Banach function algebras and certain polynomially norm-preserving maps

Abstract

Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\alpha$and $s,t\in \Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \longrightarrow B$ satisfying $\|(Tf)^s (T'g)^t-\alpha\|_Y=\|f^s g^t-\alpha \|_X$, for all $f,g \in A$, where $\|\cdot \|_X$ and $\|\cdot \|_Y$ denote the supremum norms on $X$ and $Y$, respectively. A similar result is given for the case where $T=T'$ and $T$ is defined between certain subsets of $A$ and $B$. We also show that if $T: A\longrightarrow B$ is a surjective map satisfying the stronger condition$R_\pi((Tf)^{s}(Tg)^{t}-\alpha)\cap R_\pi(f^{s}g^{t}-\alpha)\neq\varnothing $ for all $f,g \in A$, where $R_\pi(\cdot)$ denotes the peripheral range of the algebra elements, then there exists a homeomorphism $\varphi$ from the Choquet boundary $c(B)$ of $B$ onto the Choquet boundary $c(A)$ of $A$ such that $(Tf)^{d}(y)=(T1)^{d}(y)\,(f \circ \varphi(y))^{d}$ for all $f\in A$ and $y\in c(B)$,where $d$ is the greatest common divisor of $s$ and $t$.