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Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, \( \widehat{T\left( e \right)} = 1 \) and \( \widehat{T\left( {ie} \right)} = i \) on MB, then T is an algebra isomorphism.

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.