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Valdivia compacta and subspaces of \(C (K)\) spaces. (English) Zbl 0983.46020

For a set \(\Gamma\) put \(\Sigma (\Gamma)=\{x\in \mathbb R^\Gamma: \text{supp}x \) is countable\(\}\), where supp\(x=\{\gamma\in \Gamma : x(\gamma)\neq 0\}\). A compact Hausdorff space \(K\) is called a Corson compact if it is homeomorphic to a subset of \(\Sigma (\Gamma)\) for some \(\Gamma\), and a Valdivia compact if it is homeomorphic to a subset \(K'\) of \(\mathbb R^\Gamma\) for some \(\Gamma\) such that \(K'\cap \Sigma (\Gamma)\) is dense in \(K'\). Corson and Valdivia compact spaces play an important role in functional analysis. The author of the present paper [in Fundam. Math. 162, No. 2, 181-192 (1999; Zbl 1028.60001)] proved that if every continuous image of a compact \(K\) is Valdivia then \(K\) is Corson, and that there exists a Banach space \(X\) and a subspace \(Y\) of \(X\) such that \(B_{X^*}\) is Valdivia and \(B_{Y^*}\) is not. The first example of a non-Valdivia continuous image of a Valdivia compact space was given by M. Valdivia himself [Rev. Mat. Univ. Complutense Madrid 10, No. 1, 81-84 (1997; Zbl 0870.54025)]. M. Fabian and V. Zizler raised the following question: Is the dual unit ball of a Banach space Corson provided the dual unit ball of every subspace is Valdivia? The aim of the present paper is to give a partial answer to this question in the case of \(C(K)\) spaces. A compact Hausdorff space \(K\) is said to have property \((M)\) provided every Radon probability measure on \(K\) has separable support. The author proves (Theorem 1) that if \(K\) belongs to a specified class \(\mathcal{G}\Omega\) (for instance, if \(K\) has a dense subset of \(G_\delta\) points) then the following conditions are equivalent:
1. \(K\) is a Corson compact having the property \((M)\);
2. For every subspace \(Y\) of \(C(K)\) the dual unit ball \((B_{Y^*},w^*)\) is a Valdivia compact;
3. Every subspace of \(C(K)\) has a countably 1-norming Markuševič basis. If the topology of \(K\) has a basis of cardinality \(\aleph _1\) then these conditions are also equivalent to the following one:
4. Every subspace of \(C(K)\) has a projectional resolution of identity.
Theorem 2 contains similar equivalence results for a Corson compact (regardless of the property \((M)\)) using only subspaces of \(C(K)\) of the form \(C(L)\), where \(L\) is a continuous image of \(K\).

MSC:

46B26 Nonseparable Banach spaces
46B50 Compactness in Banach (or normed) spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
54C05 Continuous maps
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