The Bishop–Phelps–Bollobás property for operators on C(K)

Abstract

We provide a version for operators of the Bishop–Phelps–Bollobás theorem when the domain space is the complex space ${\mathcal{C}}_0\left(L\right)$. In fact, we prove that the pair $\left({\mathcal{C}}_0\right(L),Y)$ will satisfy the Bishop–Phelps–Bollobás property for operators for every Hausdorff locally compact space $L$ and any $\mathbb{C}$-uniformly convex space. As a consequence, this holds for $Y=L_p\left(\mu \right)$ ($1\le p<\infty$).