Peripherally Monomial-Preserving Maps between Uniform Algebras

Abstract

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.