Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ(fg) of the product of any two elements f and g in A coincides with the spectrum σ(Tf Tg). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and nonmultiplicative unital map from a commutative C*-algebra into itself such that σ(Tf Tg) = σ(fg) holds for every f, g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ(Tf Tg) ⊂ σ(fg) holds for every f, g. © 2006 Elsevier Inc. All rights reserved.