Peripherally Monomial-Preserving Maps between Uniform Algebras

Abstract.

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be uniform algebras and p(z,w) = z^mwⁿ a twovariable monomial. We characterize maps T from certain subsets of ${\mathcal{A}}$ into ${\mathcal{B}}$ such that $\sigma_{\pi}(p(T(f),T(g))) \subset \sigma_{\pi}(p(f,g))$ holds for all f and g in the domain of T; peripherally monomial-preserving maps. Furthermore ${\mathcal{A}}$ and ${\mathcal{B}}$ are proved to be isometrical isomorphic as Banach algebras. If the greatest common divisor of m and n is 1, then T is extended to an isometrical linear isomorphism; a weighted composition operator. An example of peripherally monomial-preserving surjections between uniform algebras which is not linear, nor multiplicative, nor injective is given when the greatest common divisor is strictly greater than 1.