Homomorphisms in proper Lie CQ*-algebras

Abstract

In this paper, we prove the Hyers--Ulam--Rassias stability of homomorphisms in proper Lie $CQ^*$-algebras and of Lie derivations on proper Lie $CQ^*$-algebras for the following generalized Cauchy--Jensen additive mapping: \begin{eqnarray*} 2 f(\frac{\sum_{j=1}^p x_j}{2} +\sum_{j=1}^d y_j) = \sum_{j=1}^p f(x_j) + 2\sum_{j=1}^d f(y_j) , \end{eqnarray*} which was introduced and investigated by Park \cite{pa7}. The concept of Hyers--Ulam--Rassias stability originated from the Th.M. Rassias` stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. {\bf 72} (1978), 297--300.