Norm Conditions on Maps between Certain Subspaces of Continuous Functions

Abstract

For a locally compact Hausdorff space $X$, let $C_0(X)$ be the Banach space of continuous complex-valued functions on $X$ vanishing at infinity endowed with the supremum norm $\|\cdot\|_X$. We show that for locally compact Hausdorff spaces $X$ and $Y$ and certain (not necessarily closed) subspaces $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, if $T:A \longrightarrow B$ is a surjective map satisfying one of the norm conditions i) $\|(Tf)^s (Tg)^t\|_Y=\|f^s g^t\|_X$, or ii) $\|\, |Tf|^s+|Tg|^t\,\|_Y=\|\, |f|^s+|g|^t\, \|_X$, \noindent for some $s,t\in \mathbb{N}$ and all $f,g\in A$, then there exists a homeomorphism $\varphi: \mathrm{ch}(B) \longrightarrow \mathrm{ch}(A)$ between the Choquet boundaries of $A$ and $B$ such that $|Tf(y)|=|f(\varphi(y))|$ for all $f\in A$ and $y\in \mathrm{ch}(B)$. We also give a result for the case where $A$ is closed (or, in general, satisfies a special property called Bishop's property) and $T:A \longrightarrow B$ is a surjective map satisfying the inclusion $R_\pi((Tf)^s (Tg)^t) \subseteq R_\pi(f^s g^t)$ of peripheral ranges. As an application, we characterize such maps between subspaces of the form $A_1f_1+A_2f_2+\cdots+A_nf_n$, where for each $1\le i \le n$, $A_i$ is a uniform algebra on a compact Hausdorff space $X$ and $f_i$ is a strictly positive continuous function on $X$. Our results in case (ii) improve similar results in~[30], for subspaces rather than uniform algebras, without the additional assumption that $T$ is $\mathbb{R}^+$-homogeneous.