On Banach spaces of the form $$C_0(\alpha \times L)$$ with few operators

For an exotic locally compact Hausdorff space $L$, constructed under the assumption of Ostaszewski's $\clubsuit$-principle, and a countable ordinal $\alpha$, we prove that all operators defined on $C_0(\alpha\times L)$ are as simple possible. We also investigate geometry such classify up to isomorphisms its complemented subspaces.