A $C(K)$ Banach space which does not have the Schroeder–Bernstein property
Volume 212 / 2012
Abstract
We construct a totally disconnected compact Hausdorff space $K_+$ which has clopen subsets $K_+^{\prime\prime}\subseteq K_+^{\prime}\subseteq K_+$ such that $K_+^{\prime\prime}$ is homeomorphic to $K_+$ and hence $C(K_+^{\prime\prime})$ is isometric as a Banach space to $C(K_+)$ but $C(K_+^{\prime})$ 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 $(K_{1,n}\cup K_{2,n})_{n\in \mathbb N}$ of $K$s for which $C(K)$s have few operators. We have $K_+^{\prime}=K_+\setminus K_{1,0}$ and $K_+^{\prime\prime}=K_+\setminus (K_{1,0}\cup K_{2,0}).$
