On large indecomposable Banach spaces

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko and Yost (2000) [19], , Argyros and Tolias (2004) [1]). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density 22ω. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T:C(K)→C(K) satisfies T(f)=gf+S(f) for every f∈C(K), where g∈C(K) and S:C(K)→C(K) is weakly compact (strictly singular).