# The controlled separable projection property for Banach spaces

Open Mathematics (2011)

• Volume: 9, Issue: 6, page 1252-1266
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Jesús Ferrer, and Marek Wójtowicz. "The controlled separable projection property for Banach spaces." Open Mathematics 9.6 (2011): 1252-1266. <http://eudml.org/doc/269444>.

@article{JesúsFerrer2011,
abstract = {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.},
author = {Jesús Ferrer, Marek Wójtowicz},
journal = {Open Mathematics},
keywords = {Controlled separable projection property; Weakly Lindelöf determined Banach space; Josefson-Nissenzweig sequence; Separable quotient problem; Mrówka space; controlled separable projection property; weakly Lindelöf determined Banach space; separable quotient problem; WCG-spaces},
language = {eng},
number = {6},
pages = {1252-1266},
title = {The controlled separable projection property for Banach spaces},
url = {http://eudml.org/doc/269444},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Jesús Ferrer
AU - Marek Wójtowicz
TI - The controlled separable projection property for Banach spaces
JO - Open Mathematics
PY - 2011
VL - 9
IS - 6
SP - 1252
EP - 1266
AB - 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.
LA - eng
KW - Controlled separable projection property; Weakly Lindelöf determined Banach space; Josefson-Nissenzweig sequence; Separable quotient problem; Mrówka space; controlled separable projection property; weakly Lindelöf determined Banach space; separable quotient problem; WCG-spaces
UR - http://eudml.org/doc/269444
ER -

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