\(\mathbb{N}\)-compactness and weighted composition maps (Q2781275)

Let \(X\), \(Y\) be compact Hausdorff spaces and \(C(X)\) denote the space of all continuous mappings from \(X\) into a complete commutative field \(K\) with a valuation. A linear mapping \(T\) from \(C(X)\) into \(C(Y)\) is called a weighted composition map when there is a continuous map \(h\) from \(Y\) onto a dense subset of \(X\) and \(a\in C(Y)\) such that \((Tf)(y)= a(y) f(h(y))\) for all \(f\in C(X)\) and \(y\in Y\). The Stone-Banach theorem says that if \(K\) is the real or complex numbers and j\(T\) is an isometry onto \(C(Y)\), then \(T\) is a weighted composition map with \(h\) a homeomorphism onto \(X\) and \(|a(y)|= 1\) for all \(y\in Y\). The object of the present paper is to extend the Stone-Banach theorem by weakening the assumption that \(T\) be an isometry.NEWLINENEWLINENEWLINELet \(\text{val}(f)= \{|f(x)|: f(x)\neq 0\}\) for \(f\neq 0\) and \(\text{val}(0)= \{0\}\). The author calls \(T\) infrabounded-below in case there is a constant \(M> 0\) such that \(\text{inf\;val}(Tf)\geq M\text{ inf\;val}(f)\) for all \(f\in C(X)\), and he lets infra \(T\) denote the sup of all such \(M\). He shows that if \(X\), \(Y\) are \(0\)-dimensional, then every infrabounded-below map \(T\) is a weighted composition map. Furthermore, if \(T\) is surjective, then \(h\) is a homeomorphism onto \(X\) and \(|a(y)|\geq \text{infra }T\) for all \(y\in Y\). His results are proved in a somewhat more general setting where, among other things, the field \(K\) may have a non-Archimedean valuation. They rely on earlier work by the author and \textit{J. Martinez-Maurica} [Lect. Notes Math. 1454, 64-79 (1990; Zbl 0731.46043)].