The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B

We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX(ε) such that for every Y , the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c₀-, ℓ₁- and ℓ_∞-sums of Banach spaces.