Peripherally Monomial-Preserving Maps between Uniform Algebras.

Let $${\mathcal{A}}$$ and $${\mathcal{B}}$$ be uniform algebras and p( z, w) = z ^m w ⁿ 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. [ABSTRACT FROM AUTHOR]