Maps between Banach function algebras satisfying certain norm conditions

Abstract

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.