Numerical index of vector-valued function spaces

We show that the numerical index of a c0-, l1-, or l∞-sum of Banach spaces is the infimum numerical index of the summands. Moreover, we prove that the spaces C(K, X) and L1(µ, X) (K any compact Hausdorff space, µ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.