Abstract. Let ϕ : A → B be a surjective operator between two uniform algebras with ϕ(1) = 1. We show that if ϕ satisfies the peripheral multiplicativity conditionwhere σ π (f ) is the peripheral spectrum of f , then ϕ is an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.An important question in Banach algebra theory, which still lacks a satisfactory answer, is to find criteria for an operator between two Banach algebras to be linear and multiplicative. For linear operators between semisimple algebras an answer is suggested by the theorem of Gleason-Kahane-Żelazko (e.g., [11]) in terms of spectra of algebra elements. A theorem by Kowalski and S lodkowski [6] considers alternative spectral conditions for not necessarily linear operators. N. V. Rao and A. K. Roy [9] have introduced an interesting spectral multiplicativity condition that contributes to the matter. In particular for unital, that is Φ(1) = 1, operators Φ they have proven the following:If A is a uniform algebra on X and Φ : A → A is a surjective unital operator such that where Ran (f ) = f (X) is the range of f . N. V. Rao, T. Tonev and E. Toneva [10] (see also [2]) considered a spectral additivity condition related to the peripheral
Generalized peak point Choquet boundary Shilov boundary Peripheral spectrum Homeomorphism Algebra isomorphism Norm-additive operator Norm-linear operatorLet A ⊂ C (X) and B ⊂ C (Y ) be uniform algebras with Choquet boundaries δ A and δ B.λ, μ ∈ C and all algebra elements f and g. We show that for any norm-linear surjection (y))| for every f ∈ A and y ∈ δ B. Sufficient conditions for norm-additive and norm-linear surjections, not assumed a priori to be linear, or continuous, to be unital isometric algebra isomorphisms are given. We prove that any unital norm-linear surjection T for which T (i) = i, or which preserves the peripheral spectra of C-peaking functions of A, is a unital isometric algebra isomorphism. In particular, we show that if a linear operator between two uniform algebras, which is surjective and norm-preserving, is unital, or preserves the peripheral spectra of C-peaking functions, then it is automatically multiplicative and, in fact, an algebra isomorphism.
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