The Bishop-Phelps-Bollobas property and lush spaces

Abstract

We prove that for every lush space X, the couple (l(1), X) has the Bishop-Phelps-Bollobas property for operators, that is, every lush space has the AHSP (standing for the approximate hyperplane series property). While every lush space has the alternative Daugavet property, there exists a space with the alternative Daugavet property that does not have the AHSP. We also show that there is a Banach space with both the AHSP and the alternative Daugavet property which is not lush. (C) 2012 Elsevier Inc. All rights reserved.