Universal WCG Banach spaces and universal Eberlein compacts.

*(English)*Zbl 0637.46010A 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.

Y. Benyamini, M. E. Rudin and 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 \((cof(\tau)=\omega)\).

Y. Benyamini, M. E. Rudin and 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 \((cof(\tau)=\omega)\).

Reviewer: J.Vaníček

##### MSC:

46A50 | Compactness in topological linear spaces; angelic spaces, etc. |

46B10 | Duality and reflexivity in normed linear and Banach spaces |

54D30 | Compactness |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |

##### Keywords:

topological weight; confinal cardinal numbers; Eberlein compact; weakly compactly generated
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\textit{S. A. Argyros} and \textit{Y. Benyamini}, Isr. J. Math. 58, 305--320 (1987; Zbl 0637.46010)

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