The Bishop-Phelps-Bollobás theorem for operators from ℓ1 sums of Banach spaces

We introduce a generalized approximate hyperplane series property for a pair (X,Y) of Banach spaces to characterize when (ℓ1(X),Y) has the Bishop-Phelps-Bollobás property. In particular, we show that (X,Y) has this property if X, Y are finite-dimensional, if X is a C(K) space and Y is a Hilbert space, or if X is Asplund and Y=C0(L), where K is a compact Hausdorff space and L is a locally compact Hausdorff space.