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

초록

Let $A$ be a $C^{\ast }$-algebra. A $\mathbb{C}$-linear mapping $\delta :A\to A$ is called a {\it Jordan $\ast$-derivation} if $\delta (a^2)=a\delta (a)+\delta (a^{\ast })a^{\ast }$ for all $a\in A$. Let $A$ be a $J{C}^{\ast }$-algebra. A $\mathbb{C}$-linear mapping $\delta :A\to A$ is called a {\it Jordan $\ast$-derivation} if $\delta (a^2)=a\circ \delta (a)+\delta (a^{\ast })\circ a^{\ast }$ for all $a\in A$. In this paper, we investigate Jordan $\ast$-derivations on $C^{\ast }$-algebras and Jordan $\ast$-derivations on $J{C}^{\ast }$-algebras associated with the following functional inequality $\Vert f(x)+f(y)+kf(z)\Vert \le \Vert kf\left(\frac{x+y}{k}+z\right)\Vert$ for some integer $k$ greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan $\ast$-derivations on $C^{\ast }$-algebras and of Jordan $\ast$-derivations on $J{C}^{\ast }$-algebras associated with the following functional equation $f\left(\frac{x+y}{k}+z\right)=\frac{f(x)+f(y)}{k}+f(z)$ for some integer $k$ greater than 1.