Weighted Composition Operators on $C(X)$ and $\mathrm{Lip}_c(X,\alpha)$

Abstract

Let $A$ and $B$ be subalgebras of $C(X)$ and $C(Y)$, respectively, for some topological spaces $X$ and $Y$. An arbitrary map $T: A\rightarrow B$ is said to be multiplicatively range-preserving if for every $f,g\in A$, $(fg)(X)=(TfTg)(Y)$, and $T$ is said to be separating if $TfTg=0$ whenever $fg=0$. For a given metric space $X$ and $\alpha\in (0,1]$, let Lip$_c(X,\alpha)$ be the algebra of all complex-valued functions on $X$ satisfying the Lipschitz condition of order $\alpha$ on each compact subset of $X$. In this note we first investigate the general form of multiplicatively range-preserving maps from $C(X)$ onto $C(Y)$ for realcompact spaces $X$ and $Y$ (not necessarily compact or locally compact) and then we consider such preserving maps from Lip$_c(X, \alpha)$ onto Lip$_c(Y,\beta)$ for metric spaces $X$ and $Y$ and $\alpha, \beta\in (0,1]$. We show that in both cases multiplicatively range-preserving maps are weighted composition operators which induce homeomorphisms between $X$ and $Y$. We also give a description of a linear separating map $T: A\rightarrow C(Y)$, where $A$ is either $C(X)$ for a normal space $X$ or Lip$_c(X,\alpha)$ for a metric space $X$ and $0<\alpha\le1$ and $Y$ is an arbitrary Hausdorff space.