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 operators and K contains a homeomorphic copy of βℕ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-3-4,
author = {Rog\'erio Augusto dos Santos Fajardo},
title = {Quotients of indecomposable Banach spaces of continuous functions},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {259-283},
zbl = {1270.46020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-3-4}
}
Rogério Augusto dos Santos Fajardo. Quotients of indecomposable Banach spaces of continuous functions. Studia Mathematica, Tome 209 (2012) pp. 259-283. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-3-4/