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

Author: Pandelis Dodos
Journal: Trans. Amer. Math. Soc. 361 (2009), 6407-6428
MSC (2000): Primary 03E15, 46B03, 46B07, 46B15
Published electronically: June 5, 2009
MathSciNet review: 2538598
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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.

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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

Keywords: Non-universal spaces, strongly bounded classes, Schauder bases, $\mathcal {L}_\infty $-spaces.
Received by editor(s): October 18, 2007
Published electronically: June 5, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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