Valdivia compact spaces, renormings and Asplund spaces
by
O. Kalenda
Dept. Math. Anal., Charles University Prague, Czech Republic

Valdivia compacta form a class of compact spaces which plays an important role in the theory of Banach spaces. It was studied for example by S.Argyros, S.Mercourakis, S.Negrepontis, M.Valdivia, R.Deville and G.Godefroy.

Let us call a Banach space X 1-Plichko, if there is a linear one-to-one weak* continuous mapping T:X^* --> R^I such that { x^* in X^* : supp T(x^*) is countable} is a 1-norming subspace of X^*. Each 1-Plichko space has Valdivia dual unit ball and a projectional resolution of the identity (by a result of M.Valdivia). Conversely, by M.Fabian, G.Godefroy and V.Zizler, each space of density \aleph₁ with a projective resolution of the identity (PRI) is 1-Plichko.

We obtained the following result. Let X be a Banach space such that the dual unit ball is Valdivia for every equivalent norm on X. Then X is weakly Lindelöf determined (WLD). In particular, within spaces of density \aleph₁ we get a converse to the Amir-Lindenstrauss theorem. A Banach space of density \aleph₁ is WLD provided it has PRI with respect to each equivalent norm.

Further, we say that a Banach space X belongs to the class (T) if there is a linear one-to-one weak* continuous mapping T:X^** --> R^I such that supp T(x) is countable for every x in X. If X belongs to (T), then X^* is 1-Plichko in every equivalent dual norm. It can be proved that the class (T) form a subclass of Asplund spaces which is closed to subspaces and quotients. By R.Deville and G.Godefroy each Asplund space of density \aleph₁ belongs to (T). We extend this result to all Asplund spaces with "\aleph₂-Corson" dual unit ball. Further, we give an example of Asplund space of density \aleph₂ with a non-Valdivia bidual unit ball, in answer to a question by J.Orihuela.

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Valdivia compact spaces, renormings and Asplund spaces

Date received: January 17, 2000