Approximately Ternary Homomorphisms on C * -Ternary Algebras

Abstract

Gordji et al. established the Hyers-Ulam stability and the superstability of $C^{*}$-ternary homomorphisms and $C^{*}$-ternary derivations on $C^{*}$-ternary algebras, associated with the following functional equation: $f\left(\left(x_2-x_1\right)/3\right)+f\left(\left(x_1-3x_3\right)/3\right)+f\left(\left({3x}_1+{3x}_3-x_2\right)/3\right)=f\left(x_1\right)$, by the direct method. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the conditions and prove the corrected theorems. Furthermore, we prove the Hyers-Ulam stability and the superstability of $C^{*}$-ternary homomorphisms and $C^{*}$-ternary derivations on $C^{*}$-ternary algebras by using a fixed point approach.