Maps Between Uniform Algebras Preserving Norms of Rational Functions

Abstract

Let A, B be uniform algebras. Suppose that A ₀, B ₀ 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 : A ₀ → B ₀ 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 S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B 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.