Cauchy-Rassias Stability of linear Mappings in Banach Modules Associated with a Generalized Jensen Type Mapping

Let X and Y be vector spaces. We show that a mapping f : X rightarrow Y satisfies the functional equation, if and only if the mapping f : X rightarrow Y is Cauchy additive, and prove the Cauchy-Rassias stability of the above functional equation 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, we show that every almost homomorphism h : A rightarrow B of A into B is a homomorphism when h(2nuy) = h(2nu)h(y) or h(2nu o y) = h(2nu) o h(y), for all unitaries u Є A, all y Є A, and n = 0, 1, 2, · · · . Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras or Poisson JC*-algebras.