Abstract.
Let \({\mathcal{A}}\) and \({\mathcal{B}}\) be uniform algebras and p(z,w) = zmwn 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.
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Received: February 29, 2008., Revised: June 4, 2008. Accepted: June 10, 2008.
The first, third and fourth authors were partly supported by the Grantsin-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.
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Hatori, O., Hino, K., Miura, T. et al. Peripherally Monomial-Preserving Maps between Uniform Algebras. MedJM 6, 47–59 (2009). https://doi.org/10.1007/s00009-009-0166-5
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DOI: https://doi.org/10.1007/s00009-009-0166-5