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 κ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-2,
author = {Piotr Koszmider},
title = {A space C(K) where all nontrivial complemented subspaces have big densities},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {109-127},
zbl = {1068.03042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-2}
}
Piotr Koszmider. A space C(K) where all nontrivial complemented subspaces have big densities. Studia Mathematica, Tome 166 (2005) pp. 109-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-2/