A generalized Jensen type mapping and its applications

Let X and Y be vector spaces. It is shown that a mapping f : X \rightarrow Y satisfies the functional equation \begin{aligned} (2d+1) f(\frac{\sum_{j=1}^{2d+1} (-1)^{j+1} x_j}{2d+1}) = \sum_{j=1}^{2d+1} (-1)^{j+1} f(x_j) \end{aligned} (0.1) if and only if the mapping f : X \rightarrow Y is additive, and prove the Cauchy-Rassias 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 B be unital C^* -algebras, Poisson C^* -algebras, Poisson JC^* -algebras or Lie JC^* -algebras. As an application, we show that every almost homomorphism h : A \rightarrow B is a homomorphism when or for all unitaries u \in A , all y \in A , and n = 0, 1, 2, \cdots , and that every almost linear almost multiplicative mapping h : A \rightarrow B is a homomorphism when for all x \in A . Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C^* -algebras, Poisson C^* -algebras, Poisson JC^* -algebras or Lie JC^* -algebras, and of Lie JC^* -algebra derivations in Lie JC^* -algebras.