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

摘要

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, 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: M B → M A and a closed and open subset K of M B such that $$widehat{Tleft( a ight)}left( y ight) = left{ egin{gathered} widehat{Tleft( e ight)}left( y ight)hat aleft( {phi left( y ight)} ight) y in K hfill widehat{Tleft( e ight)}left( y ight)overline {hat aleft( {phi left( y ight)} ight)} y in M_mathcal{B} ackslash K hfill end{gathered} ight.$$ for all a ∈ A, where e is unit element of A. If, in addition, $$widehat{Tleft( e ight)} = 1$$ and $$widehat{Tleft( {ie} ight)} = i$$ on M B, then T is an algebra isomorphism.