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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9939-05-07799-3
PII: S 0002-9939(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.