Quotients of indecomposable Banach spaces of continuous functions
Assuming ⋄, we construct a connected compact topological space K such that for every closed L ⊂ K the Banach space C(L) has few operators, in the sense that every operator on C(L) is multiplication by a continuous function plus a weakly compact operator. In particular, C(K) is indecomposable and has continuum many non-isomorphic indecomposable quotients, and K does not contain a homeomorphic copy of βℕ. Moreover, assuming CH we construct a connected compact K where C(K) has few...