A space $C(K)$ where all nontrivial complemented subspaces have big densities
Volume 168 / 2005
Studia Mathematica 168 (2005), 109-127
MSC: 03E35, 46B03.
DOI: 10.4064/sm168-2-2
Abstract
Using the method of forcing we prove that consistently there is a Banach space (of continuous functions on a totally disconnected compact Hausdorff space) of density $\kappa $ bigger than the continuum where all operators are multiplications by a continuous function plus a weakly compact operator and which has no infinite-dimensional complemented subspaces of density continuum or smaller. In particular no separable infinite-dimensional subspace has a complemented superspace of density continuum or smaller, consistently answering a question of Johnson and Lindenstrauss of 1974.
