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On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts

Authors: Christina Brech and Piotr Koszmider
Journal: Proc. Amer. Math. Soc. 141 (2013), 1267-1280
MSC (2010): Primary 46B26; Secondary 03E35, 46B03
Published electronically: August 10, 2012
MathSciNet review: 3008874
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Abstract: For $ \kappa $ being the first uncountable cardinal $ \omega _1$ or $ \kappa $ being the cardinality of the continuum $ \mathfrak{c}$, we prove that it is consistent that there is no Banach space of density $ \kappa $ in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gâteaux differentiable renorming or, equivalently, whose dual unit ball with the weak$ ^*$ topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, and V. Zizler which says that assuming the continuum hypothesis, there is a universal space for all Banach spaces of density $ \kappa =\mathfrak{c}=\omega _1$ that have a uniformly Gâteaux differentiable renorming. Our result implies, in particular, that $ \beta \mathbb{N}\setminus \mathbb{N}$ may not map continuously onto a compact subset of a Hilbert space with the weak topology of density $ \kappa =\omega _1$ or $ \kappa =\mathfrak{c}$ and that a $ C(K)$ space for some uniform Eberlein compact space $ K$ may not embed isomorphically into $ \ell _\infty /c_0$.

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Additional Information

Christina Brech
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970, São Paulo, Brazil

Piotr Koszmider
Affiliation: Institute of Mathematics, Technical University of Łódź, ul. Wólczańska 215, 90-924 Łódź, Poland
Address at time of publication: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland

Received by editor(s): March 14, 2011
Received by editor(s) in revised form: August 5, 2011
Published electronically: August 10, 2012
Additional Notes: The first author was partially supported by FAPESP (2010/12638-1) and Pró-reitoria de Pesquisa USP (10.1.24497.1.2).
The second author was partially supported by Polish Ministry of Science and Higher Education research grant N N201 386234.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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