Universal WCG Banach spaces and universal Eberlein compacts (Q1099032)

A compact Hausdorff space is called an Eberlein compact if it is homeomorphic to a weakly compact subset of some Banach space. A Banach space is called weakly compactly generated if it is spanned by some weakly compact subset. \textit{Y. Benyamini}, \textit{M. E. Rudin} and \textit{M. Wage} in Pac. J. Math. 70, 309-329 (1977; Zbl 0374.46011, ask whether, for a given cardinal \(\tau\), there exists an Eberlein compact of the topological weight \(\tau\), universal for all Eberlein compacts of the weight \(\tau\) and whether there exists a weakly compactly generated space of density character \(\tau\), universal for all such spaces. In the paper the following results are proved: - For cardinals \(\tau\) satisfying either \(\tau^{\omega}=\tau\) or \(\tau =\omega_ 1\), there is no universal Eberlein compact of the weight \(\tau\) of universal weakly compactly generated space of the density character \(\tau\). - If \(\tau\) is a strong limit cardinal of countable confinality, there is an Eberlein compact U of the weight \(\tau\), universal in the sense that every Eberlein compact of the weight embeds as a retract of U and a weakly compactly generated space B of density character \(\tau\) universal in the sense that every such space is isometric to a norm one complemented subspace of B. - Under GCH universal Eberlein compacts or weakly compactly generated spaces exist for \(\tau\) if and only if \(\tau\) has countable confinality \((cof(\tau)=\omega)\).