On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts
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- by Christina Brech and Piotr Koszmider PDF
- Proc. Amer. Math. Soc. 141 (2013), 1267-1280 Request permission
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 $\ell _\infty /c_0$.References
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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
- MR Author ID: 792312
- Email: christina.brech@gmail.com
- 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
- Email: \texttt{piotr.math@gmail.com}
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1267-1280
- MSC (2010): Primary 46B26; Secondary 03E35, 46B03
- DOI: https://doi.org/10.1090/S0002-9939-2012-11390-5
- MathSciNet review: 3008874