Bishop-Phelps-Bollobas Property For Positive Operators When The Domain Is L-Infinity

We prove that the class of positive operators from L-infinity(mu) to Y has the Bishop-Phelps-Bollobas property for any positive measure mu, whenever Y is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair (c0,Y ) for any uniformly monotone Banach lattice Y. Further we show that these results are optimal in case that Y is strictly monotone.