Stability for generalized Jensen functional equations and isomorphisms between $C^*$-algebras

Abstract

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.