Characterizations of peripherally multiplicative mappings between real function algebras

Let X be a compact Hausdorff space; let τ : X → X be a topological involution; and let A ⊂ C(X, τ) be a real function algebra. Given an f ∈ A, the peripheral spectrum of f is the set σπ(f) of spectral values of f of maximum modulus. We demonstrate that if T1, T2 : A → B and S1, S2 : A → A are surjective mappings between real function algebras A ⊂ C(X, τ) and B ⊂ C(Y, φ) that satisfy σπ(T1(f)T2(g)) = σπ(S1(f)S2(g)) for all f, g ∈ A, then there exists a homeomorphism ψ : Ch(B) → Ch(A) between the Choquet boundaries such that (ψ ◦ φ)(y) = (τ ◦ ψ)(y) for all y ∈ Ch(B), and there exist functions κ1, κ2 ∈ B, with κ−1 1 = κ2, such that Tj(f)(y) = κj(y)Sj(f)(ψ(y)) for all f ∈ A, all y ∈ Ch(B), and j = 1, 2. As a corollary, it is shown that if either Ch(A) or Ch(B) is a minimal boundary (with respect to inclusion) for its corresponding algebra, then the same result holds for surjective mappings T1, T2 : A → B and S1, S2 : A → A that satisfy σπ(T1(f)T2(g)) ∩ σπ(S1(f)S2(g)) 6= ∅ for all f, g ∈ A.