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). © 2013 Elsevier Inc. All rights reserved.