Bishop–Phelps–Bollobás property for positive operators when the domain is C_0(L)

Recently it was introduced the so-called Bishop–Phelps–Bollobás property for positive operators between Banach lattices. In this paper we prove that the pair (C_0(L), Y) has the Bishop–Phelps–Bollobás property for positive operators, for any locally compact Hausdorff topological space L , whenever Y is a uniformly monotone Banach lattice with a weak unit. In case that the space C_0(L) is separable, the same statement holds for any uniformly monotone Banach lattice Y . We also show the following partial converse of the main result. In case that Y is a strictly monotone Banach lattice, L is a locally compact Hausdorff topological space that contains at least two elements and the pair (C_0(L), Y ) has the Bishop–Phelps–Bollobás property for positive operators then Y is uniformly monotone.