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

Abstract

Let X, Y be Banach modules over a C*-algebra and let r(1),....r(n) epsilon (0, infinity) be given. We prove the Hyers-Ulam-Rassias stability of the following functional equation in Banach modules over a unital C*-algebra: Sigma(n)(i=1)r(i)f (Sigma(n)(j=1) rj (x(i)-x(j))) + (Sigma(n)(i=1) r(i)) f (Sigma(n)(i=1) r(i)x(i)) = (Sigma(n)(i-1)r(i)x(i)) = (Sigma(n)(i=1)r(i)) Sigma(n)(i=1)r(i)f(x(i)).(0.1) We show that if r(perpendicular to) =... = r(n) = r and odd mapping f : X -> Y satisfies the functional equation (0.1) then the odd inapping 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((nr)(d)uy) = h((nr)(d)u)h(y) for all nnitaries u epsilon A, all y epsilon A, and all d epsilon Z.