Journal of the
Korean Mathematical Society JKMS

Let $X$ and $Y$ be vector spaces. It is shown that a mapping $f : X \rightarrow Y$ satisfies the functional equation
$$ \begin{aligned} &\ mn \ {{}_{mn-2}}C_{k-2}f\bigg(\frac{x_1+\cdots + x_{mn}}{mn}\bigg) \\
&\ + m \ {{}_{mn-2}}C_{k-1}\ \sum_{i=1}^n f\bigg(\frac{x_{mi-m+1} +\cdots + x_{mi}}{m}\bigg) \\
=&\ k \ \sum_{1 \le i_1 < \cdots < i_k \le mn}f\bigg(\frac{x_{i_1}+\cdots + x_{i_k}}{k}\bigg)
\end{aligned}\tag{$*$}$$
if and only if the mapping $f : X \rightarrow Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(*)$ in Banach modules over a unital $C^*$-algebra. Let $\mathcal A$ and $\mathcal B$ be unital $C^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h : \mathcal A \rightarrow \mathcal B$ of $\mathcal A$ into $\mathcal B$ is a homomorphism when $h(2^d u y) = h(2^d u) h(y)$ or $h(2^d u \circ y) = h(2^d u)\circ h(y)$ for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and $d = 0, 1, 2, \ldots$, and that every almost linear almost multiplicative mapping $h : \mathcal A \rightarrow \mathcal B$ is a homomorphism when $h(2 x) = 2 h(x)$ for all $x \in \mathcal A$.
Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras or in Lie $JC^*$-algebras, and of Lie
$JC^*$-algebra derivations in Lie $JC^*$-algebras.

Keywords: Trif's functional equation, Cauchy-Rassias stability, $C^*$-algebra homomorphism, Lie $JC^*$-algebra homomorphism, Lie $JC^*$-algebra deriva-tion

MSC numbers: Primary 39B52, 46L05, 47B48, 17A36