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Let A and B be uniform algebras. Suppose that α ≠ 0 and A1 ⊂ A. Let ρ, τ: A1 → A and S, T: A1 → B be mappings. Suppose that ρ(A1), τ(A1) and S(A1), T(A1) 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 ∈ A1, S(e1)−1 ∈ S(A1) and S(e1) ∈ T(A1) for some e1 ∈ A1 with ρ(e1) = 1, then there exists a real-algebra isomorphism $$ \tilde S $$: A → B such that $$ \tilde S $$(ρ(f)) = S(e1)−1S(f) for every f ∈ A1. We also give some applications of this result.

Norm conditions for real-algebra isomorphisms between uniform algebras