Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between $C^*$-algebras

Abstract

Let $X, Y$ be Banach modules over a $C^*$-algebra and let $r_1, \cdots, r_n \in (0, \infty)$ be given. We prove the Hyers-Ulam-Rassias stability of the following functional equation in Banach modules over a unital $C^*$-algebra: \begin{eqnarray} \sum_{i=1}^{n} r_i f \left( \sum_{j=1}^{n}r_j(x_i-x_j) \right) + \left(\sum_{i=1}^{n} r_i\right) f \left(\sum_{i=1}^{n} r_ix_i \right) =\left(\sum_{i=1}^{n}r_i\right) \sum_{i=1}^{n} r_i f(x_i) . \end{eqnarray} We show that if $r_1=\cdots=r_n = r$ and an odd mapping $f : X \rightarrow Y$ satisfies the functional equation {\rm (0.1)} then the odd mapping $f : X \rightarrow Y$ is Cauchy additive. As an application, we show that every almost linear bijection $h : A \rightarrow B$ of a unital $C^*$-algebra $A$ onto a unital $C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h((nr)^d u y) = h((nr)^d u) h(y)$ for all unitaries $u \in A$, all $y \in A$, and all $d \in {\bf Z}$.