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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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