Approximately Linear Mappings in Banach Modules over a C-algebra

Let X and Y be vector spaces. The authors show that a mapping f : X . Y satisfies the functional equation 2df(.2dj=1(.1)j+1xj2d)=.j=12d(.1)j+1f(xj) (1) with f(0) = 0 if and only if the mapping f : X . Y is Cauchy additive, and prove the stability of the functional equation (.) in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let A and . be unital C*-algebras, Poisson C*-algebras or Poisson JC*-algebras. As an application, the authors show that every almost homomorphism h : A . . of A into . is a homomorphism when h((2d.1)n uy) = h((2d.1)n u)h(y) or h((2d.1)n u.y) = h((2d.1)n u).h(y) for all unitaries u.A , all y.A, n = 0, 1, 2, . . . . Moreover, the authors prove the stability of homomorphisms in C*-algebras, Poisson C*-algebras or Poisson JC*-algebras.