Jordan *-derivations on C*-algebras and JC*-algebras

Abstract

In this paper, we investigate Jordan $*$-derivations on $C^*$-algebras and Jordan $*$-derivations on $JC^*$-algebras associated with the following functional inequality \begin{eqnarray} \| f(x)+f(y)+kf(z) \| \le \| kf(\frac{x+y}{k}+z) \| \end{eqnarray} for some integer $k$ greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan $*$-derivations on $C^*$-algebras and of Jordan $*$-derivations on $JC^*$-algebras associated with the following functional equation \begin{eqnarray} f(\frac{x+y}{k}+z) = \frac{f(x)+f(y)}{k} + f(z) \end{eqnarray} for some integer $k$ greater than 1. The concept of generalized Hyers-Ulam 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.