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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

On classes of Banach spaces admitting ``small'' universal spaces

Author(s): Pandelis Dodos
Journal: Trans. Amer. Math. Soc. 361 (2009), 6407-6428.
MSC (2000): Primary 03E15, 46B03, 46B07, 46B15
Posted: June 5, 2009
MathSciNet review: 2538598
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Abstract | References | Similar articles | Additional information

Abstract: We characterize those classes $ \mathcal{C}$ of separable Banach spaces admitting a separable universal space $ Y$ (that is, a space $ Y$ containing, up to isomorphism, all members of $ \mathcal{C}$) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $ \mathrm{NU}$ of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture from Argyros and Dodos (2007). Our approach is based, among others, on a construction of $ \mathcal{L}_\infty$-spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family $ \{Y_\xi:\xi<\omega_1\}$ of separable, non-universal, $ \mathcal{L}_\infty$-spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well.

References:

[AD]
S. A. Argyros and P. Dodos, Genericity and amalgamation of classes of Banach spaces, Adv. Math., 209 (2007), 666-748. MR 2296312 (2008f:46014)

[Bos1]
B. Bossard, Théorie descriptive des ensembles en géométrie des espaces de Banach, Thèse, Univ. Paris VI, 1994.

[Bos2]
B. Bossard, A coding of separable Banach spaces. Analytic and co-analytic families of Banach spaces, Fund. Math., 172 (2002), 117-152. MR 1899225 (2003d:46016)

[Bou1]
J. Bourgain, On separable Banach spaces, universal for all separable reflexive spaces, Proc. AMS, 79 (1980), 241-246. MR 565347 (81f:46021)

[Bou2]
J. Bourgain, New classes of $ \mathcal{L}_p$ Spaces, Lecture Notes in Math., 889, Springer-Verlag, 1981. MR 639014 (83j:46028)

[BD]
J. Bourgain and F. Delbaen, A class of special $ \mathcal{L}_\infty$-spaces, Acta Math., 145 (1980), 155-176. MR 590288 (82h:46023)

[BP]
J. Bourgain and G. Pisier, A construction of $ \mathcal{L}_\infty$-spaces and related Banach spaces, Bol. Soc. Brasil. Mat., 14 (1983), 109-123. MR 756904 (86b:46021)

[DF]
P. Dodos and V. Ferenczi, Some strongly bounded classes of Banach spaces, Fund. Math., 193 (2007), 171-179. MR 2282714 (2008f:46016)

[Ga]
I. Gasparis, Operators on $ C[0,1]$ preserving copies of asymptotic $ \ell_1$ spaces, Math. Annalen, 333 (2005), 831-858. MR 2195147 (2007a:46006)

[J]
R. C. James, A separable somewhat reflexive Banach space with non-separable dual, Bull. AMS, 80 (1974), 738-743. MR 0417763 (54:5811)

[JRZ]
W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions and weaker structures in Banach spaces, Israel J. Math., 9 (1971), 488-506. MR 0280983 (43:6702)

[Ke]
A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer-Verlag, 1995. MR 1321597 (96e:03057)

[Ki]
S. V. Kisliakov, On spaces with ``small'' annihilators, Zap. Naucn. Sem. Leningrad. Otdel. Math. Institute Steklov (LOMI), 65 (1976), 192-195. MR 0500098 (58:17804)

[LP1]
J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $ \mathcal{L}_p$ spaces and their applications, Studia Math., 29 (1968), 275-326. MR 0231188 (37:6743)

[LP2]
J. Lindenstrauss and A. Pełczyński, Contributions to the theory of classical Banach spaces, J. Funct. Anal., 8 (1971), 225-249. MR 0291772 (45:863)

[LT]
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces vol. I: Sequence spaces, Ergebnisse, vol. 92, Springer, 1977. MR 0500056 (58:17766)

[Mi]
A. A. Milutin, Isomorphisms of spaces of continuous functions on compacts of power continuum, Tieoria Func. (Kharkov), 2 (1966), 150-156.

[OS]
E. Odell and Th. Schlumprecht, A separable reflexive space universal for the uniformly convex Banach spaces, Math. Annalen, 335 (2006), 901-916. MR 2232021 (2007f:46012)

[OSZ]
E. Odell, Th. Schlumprecht and A. Zsák, Banach spaces of bounded Szlenk index, Studia Math., 183 (2007), 63-97. MR 2360257

[P]
A. Pełczyński, Universal bases, Studia Math., 32 (1969), 247-268. MR 0241954 (39:3290)

[Pi]
G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math., 151 (1983), 181-208. MR 723009 (85m:46017)

[Ro1]
H. P. Rosenthal, On factors of $ C([0,1])$ with non-separable dual, Israel J. Math., 13 (1972), 361-378. MR 0388063 (52:8900)

[Ro2]
H. P. Rosenthal, A characterization of Banach spaces containing $ \ell_1$, Proc. Nat. Acad. Sci. USA, 71 (1974), 2411-2413. MR 0358307 (50:10773)

[Ro3]
H. P. Rosenthal, The Banach spaces $ C(K)$, Handbook of the Geometry of Banach Spaces, Volume 2, Edited by W. B. Johnson and J. Lindenstrauss, Elsevier, 2003. MR 1999613 (2004e:46001)

[Ts]
B. S. Tsirelson, Not every Banach space contains $ \ell_p$ or $ c_0$, Functional Anal. Appl., 8 (1974), 138-141. MR 0350378 (50:2871)

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Additional Information:

Pandelis Dodos
Affiliation: Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece
Email: pdodos@math.ntua.gr

DOI: 10.1090/S0002-9947-09-04913-7
PII: S 0002-9947(09)04913-7
Keywords: Non-universal spaces, strongly bounded classes, Schauder bases, $\mathcal {L}_\infty $-spaces.
Received by editor(s): October 18, 2007
Posted: June 5, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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