Norm-linear and norm-additive operators between uniform algebras

Let A ⊂ C ( X ) and B ⊂ C ( Y ) be uniform algebras with Choquet boundaries δA and δB. A map T : A → B is called norm-linear if ‖ λ T f + μ T g ‖ = ‖ λ f + μ g ‖ ; norm-additive, if ‖ T f + T g ‖ = ‖ f + g ‖ , and norm-additive in modulus, if ‖ | T f | + | T g | ‖ = ‖ | f | + | g | ‖ for each λ , μ ∈ C and all algebra elements f and g. We show that for any norm-linear surjection T : A → B there exists a homeomorphism ψ : δ A → δ B such that | ( T f ) ( y ) | = | f ( ψ ( 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.