Abstract
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. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ π (S(p)T(q)) ⊂ σ π (ρ(p) τ(q)) for all p,q ∈ I, where σ π (f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A.
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The first and second authors were partly supported by the Grant-in-Aid for Scientific Research.
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Hatori, O., Miura, T., Shindo, R. et al. Generalizations of spectrally multiplicative surjections between uniform algebras. Rend. Circ. Mat. Palermo 59, 161–183 (2010). https://doi.org/10.1007/s12215-010-0013-3
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DOI: https://doi.org/10.1007/s12215-010-0013-3