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 $E{\widehat{\otimes }}_{\epsilon }X$ 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 $\lambda \in \mathbb{K}$ 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 $C_0(\omega \times K_{\mathcal{A}})$, where $K_{\mathcal{A}}$ is a locally compact Hausdorff space induced by an almost disjoint family $\mathcal{A}$ such that $C_0(K_{\mathcal{A}})$ has few operators.