On Weakly Countably Determined Banach Spaces

• Published in: Transactions of the American Mathematical Society 1987, Vol.300 (1), p.307-327
• Main Author: Mercourakis, Sophocles
• Format: Article
• Language: English
• Subjects: Banach space; General topology; Homeomorphism; Mathematical theorems; Metric spaces; Separable spaces; Topological compactness; Topological spaces; Topological theorems
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/s0002-9947-1987-0871678-1

For a topological space X, let C1(X) denote the Banach space of all bounded functions f: X → R such that for every$\varepsilon > 0$the set {x ∈ X: |f(x)| ≥ ε} 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 Σ' of the Baire space Σ, a compact space K, and a bounded linear one-to-one operator T: C(L) → C1(Σ × K) that is pointwise to pointwise continuous. In the case where L is weakly analytic, Σ' can be replaced by Σ. 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$C_1(\Sigma' \times K) (\operatorname{resp.} C_1(\Sigma \times K))$(a compact space Ω is called Gulko or Talagrand compact if C(Ω) is WCD or a weakly K-analytic Banach space); the characterization of WCD (resp. weakly K-analytic) Banach spaces E, using one-to-one operators from E*into C1(Σ' × K) (resp. C1(Σ × K)); and the existence of equivalent "good" norms on E and E*simultaneously.