Banach spaces of continuous functions with few operators

Abstract

We present two constructions of infinite, separable, compact Hausdorff spaces K for which the Banach space C(K) of all continuous real-valued functions with the supremum norm has remarkable properties. In the first construction K is zero-dimensional and C(K) is non-isomorphic to any of its proper subspaces nor any of its proper quotients. In particular, it is an example of a C(K) space where the hyperplanes, one co-dimensional subspaces of C(K), are not isomorphic to C(K). In the second construction K is connected and C(K) is indecomposable which implies that it is not isomorphic to any C(K’) for K’ zero-dimensional. All these properties follow from the fact that there are few operators on our C(K)’s. If we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator T : C (K) → C (K) is of the form gI+S where g∈C(K) and S is weakly compact or equivalently (in C(K) spaces) strictly singular.