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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
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Full Text: Euclid