The Controlled Separable Projection Property for Banach spaces

作者：Ferrer Jesus^*; Wojtowicz Marek

来源：Central European Journal of Mathematics, 2011, 9(6): 1252-1266.

DOI：10.2478/s11533-011-0096-x

• Controlled separable projection property
• Weakly Lindelof determined Banach space
• Josefson-Nissenzweig sequence
• Separable quotient problem
• Mrowka space

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 properly) and Y is a quasi-quoteint 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-perpendicular to is weak*-separable, (b) every weak*-separable subset of B-gamma* 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)-space, where K is a Mrowka compact space, e.g., we prove that the CSPP is not a three-space property.

• 出版日期2011-12