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

TL;DR

The paper investigates the consistency of the non-existence of universal Banach spaces of certain uncountable densities that embed all spaces with specific smoothness or dual space properties, highlighting limitations in universality.

Contribution

It proves the consistency that no universal Banach space of density w_1 or continuum exists for spaces with uniformly Gâteaux differentiable renormings, extending previous results.

Findings

No universal Banach space of density w_1 or continuum exists under certain set-theoretic assumptions.

Certain compact spaces cannot be embedded into classical Banach spaces like l_infinity/c_0.

The results depend on set-theoretic assumptions beyond ZFC.

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.