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For an exotic locally compact Hausdorff space $L$, constructed under theassumption of the Ostaszewski's $\clubsuit$-principle, and a countable ordinalspace $\alpha$, 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 complementedsubspaces.

On operators on $C_0(\alpha\times L)$ under the Ostaszewski's $\clubsuit$-principle