Spectral Conditions for Composition Operators on Algebras of Functions - Johnson, J.; Tonev, T. (Communications in Mathematics and Applications, 2012)
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 $p(A)=δ A$, where $\partial A$ is the Shilov boundary, $δ A$ -- the Choquet boundary, and $p(A)$ -- the set of $p$-points of $A$. If $T\colon A → B$ is a surjective map onto a function algebra $B\subset C(Y)$ such that either $\ps(Tf· Tg)\subset\ps(fg)$ for all $f,g \in A$, or, alternatively, $\ps(fg)\subset\ps(Tf· Tg)$ for all $f,g \in A$, then there is a homeomorphism $ψ\colon δ B→δ A$ and a function $\a$ on $δ B$ so that $(Tf)(y)=\a(y)\,f(ψ(y))$ for all $f \in A$ and $y \inδ B$. If, instead, $\ps(Tf· Tg)\cap\ps(fg) eq \varnothing$ 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 $(Tf)(y)=f(ψ(y))$ for all $f \in A$ and $y \in δ B$. In particular, if $A$ and $B$ are uniform algebras and $T\colon A → B$ is a surjective map with $\ps(Tf· Tg)\cap\ps(fg) eq\varnothing$ for all $f,g \in A$, that has a limit, say $b$, at some $a\in A$ with $a^2=1$, then $(Tf)(y)=b(y)\,a(ψ(y))\, f(ψ(y))$ for every $f\in A$ and $y\inδ B$.
Keywords: Uniform algebra; Function algebra; Peripheral spectrum; Composition operator; Algebra isomorphism; Choquet boundary