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

TL;DR

The paper constructs a specific totally disconnected compact space N with subsets M and L such that C(N) and C(M) are isometric but not isomorphic to C(L), providing a counterexample to the Schroeder-Bernstein property in certain Banach spaces.

Contribution

It introduces a novel construction of a compact space N leading to Banach spaces C(N), C(M), and C(L) with unusual isomorphic and isometric relations, addressing the Schroeder-Bernstein problem.

Findings

C(N) is isometric to C(M) but not isomorphic to C(L)

Provides a counterexample to the Schroeder-Bernstein property in C(K) spaces

Constructs N as a specific compactification of disjoint unions of K spaces

Abstract

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.