  We construct a totally disconnected compact Hausdorff space N which has
clopen subsets M included in L included in N such that N is homeomorphic to M
and hence C(N) is isometric as a Banach space to C(M) but C(N) is not
isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form C(K)
which are isomorphic to complemented subspaces of each other (even in the above
strong isometric sense), providing a solution to the Schroeder-Bernstein
problem for Banach spaces of the form C(K). N is obtained as a particular
compactification of the pairwise disjoint union of a sequence of Ks for which
C(K)s have few operators.
