  For an exotic locally compact Hausdorff space $L$, constructed under the
assumption of the Ostaszewski's $\clubsuit$-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.
