We say that an infinite, zero dimensional, compact Hausdorff space K has property (*) if for every nonempty open subset U of K there exists an open and closed subset V of U which is homeomorphic to K. We show that if K is a compact Hausdorff space with property (*) and X is a Banach space which contains a subspace isomorphic to the space C(K) of all scalar (real or complex) continuous functions on K and Y is a closed linear subspace of X which does not contain any subspace isomorphic to the space C([0,1]), then the quotient space X/Y contains a subspace isomorphic to the space C(K).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1180,
author = {Artur Michalak},
title = {On some properties of quotients of homogeneous C(K) spaces},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {36},
year = {2016},
pages = {33-43},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1180}
}
Artur Michalak. On some properties of quotients of homogeneous C(K) spaces. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 36 (2016) pp. 33-43. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1180/
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