Cauchy-Rassias Stability of a generalized additive mapping in Banach modules and isomorphisms in C*-algebras

Abstract

Let X, Y be vector spaces, and let r be 2 or 4. It is shown that if an odd mapping f : X \rightarrow Y satisfies the functional equation [수식] then the odd mapping f : X → Y is additive, and we prove the Cauchy-Rassias stability of the functional equation in Banach modules over a unital C^*-algebra. 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^n u y) = h(2^n u) h(y) for all unitaries u ∈ A, all y ∈ A, and n=0, 1, 2, ….