Operators on injective tensor products of separable Banach spaces and spaces with few operators

We give a characterization of the operators on the injective tensor product $E{\hat{\otimes }}_{\epsilon }X$ for any separable Banach space E and any (non-separable) Banach space X with few operators, in the sense that any operator $T:X\to X$ takes the form $T=\lambda I+S$ for a scalar $\lambda \in K$ and an operator S with separable range. This is used to give a classification of the complemented subspaces and closed operator ideals of spaces of the form $C_0(\omega \times K_A)$, where $K_A$ is a locally compact Hausdorff space induced by an almost disjoint family $A$ such that $C_0\left(K_A\right)$ has few operators.