Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable iff its dual W ⊥ is weak*-separable, (b) every weak*-separable subset of B Y* is weak*-metrizable, (c) every weak*-null sequence in the unit sphere of Y* contains a “nice“ subsequence; and (d) if U is separable, then X/U also has the CSPP. Property (a) yields an easy way of obtaining separable quotients in a class of Banach spaces. We also study the CSPP for C(K)-spaces, where K is a Mrówka compact space, e.g., we prove that the CSPP is not a three-space property.
@article{bwmeta1.element.doi-10_2478_s11533-011-0096-x,
author = {Jes\'us Ferrer and Marek W\'ojtowicz},
title = {The controlled separable projection property for Banach spaces},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {1252-1266},
zbl = {1253.46024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0096-x}
}
Jesús Ferrer; Marek Wójtowicz. The controlled separable projection property for Banach spaces. Open Mathematics, Tome 9 (2011) pp. 1252-1266. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0096-x/
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