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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.

Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras