Noncommutative Valdivia compacta

Marek Cúth

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 1, page 53-72
  • ISSN: 0010-2628

Abstract

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We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the dual unit ball of a Banach space is Corson provided the dual unit ball of every equivalent norm has a retractional skeleton. Another result to be mentioned is the following. Having a compact space K , we show that K is Corson if and only if every continuous image of K has a retractional skeleton. We also present some open problems in this area.

How to cite

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Cúth, Marek. "Noncommutative Valdivia compacta." Commentationes Mathematicae Universitatis Carolinae 55.1 (2014): 53-72. <http://eudml.org/doc/260834>.

@article{Cúth2014,
abstract = {We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the dual unit ball of a Banach space is Corson provided the dual unit ball of every equivalent norm has a retractional skeleton. Another result to be mentioned is the following. Having a compact space $K$, we show that $K$ is Corson if and only if every continuous image of $K$ has a retractional skeleton. We also present some open problems in this area.},
author = {Cúth, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {retractional skeleton; projectional skeleton; Valdivia compacta; Plichko spaces; retractional skeleton; projectional skeleton; Valdivia compacta; Plichko spaces},
language = {eng},
number = {1},
pages = {53-72},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Noncommutative Valdivia compacta},
url = {http://eudml.org/doc/260834},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Cúth, Marek
TI - Noncommutative Valdivia compacta
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 1
SP - 53
EP - 72
AB - We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the dual unit ball of a Banach space is Corson provided the dual unit ball of every equivalent norm has a retractional skeleton. Another result to be mentioned is the following. Having a compact space $K$, we show that $K$ is Corson if and only if every continuous image of $K$ has a retractional skeleton. We also present some open problems in this area.
LA - eng
KW - retractional skeleton; projectional skeleton; Valdivia compacta; Plichko spaces; retractional skeleton; projectional skeleton; Valdivia compacta; Plichko spaces
UR - http://eudml.org/doc/260834
ER -

References

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  1. Cúth M., 10.4064/fm219-3-1, Fund. Math. 219 (2012), 191–222. Zbl1270.46015MR3001239DOI10.4064/fm219-3-1
  2. Cúth M., 10.1016/j.jmaa.2013.09.020, J. Math. Anal. Appl. 411 (2014), 19–29; DOI: 10.1016/j.jmaa.2013.09.020. MR3118464DOI10.1016/j.jmaa.2013.09.020
  3. Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, 64, Longman Scientific and Technical, New York, 1993. Zbl0782.46019MR1211634
  4. Engelking R., General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. Zbl0684.54001MR1039321
  5. Hájek P., Montesinos V., Vanderwerff J., Zizler V., Biorthogonal Systems in Banach Spaces, CMS Books in Mathematics, 26, Springer, New York, 2008. Zbl1136.46001MR2359536
  6. Kalenda O., Valdivia compacta and subspaces of 𝒞 ( K ) spaces, Extracta Math. 14 (1999), no. 3, 355–371. MR1759476
  7. Kalenda O., Continuous images and other topological properties of Valdivia compacta, Fund. Math. 162 (1999), no. 2, 181–192. Zbl0989.54019MR1734916
  8. Kalenda O., Embedding the ordinal segment [ 0 , ω 1 ] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (1999), no. 4, 777–783. MR1756552
  9. Kalenda O., Valdivia compacta and equivalent norms, Studia Math. 138 (2000), 179–191. Zbl1073.46009MR1749079
  10. Kalenda O., A characterization of Valdivia compact spaces, Collect. Math. 51 (2000), no. 1, 59–81. Zbl0949.46004MR1757850
  11. Kalenda O., :, Valdivia compact spaces in topology and Banach space theory, Extracta Math. 15 (2000), no. 1, 1–85. MR1792980
  12. Kalenda O.F.K., 10.4064/cm92-2-3, Colloq. Math. 92 (2002), no. 2, 179–187. Zbl1029.46006MR1899436DOI10.4064/cm92-2-3
  13. Kubiś W., Michalewski H., 10.1016/j.topol.2005.09.010, Topology Appl. 153 (2006), 2560–2573. Zbl1138.54024MR2243734DOI10.1016/j.topol.2005.09.010
  14. Banakh T., Kubiś W., Spaces of continuous functions over Dugundji compacta, preprint, arXiv:math/0610795v2, 2008. 
  15. Kubiś W., 10.1016/j.jmaa.2008.07.006, J. Math. Anal. Appl. 350 (2009), no. 2, 758–776. Zbl1166.46008MR2474810DOI10.1016/j.jmaa.2008.07.006
  16. Kakol J., Kubiś W., López-Pellicer M., 10.1007/978-1-4614-0529-0, Developments in Mathematics, 24, Springer, New York, 2011. MR2953769DOI10.1007/978-1-4614-0529-0
  17. Kunen K., Set Theory, Studies in Logic and the Foundations of Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1983. Zbl0960.03033MR0756630

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