The controlled separable projection property for Banach spaces

• Autorzy: Jesús Ferrer , Marek Wójtowicz
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Controlled separable projection property; Weakly Lindelöf determined Banach space; Josefson-Nissenzweig sequence; Separable quotient problem; Mrówka space

Kategorie tematyczne

• 54D30: Compactness
• 54C35: Function spaces
• 46B10: Duality and reflexivity
• 46B26: Nonseparable Banach spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 6
• Strony: 1252-1266

Daty

wydano

2011-12-01

online

2011-09-23

Twórcy

autor

Jesús Ferrer

• Universidad de Valencia

autor

Marek Wójtowicz

• Uniwersytet Kazimierza Wielkiego

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Identyfikatory

DOI

10.2478/s11533-011-0096-x