A space C(K) where all nontrivial complemented subspaces have big densities
Using the method of forcing we prove that consistently there is a Banach space (of continuous functions on a totally disconnected compact Hausdor space) of den- sity bigger than the continuum where all operators are multiplications by a continuous function plus a weakly compact operator and which has no innite-dimension al comple- mented subspaces of density continuum or smaller. In particular no separable innite- dimensional subspace has a complemented superspace of density continuum or smaller, consistently answering a question of Johnson and Lindenstrauss of 1974.