The controlled separable projection property for Banach spaces

Jesús Ferrer; Marek Wójtowicz

Open Mathematics (2011)

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

Abstract

top
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

top

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 -

References

top
  1. [1] Argyros S., Mercourakis S., On weakly Lindelöf Banach spaces, Rocky Mountain J. Math., 1993, 23(2), 395–446 http://dx.doi.org/10.1216/rmjm/1181072569 Zbl0797.46009
  2. [2] Banakh T., Plichko A., Zagorodnyuk A., Zeros of quadratic functionals on non-separable spaces, Colloq. Math., 2004, 100(1), 141–147 http://dx.doi.org/10.4064/cm100-1-13 Zbl1066.46040
  3. [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 Zbl0914.46015
  4. [4] Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math., 64, Scientific & Technical, Harlow, 1993 Zbl0782.46019
  5. [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. [6] Dow A., Vaughan J.E., Mrówka maximal almost disjoint families for uncountable cardinals, Topology Appl., 2010, 157(8), 1379–1394 http://dx.doi.org/10.1016/j.topol.2009.08.024 Zbl1196.54060
  7. [7] Fajardo R.A.S., An indecomposable Banach space of continuous functions which has small density, Fund. Math., 2009, 202(1), 43–63 http://dx.doi.org/10.4064/fm202-1-2 Zbl1159.03034
  8. [8] Ferrer J., Zeros of real polynomials on C(K)-spaces, J. Math. Anal. Appl., 2007, 336(2), 788–796 http://dx.doi.org/10.1016/j.jmaa.2007.02.083 Zbl1161.46024
  9. [9] Ferrer J., On the controlled separable projection property for some C(K) spaces, Acta Math. Hungar., 2009, 124(1–2), 145–154 http://dx.doi.org/10.1007/s10474-009-8165-3 Zbl1265.26009
  10. [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 Zbl1234.46018
  11. [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 Zbl1098.46015
  12. [12] Frankiewicz R., Zbierski P., Hausdorff Gaps and Limits, Stud. Logic Found. Math., 132, North-Holland, Amsterdam, 1994 http://dx.doi.org/10.1016/S0049-237X(08)70177-0 Zbl0821.54001
  13. [13] Hagler J., Sullivan F., Smoothness and weak* sequential compactness, Proc. Amer. Math. Soc., 1980, 78(4), 497–503 Zbl0463.46010
  14. [14] Hrušák H., Szeptycki P.J., Tamariz-Mascarúa Á., Spaces of continuous functions defined on Mrówka spaces, Topology Appl., 2005, 148, 239–252 http://dx.doi.org/10.1016/j.topol.2004.09.009 Zbl1068.54022
  15. [15] Johnson W.B., On quasi-complements, Pacific J. Math., 1973, 48(1), 113–118 Zbl0283.46008
  16. [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. [17] Josefson B., Weak sequential convergence in the dual of a Banach space does not imply norm convergence, Ark. Math., 1975, 13, 79–89 http://dx.doi.org/10.1007/BF02386198 Zbl0303.46018
  18. [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 http://dx.doi.org/10.1090/S0002-9947-1979-0542869-X Zbl0424.46004
  19. [19] Koszmider P., Banach spaces of continuous functions with few operators, Math. Ann., 2004, 330(1), 151–183 http://dx.doi.org/10.1007/s00208-004-0544-z Zbl1064.46009
  20. [20] Kuratowski K., Mostowski A., Set Theory, 2nd ed., Polish Scientific Publishers, Warsaw, 1976 
  21. [21] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. I, Ergeb. Math. Grenzgeb., 92, Springer, Berlin, 1977 Zbl0362.46013
  22. [22] Mrówka S., On completely regular spaces, Fund. Math., 1954, 41, 105–106 Zbl0055.41304
  23. [23] Mrówka S., Some set-theoretic constructions in topology, Fund. Math., 1977, 94(2), 83–92 Zbl0348.54017
  24. [24] Mujica J., Separable quotients of Banach spaces, Rev. Mat. Univ. Complut. Madrid, 1997, 10(2), 299–330 Zbl0908.46007
  25. [25] Nissenzweig A., w* sequential convergence, Israel J. Math., 1975, 22(3–4), 266–272 http://dx.doi.org/10.1007/BF02761594 
  26. [26] Pełczynski A., Projections in certain Banach spaces, Studia Math., 1960, 19, 209–228 Zbl0104.08503
  27. [27] Plebanek G., A construction of a Banach space C(K) with few operators, Topology Appl., 2004, 143, 217–239 http://dx.doi.org/10.1016/j.topol.2004.03.001 Zbl1064.46011
  28. [28] Rosenthal H.P., On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math., 1970, 37, 13–36 Zbl0227.46027
  29. [29] Rudin W., Functional Analysis, 2nd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991 
  30. [30] Saxon S.A., Wilansky A., The equivalence of some Banach space problems, Colloq. Math., 1977, 37(2), 217–226 Zbl0373.46027
  31. [31] Semadeni Z., Banach Spaces of Continuous Functions. I, Monogr. Mat., 55, Polish Scientific Publishers, Warszawa, 1971 Zbl0225.46030
  32. [32] Sliwa W., (LF)-Spaces and the Separable Quotient Problem, Ph.D. thesis, Adam Mickiewicz University, Poznan, 1996 (in Polish) 
  33. [33] Valdivia M., Resolutions of the identity in certain Banach spaces, Collect. Math., 1988, 39(2), 127–140 Zbl0718.46006
  34. [34] Vašák L., On one generalization of weakly compactly generated Banach spaces, Studia Math., 1981, 70(1), 11–19 Zbl0376.46012
  35. [35] Walker R.C., The Stone-Čech Compactification, Ergeb. Math. Grenzgeb., 83, Springer, Berlin-Heidelberg-New York, 1974 Zbl0292.54001
  36. [36] Wójtowicz M., Effective constructions of separable quotients of Banach spaces, Collect. Math., 1997, 48(4–6), 809–815 Zbl0903.46016
  37. [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. [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 Zbl1031.46009
  39. [39] Zizler V., Nonseparable Banach Spaces, In: Handbook of the Geometry of Banach Spaces, 2, North-Holland, Amsterdam, 2003, 1743–1816 http://dx.doi.org/10.1016/S1874-5849(03)80048-7 Zbl1041.46009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.