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

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 $Y= L_p (\mu)$ ($1 \le p < \infty$).