### 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 $y\in\delta B$.

#### Keywords

Uniform algebra; Function algebra; Peripheral spectrum; Composition operator; Algebra isomorphism; Choquet boundary

#### Full Text:

PDFDOI: http://dx.doi.org/10.26713%2Fcma.v3i1.144

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eISSN 0975-8607; pISSN 0976-5905