The Bishop-Phelps-Bollob\'as property for operators between spaces of continuous functions

TL;DR

This paper proves that certain classes of operators between spaces of continuous functions possess the Bishop-Phelps-Bollobás property, extending known results to new operator classes and spaces.

Contribution

It establishes the Bishop-Phelps-Bollobás property for operators between spaces of continuous functions and related classes of compact operators in new settings.

Findings

The space of bounded linear operators between continuous function spaces has the Bishop-Phelps-Bollobás property.

Compact operators from continuous functions vanishing at infinity to uniformly convex spaces have this property.

Compact operators into preduals of L1-spaces also exhibit the Bishop-Phelps-Bollobás property.

Abstract

We show that the space of bounded and linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollob\'as property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an $L_1$-space.