# zbMATH — the first resource for mathematics

Almost derivations on $$C^*$$-ternary rings. (English) Zbl 1132.39026
The author presents generalized stability results, in the spirit of P. Găvruţa [J. Math. Anal. Appl. 184, No. 3, 431–436 (1994; Zbl 0818.46043)], for derivations in $$C^*$$-ternary rings. In particular, the following result is obtained: Let $${\mathcal A}$$ denote a $$C^*$$-ternary ring, $$\varepsilon > 0$$, $$0 \leq p < 1$$, and suppose that $$f : {\mathcal A} \to {\mathcal A}$$ fulfils $$f(0) = 0$$ and
$\begin{split} \| f(\mu x + \mu y + [u \, v \, w]) - \mu f(x) - \mu f(y) - [f(u) \, v \, w] - [u \, f(v) \, w] - [u \, v \, f(w)] \| \\ \leq \varepsilon ( {\| x \|}^p + {\| y \|}^p + {\| u \|}^p + {\| v \|}^p + {\| w \|}^p ) \end{split}$ for every $$x,y,u,v,w \in {\mathcal A}$$ and every complex number $$\mu$$ with $$| \mu | = 1$$. Then $$f$$ is a derivation.

##### 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
Full Text: