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On the density of Banach spaces $ C(K)$ with the Grothendieck property

Author: Christina Brech
Journal: Proc. Amer. Math. Soc. 134 (2006), 3653-3663
MSC (2000): Primary 03E35; Secondary 46B20
Published electronically: May 18, 2006
MathSciNet review: 2240680
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Abstract: Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that ``no nontrivial complemented subspace of a Grothendieck space $ C(K)$ is separable'' cannot be strengthened by replacing ``is separable'' by ``has density less than that of $ l_\infty$'', without using an additional set-theoretic assumption. Such a strengthening was proved by Haydon, Levy and Odell, assuming Martin's axiom and the negation of the continuum hypothesis. Moreover, our example shows that certain separation properties of Boolean algebras are quite far from the Grothendieck property.

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Christina Brech
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, CEP: 05315-970, São Paulo, Brasil

Received by editor(s): September 26, 2004
Received by editor(s) in revised form: June 14, 2005
Published electronically: May 18, 2006
Additional Notes: The author was supported by a scholarship from FAPESP (02/04531-6). This paper is part of the author’s M.A. thesis at the University of São Paulo, under the guidance of Professor Piotr Koszmider. The author thanks him for his guidance and assistance during the preparation of this paper.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.