Approximately linear mappings in banach modules over a C*-algebra

Abstract

Let X and Y be vector spaces. The authors show that a mapping f: X -> Y satisfies the functional equation [GRAPHICS] with f(0) = 0 if and only if the mapping f: X -> Y is Cauchy additive, and prove the stability of the functional equation (double dagger) in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let A and B be unital C*-algebras, Poisson C*-algebras or Poisson JC*-algebras. As an application, the authors show that every almost homomorphism h: A -> B of A into B 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 is an element of A, all y is an element of A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C*-algebras, Poisson C*-algebras or Poisson JC*-algebras.