On decompositions of Banach spaces of continuous functions on Mrówka’s spaces
HTML articles powered by AMS MathViewer
- by Piotr Koszmider
- Proc. Amer. Math. Soc. 133 (2005), 2137-2146
- DOI: https://doi.org/10.1090/S0002-9939-05-07799-3
- Published electronically: February 25, 2005
- PDF | Request permission
Abstract:
It is well known that if $K$ is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space $C(K)$ of continuous functions on $K$ has complemented copies of $c_{0}$, i.e., $C(K)\sim c_{0} \oplus X\sim c_{0}\oplus c_{0} \oplus X\sim c_{0}\oplus C(K)$. We address the question if this could be the only type of decompositions of $C(K)\not \sim c_{0}$ into infinite-dimensional summands for $K$ infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin’s axiom we construct an example of Mrówka’s space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.References
- D. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals, Israel J. Math. 27 (1977), no. 1, 64–92. MR 440349, DOI 10.1007/BF02761606
- Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
- Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
- W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
- W. B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219–230. MR 417760, DOI 10.1007/BF02882239
- P. Koszmider; Banach spaces of large densities but few operators. Preprint.
- Piotr Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004), no. 1, 151–183. MR 2091683, DOI 10.1007/s00208-004-0544-z
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Functional Analysis 8 (1971), 225–249. MR 0291772, DOI 10.1016/0022-1236(71)90011-5
- N. N. Luzin, On subsets of the series of natural numbers, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 403–410 (Russian). MR 0021576
- A. Miller; private notes, 2003.
- Aníbal Moltó, On a theorem of Sobczyk, Bull. Austral. Math. Soc. 43 (1991), no. 1, 123–130. MR 1086724, DOI 10.1017/S0004972700028835
- S. Mrówka, Some set-theoretic constructions in topology, Fund. Math. 94 (1977), no. 2, 83–92. MR 433388, DOI 10.4064/fm-94-2-83-92
- A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
- A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $C(\Omega )$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222. MR 107806, DOI 10.4064/sm-18-2-211-222
- Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
- Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, Monografie Matematyczne, Tom 55, PWN—Polish Scientific Publishers, Warsaw, 1971. MR 0296671
- Saharon Shelah, A Banach space with few operators, Israel J. Math. 30 (1978), no. 1-2, 181–191. MR 508262, DOI 10.1007/BF02760838
- Saharon Shelah and Juris Steprāns, A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 101–105. MR 958051, DOI 10.1090/S0002-9939-1988-0958051-9
- H. M. Wark, A non-separable reflexive Banach space on which there are few operators, J. London Math. Soc. (2) 64 (2001), no. 3, 675–689. MR 1865556, DOI 10.1112/S0024610701002393
Bibliographic Information
- Piotr Koszmider
- Affiliation: Departamento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, São Paulo, Sp CEP: 05315-970, Brazil
- Email: piotr@ime.usp.br
- Received by editor(s): July 24, 2003
- Received by editor(s) in revised form: April 15, 2004
- Published electronically: February 25, 2005
- Additional Notes: The author acknowledges support from CNPQ, Processo Número 300369/01-8, from FAPESP, Processo Número 02/03677-7 and from Centre de Recerca Matemática at Universidad Autonoma de Barcelona.
- Communicated by: Alan Dow
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2137-2146
- MSC (2000): Primary 03E50, 46E15, 54G12
- DOI: https://doi.org/10.1090/S0002-9939-05-07799-3
- MathSciNet review: 2137881