This is an announcement for the paper "On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts" by Christina Brech and Piotr Koszmider. Abstract: For k being the first uncountable cardinal w_1 or k being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density k in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly G\^ateaux 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, V. Zizler that assuming the continuum hypothesis, there is a universal space for all Banach spaces of density k=c=w_1 which have a uniformly G\^ateaux differentiable renorming. Our result implies, in particular, that \beta N-N may not map continuously onto a compact subset of a Hilbert space with the weak topology of density k=w_1 or k=c and that a C(K) space for some uniform Eberlein compact space K may not embed isomorphically into l_\infty/c_0. Archive classification: math.FA math.GN math.LO Mathematics Subject Classification: Primary 46B26, Secondary 03E35, 46B03 Submitted from: piotr.math@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.4259 or http://arXiv.org/abs/1103.4259