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On weakly countably determined Banach spaces. (English) Zbl 0621.46018
For a topological X, let \(C_ 1(X)\) denote the Banach space of all bounded functions f:X\(\to {\mathbb{R}}\) such for every \(\epsilon >0\) the set \(\{\) \(x\in X:| f(x)| \geq \epsilon \}\) is closed and discrete in X, endowed with the supremum norm. The main theorem is the following:
Let L be a weakly countably determined subset of a Banach space; then there exist a subset \(\Sigma\) ’ of the Baire space \(\Sigma\), a compact space K, and a bounded linear one-to-one operator \(T:C(L)\to C_ 1(\Sigma '\times K)\) that is pointwise to pointwise continuous. In the case where L is weakly \({\mathcal K}\)-analytic \(\Sigma\) ’ can be replaced by \(\Sigma\). This theorem is connected with the basic result of Amir- Lindenstrauss on WCG Banach spaces and has corresponding consequences such as: the representation of Gulko (resp. Talagrand) compact spaces as pointwise compact subsets of \(C_ 1(\Sigma '\times K)\) (resp. \(C_ 1(\Sigma \times K))\) (a compact space \(\Omega\) is called Gulko or Talagrand compact if C(\(\Omega)\) is WCD or a weakly \({\mathcal K}\)-analytic Banach space); and the characterization of WCD (resp. weakly \({\mathcal K}\)- analytic) Banach spaces E, using one-to-one operators from \(E^*\) into \(C_ 1(\Sigma '\times K)\) (resp. \(C_ 1(\Sigma \times K))\); It is also proved that any space of the form \(C_ 1(\Sigma '\times K)\) admits an equivalent strictly convex norm, which is moreover pointwise lower semicontinuous; combining this result with the main theorem, we obtain equivalent ”good” norms on E and \(E^*\) simultaneously.

MSC:
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46A50 Compactness in topological linear spaces; angelic spaces, etc.
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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