Bishop-Phelps-Bollob\'as property for positive operators when the domain is $C_0(L) $

TL;DR

This paper establishes the Bishop-Phelps-Bollobás property for positive operators from $C_0(L)$ to certain Banach lattices, expanding understanding of operator approximation in Banach lattice theory.

Contribution

It proves the property holds for pairs $(C_0(L), Y)$ with $Y$ uniformly monotone, and explores conditions under which the property implies uniform monotonicity.

Findings

The property holds for pairs $(C_0(L), Y)$ with $Y$ uniformly monotone.

Separable $C_0(L)$ spaces extend the result to all uniformly monotone $Y$.

A partial converse links the property to uniform monotonicity of $Y$.

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.