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On decompositions of Banach spaces of continuous functions on Mrówka's spaces


Author: Piotr Koszmider
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
Published electronically: February 25, 2005
MathSciNet review: 2137881
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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.


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  • 1. D. Alspach, Y. Benyamini; Primariness of spaces of continuous functions on ordinals; Israel J. Math. vol 27, No 1, 1977, pp. 64 - 92. MR 0440349 (55:13224)
  • 2. J. Diestel; Sequences and series in Banach spaces; Springer-Verlag 1984. MR 0737004 (85i:46020)
  • 3. E. van Douwen; The Integers and Topology; Handbook of Set-theoretic Topology; eds K. Kunen, J. Vaughan, North Holland 1984; pp. 111 - 167. MR 0776622 (87f:54008)
  • 4. W. T. Gowers, B. Maurey; The unconditional basic sequence problem; Journal A. M. S. 6 (1993), pp. 851-874. MR 1201238 (94k:46021)
  • 5. W. Johnson, J. Lindenstrauss; Some remarks on weakly compactly generated Banach spaces; Israel J. Math. 17, 1974, pp. 219 - 230. and Israel J. Math. 32 (1979), no. 4, pp. 382 - 383. MR 0417760 (54:5808); MR 0571092 (81g:46015)
  • 6. P. Koszmider; Banach spaces of large densities but few operators. Preprint.
  • 7. P. Koszmider; Banach spaces of continuous functions with few operators; Math. Annalen. 330, 2004, pp. 151 - 183. MR 2091683
  • 8. K. Kunen; Set Theory. An Introduction to Independence Proofs; North Holland, 1980. MR 0597342 (82f:03001)
  • 9. Y. Lindenstrauss, A. Pe\lczynski; Contributions to the theory of the classical Banach spaces; J. Funct. Anal. 8, 1971, pp. 225 - 249. MR 0291772 (45:863)
  • 10. N. Luzin; On subsets of the series of natural numbers; Izv. Akad. Nauk SSSR, Ser. Mat.., 11, pp. 403 - 411. MR 0021576 (9:82c)
  • 11. A. Miller; private notes, 2003.
  • 12. A. Moltó; On a theorem of Sobczyk; Bull. Austral. Math. Soc. 43, 1991, 123 - 130. MR 1086724 (92d:46043)
  • 13. S. Mrówka; Some set-theoretic constructions in topology; Fund. Math. vol. XCIV, 1977, pp. 83 - 92. MR 0433388 (55:6364)
  • 14. A. Pe\lczynski; Projections in certain Banach spaces; Studia Math. vol. XIX, 1960, pp. 209 - 228. MR 0126145 (23:A3441)
  • 15. A. Pe\lczynski, Z. Semadeni; Spaces of continuous functions (III) (Spaces $C(\Omega )$ for $\Omega $ without perfect subsets); Studia Math. 18, 1959, pp. 211 - 222. MR 0107806 (21:6528)
  • 16. H. Rosenthal; On relatively disjoint families of measures with some applications to Banach space theory; Studia Math. 37, (1970), pp. 13-36. MR 0270122 (42:5015)
  • 17. Z. Semadeni; Banach spaces of continuous functions; Panstwowe Wydawnictwo Naukowe, 1971. MR 0296671 (45:5730)
  • 18. S. Shelah; A Banach space with few operators; Israel J. Math. 30 (1978), pp. 181-191. MR 0508262 (80b:46033)
  • 19. S. Shelah, J. Steprans; A Banach space on which there are few operators; Proc. Amer. Math. Soc. 104 (1988), pp. 101-105. MR 0958051 (90a:46047)
  • 20. H. Wark; A non-separable reflexive Banach space on which there are few operators. J. London Math. Soc. (2) 64 (2001), no. 3, pp. 675 - 689. MR 1865556 (2003a:46031)

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Additional 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

DOI: https://doi.org/10.1090/S0002-9939-05-07799-3
Keywords: Banach spaces of continuous functions, few operators, scattered spaces, almost disjoint families
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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