The Bishop-Phelps-Bollob\'{a}s property for operators on $C(K)$

TL;DR

This paper extends the Bishop-Phelps-Bollobás property to operators with domain space C_0(L), showing it holds for all Hausdorff locally compact spaces L and any complex uniformly convex target space, including L_p spaces.

Contribution

It establishes the Bishop-Phelps-Bollobás property for operators on C_0(L) spaces with complex uniformly convex targets, generalizing previous results.

Findings

The property holds for all C_0(L) spaces with locally compact L.

It applies to any complex uniformly convex space Y.

Includes L_p spaces as specific cases.

Abstract

We provide a version for operators of the Bishop-Phelps-Bollob\'{a}s Theorem when the domain space is the complex space $C_0(L)$. In fact we prove that the pair $(C_0(L), Y)$ satisfies the Bishop-Phelps-Bollob\'{a}s property for operators for every Hausdorff locally compact space $L$ and any $\mathbb{C}$-uniformly convex space. As a consequence, this holds for $Y= L_p (\mu)$ ($1 \le p < \infty $).