Norm conditions for real-algebra isomorphisms between uniform algebras
Tóm tắt
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.
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