A C(K) Banach space which does not have the Schroeder–Bernstein property

Abstract

We construct a totally disconnected compact Hausdorff space K 0 which has clopen subsets K 2 ⊆ K 1 ⊆ K 0 such that K 0 is homeomorphic to K 2 and hence C(K 0 ) is isometric as a Banach space to C(K 2 ) but C(K 0 ) is not isomorphic to C(K 1 ). 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). K 0 is obtained as a particular compactification of the pairwise disjoint union of a sequence of Ks for which C(K)s have few operators.

Banach spaces widely complemented in each other

“…In 1996, W. T. Gowers [14] gave a first negative solution to this problem (see also [1]- [6] and [15]. More recently [16] a C(K) space was introduced which is also a solution to this problem.…”