Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Corson compacta and embeddings of $ C(K)$ spaces


Authors: Witold Marciszewski and Grzegorz Plebanek
Journal: Proc. Amer. Math. Soc. 138 (2010), 4281-4289
MSC (2010): Primary 46B26, 46E15; Secondary 46E27
DOI: https://doi.org/10.1090/S0002-9939-2010-10403-3
Published electronically: May 17, 2010
MathSciNet review: 2680054
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate properties of those compact spaces $ K$ for which the Banach space $ C(K)$ can be isomorphically embedded into a space $ C(L)$, where $ L$ is Corson compact. We show that in such a case $ K$ must be Corson compact provided $ K$ has some additional measure-theoretic property. The result is applicable to Rosenthal compacta and several other classes of compact spaces $ K$.


References [Enhancements On Off] (What's this?)

  • 1. S. Argyros, S. Mercourakis, S. Negrepontis, Functional analytic properties of Corson-compact spaces, Studia Math. 89 (1988), 197-228. MR 956239 (90e:46020)
  • 2. G. Borodulin-Nadzieja, Measures on minimally generated Boolean algebras, Topology Appl. 154 (2007), 3107-3124. MR 2364639 (2009c:28008)
  • 3. J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets of measurable functions, Amer. J. Math. 100 (1978), 845-886. MR 509077 (80b:54017)
  • 4. J. Diestel, Sequences and series in Banach spaces, Springer, 1984. MR 737004 (85i:46020)
  • 5. M. Džamonja, G. Plebanek, Precalibre pairs of measure algebras, Topology Appl. 144 (2004), 67-94. MR 2097129 (2005j:03041)
  • 6. M. Džamonja, G. Plebanek, On Efimov spaces and Radon measures, Topology Appl. 154 (2007), 2063-2072. MR 2324916 (2008j:54028)
  • 7. M. Džamonja, G. Plebanek, Strictly positive measures on Boolean algebras, J. Sym. Logic 73 (2008), 1416-1432. MR 2467227 (2010b:03077)
  • 8. E. Medina Galego, On isomorphic classification of $ C({\bf 2}^{\rm m} \oplus [0, \alpha])$, Fund. Math. 204 (2009), 87-95. MR 2507691
  • 9. G. Godefroy, Compacts de Rosenthal, Pacific J. Math. 91 (1980), 293-306. MR 615679 (82f:54030)
  • 10. J. Hagler, On the structure of $ S$ and $ C(S)$ for $ S$ dyadic, Trans. Amer. Math. Soc. 213 (1975), 415-428. MR 0388062 (52:8899)
  • 11. R. Haydon, On dual $ L^{1}$-spaces and injective bidual Banach spaces, Israel J. Math. 31 (1978), 142-152. MR 516250 (80e:46013)
  • 12. O. Kalenda, Valdivia compact spaces in topology and Banach space theory, Extracta Math. 15 (2000), 1-85. MR 1792980 (2001k:46024)
  • 13. P. Koszmider, The interplay between compact spaces and the Banach spaces of their continuous functions, in: Open Problems in Topology II, E. Pearl (ed.), Elsevier, 2007. MR 2367385 (2008j:54001)
  • 14. K. Kunen, A compact L-space under CH, Topology Appl. 12 (1981), 283-287. MR 623736 (82h:54065)
  • 15. K. Kunen, J. van Mill, Measures on Corson compact spaces, Fund. Math. 147 (1995), 61-72. MR 1330107 (96c:54040)
  • 16. W. Marciszewski, Rosenthal compacta, in: Encyclopedia of General Topology, Elsevier, 2004, 142-144. MR 2049453 (2005d:54001)
  • 17. W. Marciszewski, G. Plebanek, On measures on Rosenthal compacta, preprint (2010).
  • 18. S. Mercourakis, Some remarks on countably determined measures and uniform distribution of sequences, Monats. Math. 121 (1996), 79-101. MR 1375642 (97j:28029)
  • 19. S. Negrepontis, Banach spaces and topology, in: Open problems in topology, J. van Mill and G.M. Reed (eds.), Chapter 23, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636 (92c:54001)
  • 20. G. Plebanek, On some properties of Banach spaces of continuous functions, Seminaire d'Initiation à l'Analyse (G. Choquet et al.), 31 (1991/92), Université Paris VI.
  • 21. G. Plebanek, Nonseparable Radon measures and small compact spaces, Fund. Math. 153 (1997), 25-40. MR 1450994 (98m:28025)
  • 22. G. Plebanek, Approximating Radon measures on first-countable compact spaces, Colloq. Math. 86 (2000), 15-23. MR 1799884 (2002a:28014)
  • 23. G. Plebanek, Convex Corson compacta and Radon measures, Fund. Math. 175 (2002), 143-154. MR 1969632 (2004d:28027)
  • 24. R. Pol, Note on the spaces of regular probability measures whose topology is determined by countable subsets, Pacific J. Math. 100 (1982), 185-201. MR 661448 (83g:54024)
  • 25. M. Talagrand, Pettis integral and measure theory. Mem. Amer. Math. Soc. 307 (1984). MR 756174 (86j:46042)
  • 26. S. Todorčević, Compact sets of the first Baire class, J. Amer. Math. Soc. 12 (1999), 1179-1212. MR 1685782 (2000d:54028)
  • 27. S. Todorčević, Chain-condition methods in topology, Topology Appl. 101 (2000), 45-82. MR 1730899 (2001a:54055)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B26, 46E15, 46E27

Retrieve articles in all journals with MSC (2010): 46B26, 46E15, 46E27


Additional Information

Witold Marciszewski
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02–097 Warszawa, Poland
Email: wmarcisz@mimuw.edu.pl

Grzegorz Plebanek
Affiliation: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: grzes@math.uni.wroc.pl

DOI: https://doi.org/10.1090/S0002-9939-2010-10403-3
Received by editor(s): September 11, 2009
Received by editor(s) in revised form: January 24, 2010
Published electronically: May 17, 2010
Additional Notes: Research of the first author was partially supported by MNiSW Grant No. N N201 382034.
The second author was partially supported by grant 2191/W/IM/09 from the University of Wrocław.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society