On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts

Abstract:

For $\kappa$ being the first uncountable cardinal $\omega _1$ or $\kappa$ being the cardinality of the continuum $\mathfrak {c}$, we prove that it is consistent that there is no Banach space of density $\kappa$ 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 $\kappa =\mathfrak {c}=\omega _1$ that have a uniformly Gâteaux differentiable renorming. Our result implies, in particular, that $\beta \mathbb {N}\setminus \mathbb {N}$ may not map continuously onto a compact subset of a Hilbert space with the weak topology of density $\kappa =\omega _1$ or $\kappa =\mathfrak {c}$ and that a $C(K)$ space for some uniform Eberlein compact space $K$ may not embed isomorphically into $\ell _\infty /c_0$.