Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between C*-algebras

Abstract

Let X, Y be Banach modules over a C *-algebra. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach modules over a unital C *-algebra: equation image It is shown that a mapping f: X → Y satisfies the above functional equation and f (0) = 0 if and only if the mapping f: X → Y is Cauchy additive. As an application, we show that every almost linear bijection h: A → B of a unital C *-algebra A onto a unital C *-algebra B is a C *-algebra isomorphism when h (2d uy) = h (2d u) h (y) for all unitaries u ∈ A, all y ∈ A, and all d ∈ Z.