Spectral Conditions for Composition Operators on Algebras of Functions

Abstract

We establish general sufficient conditions for maps between function algebras to be composition or weighted composition operators, which extend previous results in [2,4,6,7]. Let $X$ be a locally compact Hausdorff space and $A \subset C(X)$ a dense subalgebra of a function algebra, not necessarily with unit, such that $X = \partial A$ and $p(A)=\delta A$, where $\partial A$ is the Shilov boundary, $\delta A$ -- the Choquet boundary, and $p(A)$ -- the set of $p$-points of $A$. If $T\colon A \to B$ is a surjective map onto a function algebra $B\subset C(Y)$ such that either $\ps(Tf\cdot Tg)\subset\ps(fg)$ for all $f,g \in A$, or, alternatively, $\ps(fg)\subset\ps(Tf\cdot Tg)$ for all $f,g \in A$, then there is a homeomorphism $\psi\colon \delta B\to\delta A$ and a function $\a$ on $\delta B$ so that $(Tf)(y)=\a(y)\,f(\psi(y))$ for all $f \in A$ and $y \in\delta B$. If, instead, $\ps(Tf\cdot Tg)\cap\ps(fg)\neq \varnothing$ for all $f,g\in A$, and either $\ps(f)\subset \ps(Tf)$ for all $f\in A$, or, alternatively, $\ps(Tf)\subset \ps(f)$ for all $f\in A$, then $(Tf)(y)=f(\psi(y))$ for all $f \in A$ and $y \in \delta B$. In particular, if $A$ and $B$ are uniform algebras and $T\colon A \to B$ is a surjective map with $\ps(Tf\cdot Tg)\cap\ps(fg)\neq\varnothing$ for all $f,g \in A$, that has a limit, say $b$, at some $a\in A$ with $a^2=1$, then $(Tf)(y)=b(y)\,a(\psi(y))\, f(\psi(y))$ for every $f\in A$ and $y\in\delta B$.