Banach spaces of continuous functions with few operators

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 gC(K) and S is weakly compact or equivalently (in C(K) spaces) strictly singular.While conducting research leading to the results presented in this paper, the author was partially supported by a fellowship Produtividade em Pesquisa from National Research Council of Brazil (Conselho Nacional de Pesquisa, Processo Número 300369/01-8). The final stage of the research was realized at the Fields Institute in Toronto where the author was supported by the State of São Paulo Research Assistance Foundation (Fundação de Amparoá Pesquisa do Estado de São Paulo), Processo Número 02/03677-7 and by the Fields Institute.Revised version: 29 January 2004