On classes of Banach spaces admitting “small” universal spaces
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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.References
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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
- Received by editor(s): October 18, 2007
- Published electronically: June 5, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 6407-6428
- MSC (2000): Primary 03E15, 46B03, 46B07, 46B15
- DOI: https://doi.org/10.1090/S0002-9947-09-04913-7
- MathSciNet review: 2538598