Abstract
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let \(\bar A\) and \(\bar B\) be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖ x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and a homeomorphism ϕ: M B → M A between the maximal ideal spaces of B and A such that for all f ∈ A,
where \(\hat \cdot\) denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from \(\bar A\) onto \(\bar B\) inducing a homeomorphism between \(M_{\bar B}\) and \(M_{\bar A}\). We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.
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Hosseini, M., Sady, F. Maps between Banach function algebras satisfying certain norm conditions. centr.eur.j.math. 11, 1020–1033 (2013). https://doi.org/10.2478/s11533-013-0224-x
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DOI: https://doi.org/10.2478/s11533-013-0224-x