Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Chun-Gil Park

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 $d \in {\bf Z}$.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 4 (2006), 619-631.

Dates
First available in Project Euclid: 16 January 2007

https://projecteuclid.org/euclid.bbms/1168957339

Digital Object Identifier
doi:10.36045/bbms/1168957339

Mathematical Reviews number (MathSciNet)
MR2300619

Zentralblatt MATH identifier
1125.39027

Citation

Park, Chun-Gil. Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between $C^*$-algebras. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 619--631. doi:10.36045/bbms/1168957339. https://projecteuclid.org/euclid.bbms/1168957339