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

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.