Maps Between Uniform Algebras Preserving Norms of Rational Functions

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 alpha 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 parallel to T(f)(m)T(g)(n) - alpha parallel to(infinity) = parallel to f(m)g(n) - alpha parallel to(infinity) for all f, g is an element of A(0), then there exists a real-algebra isomorphism (T) over tilde : A -> B such that (T) over tilde (f)(d) = (T(f)/T(1))(d) for every f is an element of 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 parallel to T(f)(m)T(g)(n) - alpha parallel to(infinity) = parallel to f(m)g(n) - alpha parallel to(infinity) for all f, g is an element of S(A), then there exists a real-algebra isomorphism (T) over tilde : A -> B such that (T) over tilde (f)(d) = (T(f)/T(1))(d) for every f is an element of S(A). Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.