On operators on $C_0(\alpha \times L)$ under the Ostaszewski's $\mathrm{♣}$-principle

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