EN
We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊'' ⊆ K₊' ⊆ K₊ such that K₊'' is homeomorphic to K₊ and hence C(K₊'') is isometric as a Banach space to C(K₊) but C(K₊') is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) 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). The subset K₊ is obtained as a particular compactification of the pairwise disjoint union of an appropriately chosen sequence $(K_{1,n} ∪ K_{2,n})_{n ∈ ℕ}$ of Ks for which C(K)s have few operators. We have $K₊' = K₊∖K_{1,0}$ and $K₊'' = K₊∖(K_{1,0} ∪ K_{2,0})$.