Poisson C ^*-algebra derivations in Poisson C ^*-algebras

Abstract

In this study, we introduce the following additive functional equation: g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y ) g\left(\lambda u+v+2y)=\lambda g\left(u)+g\left(v)+2g(y) for all λ ∈ C \lambda \in {\mathbb{C}} , all unitary elements u , v u,v in a unital Poisson C * {C}^{* } -algebra P P , and all y ∈ P y\in P . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the aforementioned additive functional equation in unital Poisson C * {C}^{* } -algebras. Furthermore, we apply to study Poisson C * {C}^{* } -algebra homomorphisms and Poisson C * {C}^{* } -algebra derivations in unital Poisson C * {C}^{* } -algebras.