Bishop-Phelps-Bollobás property for positive operators when the domain is $L_{\infty }$

Abstract

We prove that the class of positive operators from $L_{\infty }(\mu )$ to $Y$ has the Bishop-Phelps-Bollobás 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 $(c_0,Y)$ for any uniformly monotone Banach lattice $Y.$ Further we show that these results are optimal in case that $Y$ is strictly monotone.

18 pages. arXiv admin note: text overlap with arXiv:1905.12972