×

Order isomorphisms on function spaces. (English) Zbl 1295.46019

Let \(X\) be a compact Hausdorff space. A subspace \(A\) of continuous functions is said to precisely separate points from closed sets if, given a closed set \(F \subset X\) and \(x \notin F\), there exists an \(f \in A\), \(0\leq f \leq 1\), \(f = 0\) on \(F\) and \(f(x) = 1\). For compact spaces \(X,Y\) and subspaces \(A \subset C(X)\) and \(B \subset C(Y)\), containing constants and precisely separating points from closed sets, the authors show that any order preserving linear onto isomorphism \(T: A \rightarrow B\) is of the form \(T(f) = T(1)f \circ h^{-1}\) for a surjective homeomorphism \(h: X \rightarrow Y\). The proof follows a standard Banach-Stone theorem-type argument (see [E. Behrends, M-structure and the Banach-Stone theorem. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0436.46013)]) by analysing the zero sets.
The paper also contains an analysis of these aspects for metric and completely regular topological spaces.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions

Citations:

Zbl 0436.46013
PDFBibTeX XMLCite
Full Text: DOI arXiv