Stability of homomorphisms and derivations in $C^*$-ternary algebras

TL;DR

This paper studies the stability of homomorphisms and derivations in $C^*$-ternary algebras, proving generalized Hyers-Ulam-Rassias stability for these algebraic structures and their associated functional equations.

Contribution

It introduces new stability results for homomorphisms and derivations in $C^*$-ternary algebras related to a specific functional equation.

Findings

Established generalized Hyers-Ulam-Rassias stability for homomorphisms.

Proved stability of derivations in $C^*$-ternary algebras.

Analyzed the functional equation related to these algebraic structures.

Abstract

In this paper, we investigate homomorphisms between $C^*$-ternary algebras and derivations on $C^*$-ternary algebras, associated with the following functional equation $$f(\frac{x_2-x_1}{3})+f(\frac{x_1-3x_3}{3})+f(\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$ Moreover, we prove the generalized Hyers-Ulam -Rassias stability of homomorphisms in $C^*$-ternary algebras and of derivations on $C^*$-ternary algebras.