zbMATH — the first resource for mathematics

Universal WCG Banach spaces and universal Eberlein compacts. (English) Zbl 0637.46010
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.
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)
Full Text:
References:
 [1] D. Amir and J. Lindenstrauss,The structure of weakly compact sets in Banach spaces, Ann. of Math.88 (1968), 34–46. · Zbl 0164.14903 [2] Y. Benyamini, M. E. Rudin and M. Wage,Continuous images of weakly compact subsets of Banach spaces, Pac. J. Math.70 (1977), 309–324. · Zbl 0374.46011 [3] Y. Benyamini and T. Starbird,Embedding weakly compact sets into Hilbert space, Israel J. Math.23 (1976), 137–141. · Zbl 0325.46023 [4] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński,Factoring weakly compact operators, J. Func. Anal.17 (1974), 311–327. · Zbl 0306.46020 [5] S. P. Gulko,On the structure of spaces of continuous functions and their complete paracompactness, Russian Math. Surveys34:6 (1979), 36–44 (English translation). · Zbl 0446.46014 [6] A. G. Leiderman and G. A. Sokolov,Adequate families of sets and Corson compacts, Comm. Math. Univ. Carolinae25 (1984), 233–245. · Zbl 0586.54022 [7] J. Lindenstrauss,Weakly compact sets, their topological properties and the Banach spaces they generate, Ann. of Math. Studies69, Princeton Univ. Press, 1972, pp. 235–273. · Zbl 0232.46019 [8] I. Namioka,Separate continuity and joint continuity, Pac. J. Math.51 (1974), 515–531. · Zbl 0294.54010 [9] W. Szlenk,The non-existence of a separable reflexive Banach space, universal for all separable reflexive Banach spaces, Studia Math.30 (1968), 53–61. · Zbl 0169.15303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.