On weakly countably determined Banach spaces

Abstract:

For a topological space $X$, let ${C_1}(X)$ denote the Banach space of all bounded functions $f:X \to {\mathbf {R}}$ such that for every $\varepsilon > 0$ the set $\{ x \in X:|f(x)| \geqslant \varepsilon \}$ 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 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 $K$-analytic Banach space); the characterization of WCD (resp. weakly $K$-analytic) Banach spaces $E$, using one-to-one operators from ${E^{\ast }}$ into ${C_1}(\Sigma ’ \times K)$ (resp. ${C_1}(\Sigma \times K)$); and the existence of equivalent "good" norms on $E$ and ${E^{\ast }}$ simultaneously.