Automorphisms on a C*-algebra and isomorphisms between lie JC*-algebras associated with a generalized additive mapping

Abstract

Let X, Y be vector spaces, and let r be 1 or 3. It is shown that if an odd mapping f : X -> Y satisfies the functional equation [GRAPHICS] then the odd mapping f : X -> Y is additive, and we prove the Cauchy-Rassias stability of the functional equation (0.1) in Banach modules over a unital C*-algebra. As an application, we show that every almost linear bijective mapping h : A -> A on a unital C*-algebra A is an automorphism when h(3(n)uy) = h(3(n)u)h(y) for all unitaries u epsilon A, all y epsilon A and all n epsilon Z, and that every almost linear bijective mapping h : A -> B of a unital Lie JC*-algebra A onto a unital Lie JC*-algebra B is a Lie JC*-algebra isomorphism when h(3(n)u circle y) = h(3(n)u) circle h(y) for all y epsilon A, all unitaries u epsilon A and all n epsilon Z.