Cent. Eur. J. Math. • 9(6) • 2011 • 1252-1266 DOI: 10.2478/s11533-011-0096-x The Controlled Separable Projection Property for Banach spaces Jesús Ferrer 1∗ , Marek Wójtowicz 2† 1 Departamento de Anàlisis Matemático, Universidad de Valencia, Dr. Moliner, 50, 46100 Burjasot (Valencia), Spain 2 Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland 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. 46B10, 46B26, 54C35, 54D30 Controlled separable projection property • Weakly Lindelöf determined Banach space • Josefson–Nissenzweig sequence • Separable quotient problem • Mrówka space © Versita Sp. z o.o. 1. Introduction We follow standard notations and for notions undefined here we refer the reader to the survey article by Zizler [39]. ∗ E-mail: Jesus.Ferrer@uv.es † E-mail: mwojt@ukw.edu.pl