A polynomially spectrum preserving map between uniform algebras
Let \(A\) be a uniform algebra on a compact Hausdorff space \(X\) and \(f \in A\). The subset \(\{z \in \sigma(f): |z|= \|f\|\}\) of the spectrum \(\sigma(f)\) is called the \textit{peripheral spectrum} and denoted by \(\sigma_\pi(f)\). The authors characterized surjective maps \(T\) between two uniform algebras \(A\) and \(B\) such that \[ \sigma_\pi(p(T(f), T(g))) \subset \sigma_\pi(p(f, g)) \quad \text{for all} \quad f, g \in A, \] where \(p\) is a polynomial in two variables with complex coefficients of the form \(p(z, w)= zw+ az+ bw+ c\). They show that if \(a \neq b\), then \(T\) is an algebraic isomorphism induced by a homeomorphism from the maximal ideal space \(M_B\) onto \(M_A\); if \(a=b\), then \(T(f)(y)= \eta(y)f(\Phi(y))+ a(\eta(y)-1)\), where \(\eta: M_B \to \{-1, 1\}\) is a continuous map and \(\Phi: M_B \to M_A\) is an onto homeomorphism. Such polynomially spectrum preserving maps were first considered in an unpublished result of O.\,Hatori, T.\,Miura and H.\,Takagi, where the polynomials are more restricted as \(p(z, w)= zw+ az+ bw+ ab\).
scientific article; zbMATH DE number 5883056