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

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: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 (2 d uy ) = h (2 d u ) h ( y ) for all unitaries u ∈ A , all y ∈ A , and all d ∈ Z . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)