Let A and B be unital semisimple commutative Banach algebras. It is shown that if surjections S, T : A → B with S(1) = T (1) = 1 and α ∈ C \ {0} satisfy r(S(a)T (b) − α) = r(ab − α) for all a, b ∈ A , then S = T and S is a real algebra isomorphism, where r(a) is the spectral radius of a.

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Suppose that A and B are uniform algebras on compact Hausdorff spaces X and Y , respectively. Let ρ, τ : Λ → A and S, T : Λ → B be mappings on a nonempty set Λ. Suppose that ρ(Λ), τ (Λ) and S(Λ), T (Λ) are closed under multiplications and contain exp A and exp B respectively and that S(e∅ for all f, g ∈ Λ and there exists a first-countable dense subset D B in Ch(B), or a first-countable dense subset D A in Ch(A), then there exists an algebra isomorphism S : A → B such that S(ρ(f )) = S(e1) −1 S(f ) and S(τ (f )) = T (e 2 ) −1 T (f ) for every f ∈ Λ.

Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism $$ \tilde S $$: A → B such that $$ \tilde S $$(ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.

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. (1)) d for every f ∈ A0. 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 :Note that in these results and elsewhere in this paper we do not assume that T (exp A) = exp B.Mathematics Subject Classification (2010). Primary 46J10, 47B48; Secondary 46H40, 46J20.