Let $\mathcal{A}$ be a unital $C^*$-algebra and let $M_1$ and $M_2$ be Banach left $\mathcal{A}$-modules.
In this paper, we prove the generalized Hyers-Ulam-Rassias stability for a generalized form,
\begin{eqnarray}
g\Big( \sum_{i=1}^{n}r_i x_i \Big)= \sum_{i=1}^{n} s_i g(x_i)
\end{eqnarray}
of a Cauchy-Jensen functional equation $2g(\frac{x+y}{2})=g(x)+g(y)$
for a mapping $g : M_1 \rightarrow M_2.$
As an application, we show that every approximate $C^*$-algebra isomorphism $h:\mathcal{A} \rightarrow \mathcal{B}$
between unital $C^*$-algebras is a $C^*$-algebra isomorphism when $h$ satisfies some regular conditions.
@article{1172852240,
author = {Kim, Hark-Mahn},
title = {Stability for generalized Jensen functional equations
and isomorphisms between $C^*$-algebras},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {13},
number = {5},
year = {2007},
pages = { 1-14},
language = {en},
url = {http://dml.mathdoc.fr/item/1172852240}
}
Kim, Hark-Mahn. Stability for generalized Jensen functional equations
and isomorphisms between $C^*$-algebras. Bull. Belg. Math. Soc. Simon Stevin, Tome 13 (2007) no. 5, pp. 1-14. http://gdmltest.u-ga.fr/item/1172852240/