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

Abstract: Recently it was introduced the so-called Bishop-Phelps-Bollob{\'a}s property for positive operators between Banach lattices. In this paper we prove that the pair $(C_0(L), Y) $ has the Bishop-Phelps--Bollob{\'a}s 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--Bollob{\'a}s property for positive operators then $Y$ is uniformly monotone.