Approximately ternary homomorphisms on \(C^*\)-ternary algebras. (English) Zbl 1470.39070
Summary: 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: \(f \left(\left(x_2 - x_1\right) / 3\right) + f \left(\left(x_1 - 3 x_3\right) / 3\right) + f \left(\left(3 x_1 + 3 x_3 - x_2\right) / 3\right) = f \left(x_1\right)\), 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.
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 46L05 | General theory of \(C^*\)-algebras |
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