# Bishop-Phelps-Bollob\'as property for positive operators when the domain is $L_\infty $

## TL;DR

This paper establishes that positive operators from $L_
abla$ to certain Banach lattices possess the Bishop-Phelps-Bollobás property, extending to pairs like $(c_0, Y)$, with optimality results for strictly monotone lattices.

## Contribution

It proves the Bishop-Phelps-Bollobás property for positive operators from $L_
abla$ and $c_0$ to uniformly monotone Banach lattices, including optimality conditions.

## Key findings

- Positive operators from $L_
abla$ to $Y$ have the Bishop-Phelps-Bollobás property.
- The property also holds for the pair $(c_0, Y)$.
- Results are optimal for strictly monotone Banach lattices.

## Abstract

We prove that the class of positive operators from $L_\infty (\mu)$ to $Y$ has the Bishop-Phelps-Bollob\'as 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.