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2011 | 9 | 6 | 1252-1266
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The controlled separable projection property for Banach spaces

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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.
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  • [1] Argyros S., Mercourakis S., On weakly Lindelöf Banach spaces, Rocky Mountain J. Math., 1993, 23(2), 395–446
  • [2] Banakh T., Plichko A., Zagorodnyuk A., Zeros of quadratic functionals on non-separable spaces, Colloq. Math., 2004, 100(1), 141–147
  • [3] Castillo J.M.F., González M., Three-Space Problems in Banach Space Theory, Lecture Notes in Math., 1667, Springer, Berlin-Heidelberg-New York, 1997
  • [4] Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math., 64, Scientific & Technical, Harlow, 1993
  • [5] van Douwen E.K., The integers and topology, In: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam-New York-Oxford, 1984, 111–167
  • [6] Dow A., Vaughan J.E., Mrówka maximal almost disjoint families for uncountable cardinals, Topology Appl., 2010, 157(8), 1379–1394
  • [7] Fajardo R.A.S., An indecomposable Banach space of continuous functions which has small density, Fund. Math., 2009, 202(1), 43–63
  • [8] Ferrer J., Zeros of real polynomials on C(K)-spaces, J. Math. Anal. Appl., 2007, 336(2), 788–796
  • [9] Ferrer J., On the controlled separable projection property for some C(K) spaces, Acta Math. Hungar., 2009, 124(1–2), 145–154
  • [10] Ferrer J., Kakol J., López Pellicer M., Wójtowicz M., On a three-space property for Lindelöf Σ-spaces, (WCG)-spaces, and the Sobczyk property, Funct. Approx. Comment. Math., 2011, 44(2), 289–306
  • [11] Finol C., Wójtowicz M., The structure of nonseparable Banach spaces with uncountable unconditional bases, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RASCAM, 2005, 99(1), 15–22
  • [12] Frankiewicz R., Zbierski P., Hausdorff Gaps and Limits, Stud. Logic Found. Math., 132, North-Holland, Amsterdam, 1994
  • [13] Hagler J., Sullivan F., Smoothness and weak* sequential compactness, Proc. Amer. Math. Soc., 1980, 78(4), 497–503
  • [14] Hrušák H., Szeptycki P.J., Tamariz-Mascarúa Á., Spaces of continuous functions defined on Mrówka spaces, Topology Appl., 2005, 148, 239–252
  • [15] Johnson W.B., On quasi-complements, Pacific J. Math., 1973, 48(1), 113–118
  • [16] Johnson W.B., Rosenthal H.P., On Ω*-basic sequences and their applications to the study of Banach spaces, Studia Math., 1972, 43, 77–92
  • [17] Josefson B., Weak sequential convergence in the dual of a Banach space does not imply norm convergence, Ark. Math., 1975, 13, 79–89
  • [18] Kalton N.J., Peck N.T., Twisted sums of sequence spaces and the three-space problem, Trans. Amer. Math. Soc., 1979, 255, 1–30
  • [19] Koszmider P., Banach spaces of continuous functions with few operators, Math. Ann., 2004, 330(1), 151–183
  • [20] Kuratowski K., Mostowski A., Set Theory, 2nd ed., Polish Scientific Publishers, Warsaw, 1976
  • [21] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. I, Ergeb. Math. Grenzgeb., 92, Springer, Berlin, 1977
  • [22] Mrówka S., On completely regular spaces, Fund. Math., 1954, 41, 105–106
  • [23] Mrówka S., Some set-theoretic constructions in topology, Fund. Math., 1977, 94(2), 83–92
  • [24] Mujica J., Separable quotients of Banach spaces, Rev. Mat. Univ. Complut. Madrid, 1997, 10(2), 299–330
  • [25] Nissenzweig A., w* sequential convergence, Israel J. Math., 1975, 22(3–4), 266–272
  • [26] Pełczynski A., Projections in certain Banach spaces, Studia Math., 1960, 19, 209–228
  • [27] Plebanek G., A construction of a Banach space C(K) with few operators, Topology Appl., 2004, 143, 217–239
  • [28] Rosenthal H.P., On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math., 1970, 37, 13–36
  • [29] Rudin W., Functional Analysis, 2nd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991
  • [30] Saxon S.A., Wilansky A., The equivalence of some Banach space problems, Colloq. Math., 1977, 37(2), 217–226
  • [31] Semadeni Z., Banach Spaces of Continuous Functions. I, Monogr. Mat., 55, Polish Scientific Publishers, Warszawa, 1971
  • [32] Sliwa W., (LF)-Spaces and the Separable Quotient Problem, Ph.D. thesis, Adam Mickiewicz University, Poznan, 1996 (in Polish)
  • [33] Valdivia M., Resolutions of the identity in certain Banach spaces, Collect. Math., 1988, 39(2), 127–140
  • [34] Vašák L., On one generalization of weakly compactly generated Banach spaces, Studia Math., 1981, 70(1), 11–19
  • [35] Walker R.C., The Stone-Čech Compactification, Ergeb. Math. Grenzgeb., 83, Springer, Berlin-Heidelberg-New York, 1974
  • [36] Wójtowicz M., Effective constructions of separable quotients of Banach spaces, Collect. Math., 1997, 48(4–6), 809–815
  • [37] Wójtowicz M., Generalizations of the c 0-ℓ 1-ℓ ∞ theorem of Bessaga and Pełczynski, Bull. Polish Acad. Sci. Math., 2002, 50(4), 373–382
  • [38] Wójtowicz M., Reflexivity and the separable quotient problem for a class of Banach spaces, Bull. Polish Acad. Sci. Math., 2002, 50(4), 383–394
  • [39] Zizler V., Nonseparable Banach Spaces, In: Handbook of the Geometry of Banach Spaces, 2, North-Holland, Amsterdam, 2003, 1743–1816
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