On classes of Banach spaces admitting “small” universal spaces

Dodos, Pandelis

doi:10.1090/S0002-9947-09-04913-7

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.

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