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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11390-5

Published electronically:
August 10, 2012

MathSciNet review:
3008874

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Abstract: For being the first uncountable cardinal or being the cardinality of the continuum , we prove that it is consistent that there is no Banach space of density 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 that have a uniformly Gâteaux differentiable renorming. Our result implies, in particular, that may not map continuously onto a compact subset of a Hilbert space with the weak topology of density or and that a space for some uniform Eberlein compact space may not embed isomorphically into .

**1.**D. Amir and J. Lindenstrauss,*The structure of weakly compact sets in Banach spaces*. Ann. of Math. (2) 88 (1968) 35-46. MR**0228983 (37:4562)****2.**S. Argyros and Y. Benyamini,*Universal WCG Banach spaces and universal Eberlein compacts.*Israel J. Math. 58 (1987), no. 3, 305-320. MR**917361 (89d:46016)****3.**M. Bell,*Universal uniform Eberlein compact spaces*. Proc. Amer. Math. Soc. 128 (2000), no. 7, 2191-2197. MR**1676311 (2000m:54019)****4.**M. Bell and W. Marciszewski,*Universal spaces for classes of scattered Eberlein compact spaces*. J. Symbolic Logic 71 (2006), no. 3, 1073-1080. MR**2251557 (2007g:46012)****5.**J. Clarkson,*Uniformly convex spaces*, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396-414. MR**1501880****6.**Y. Benyamini, M. Rudin and M. Wage,*Continuous images of weakly compact subsets of Banach spaces.*Pacific J. Math. 70 (1977), no. 2, 309-324. MR**0625889 (58:30065)****7.**C. Brech and P. Koszmider,*On universal Banach spaces of density continuum*. To appear in Israel J. Math.**8.**M. Džamonja and L. Soukup,*Some Notes*, circulated notes, 2010.**9.**M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucia, J. Pelant, and V. Zizler,*Functional analysis and infinite-dimensional geometry*, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8, Springer-Verlag, New York, 2001. MR**1831176 (2002f:46001)****10.**M. Fabian, G. Godefroy, and V. Zizler,*The structure of uniformly Gateaux smooth Banach spaces*. Israel J. Math. 124 (2001), 243-252. MR**1856517 (2002g:46015)****11.**K. Kunen,*Set Theory. An introduction to independence proofs*, Studies in Logic and the Foundations of Mathematics, 102, North-Holland, Amsterdam, 1980. MR**597342 (82f:03001)****12.**Z. Semadeni,*Banach spaces of continuous functions*, Monografie Matematyczne, Tom 55, Państwowe Wydawnictwo Naukowe, Warsaw, 1971. MR**0296671 (45:5730)****13.**S. Todorcevic,*Embedding function spaces into*, J. Math. Anal. Appl. 384 (2011), no. 2, 246-251. MR**2825178**

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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

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:
piotr.math@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-2012-11390-5

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.