Maps Between Uniform Algebras Preserving Norms of Rational Functions

Abstract

Let A, B be uniform algebras. Suppose that A0, B0 are subgroups of A−1, B−1 that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A0 → B0 is a surjection with \({\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}}\) for all \({f,g \in A_0}\), then there exists a real-algebra isomorphism \({\tilde{T} : A \rightarrow B}\) such that \({\tilde{T}(f)^d = (T(f)/T(1))^d}\) for every \({f \in A_0}\). This result leads to the following assertion: Suppose that SA, SB are subsets of A, B that contain A−1, B−1 respectively. If m, n > 0 and a surjection T : SA → SB satisfies \({\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}}\) for all \({f, g \in S_A}\), then there exists a real-algebra isomorphism \({\tilde{T} : A \rightarrow B}\) such that \({\tilde{T}(f)^d = (T(f)/T(1))^d}\) for every \({f \in S_A}\). Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.