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), 12671280
MSC (2010):
Primary 46B26; Secondary 03E35, 46B03
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.
Spiros
A. Argyros and Yoav
Benyamini, Universal WCG Banach spaces and universal Eberlein
compacts, Israel J. Math. 58 (1987), no. 3,
305–320. MR
917361 (89d:46016), 10.1007/BF02771694
 3.
M.
Bell, Universal uniform Eberlein compact
spaces, Proc. Amer. Math. Soc.
128 (2000), no. 7,
2191–2197. MR 1676311
(2000m:54019), 10.1090/S0002993900054034
 4.
Murray
Bell and Witold
Marciszewski, Universal spaces for classes of scattered Eberlein
compact spaces, J. Symbolic Logic 71 (2006),
no. 3, 1073–1080. MR 2251557
(2007g:46012), 10.2178/jsl/1154698593
 5.
James
A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. MR
1501880, 10.1090/S00029947193615018804
 6.
Y.
Benyamini, M.
E. 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.
Marián
Fabian, Petr
Habala, Petr
Hájek, Vicente
Montesinos Santalucía, Jan
Pelant, and Václav
Zizler, Functional analysis and infinitedimensional geometry,
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8,
SpringerVerlag, New York, 2001. MR 1831176
(2002f:46001)
 10.
Marián
Fabian, Gilles
Godefroy, and Václav
Zizler, The structure of uniformly Gateaux smooth Banach
spaces, Israel J. Math. 124 (2001), 243–252. MR 1856517
(2002g:46015), 10.1007/BF02772620
 11.
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, NorthHolland Publishing Co., AmsterdamNew
York, 1980. An introduction to independence proofs. MR 597342
(82f:03001)
 12.
Zbigniew
Semadeni, Banach spaces of continuous functions. Vol. I,
PWN—Polish Scientific Publishers, Warsaw, 1971. Monografie
Matematyczne, Tom 55. MR 0296671
(45 #5730)
 13.
Stevo
Todorcevic, Embedding function spaces into
ℓ_{∞}/𝑐₀, J. Math. Anal. Appl.
384 (2011), no. 2, 246–251. MR 2825178
(2012k:54028), 10.1016/j.jmaa.2011.05.045
 1.
 D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces. Ann. of Math. (2) 88 (1968) 3546. 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, 305320. MR 917361 (89d:46016)
 3.
 M. Bell, Universal uniform Eberlein compact spaces. Proc. Amer. Math. Soc. 128 (2000), no. 7, 21912197. 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, 10731080. MR 2251557 (2007g:46012)
 5.
 J. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396414. 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, 309324. 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 infinitedimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8, SpringerVerlag, 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), 243252. MR 1856517 (2002g:46015)
 11.
 K. Kunen, Set Theory. An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, 102, NorthHolland, 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, 246251. 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, 05314970, São Paulo, Brazil
Email:
christina.brech@gmail.com
Piotr Koszmider
Affiliation:
Institute of Mathematics, Technical University of Łódź, ul. Wólczańska 215, 90924 Łódź, Poland
Address at time of publication:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00956 Warszawa, Poland
Email:
piotr.math@gmail.com
DOI:
http://dx.doi.org/10.1090/S000299392012113905
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 (2010/126381) and Próreitoria 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.
