/j.jmaa.2004.02.006

M3 - Article

AN - SCOPUS:3242729171

VL - 294

SP - 158

EP - 180

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -