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Generalized stability of $$C^{\ast}$$-ternary quadratic mappings. (English) Zbl 1158.39020
A $$C^{*}$$-ternary ring is a complex Banach space $$A$$ equipped with a ternary product $$(x,y,z) \to [x,y,z]$$ of $$A^{3}$$ into $$A$$ which is $$\mathbb C$$-linear in the outer variables and conjugate $$\mathbb C$$-linear in the middle variable, and associative in the following sense
$[x,y,[z,w,v]] = [x,[w,z,y],v] = [[x,y,z],w,v]$ and satisfies conditions
$\|[x,y,z] \| \leq \|x \| \|y \| \|z \|, \;\|[x,x,x] \| = \|x \|^{3}.$ Let $$A$$ and $$B$$ be two $$C^{*}$$-ternary rings. A mapping $$Q: A \to B$$ is called a quadratic mapping if $Q(x + y) + Q(x - y) = 2Q(x) + 2Q(y)$ for $$x, y \in A$$. A quadratic mapping $$Q: A \to B$$ is said to be a $$C^{*}$$-ternary quadratic mapping if
$Q([x,y,z]) = [Q(x), Q(y), Q(z)]$ for all $$x,y,z \in A$$.
The authors prove the following stability theorems.
Let $$f: A \to B$$ be a mapping for which there exists a function $$\varphi: A^{3} \to [0,\infty)$$ such that
$\sum_{j=0}^{\infty} 4^{3j} \varphi\bigg(\frac{x}{2^{j}}, \frac{y}{2^{j}}, \frac{z}{2^{j}}\bigg) < \infty,$
$\|f(x + y) + f(x - y) - 2f(y) \| \leq \varphi(x, y, 0), \tag{1}$
$f([x,y,z]) - [f(x), f(y), f(z)] \| \leq \varphi(x,y,z) \tag{2}$ for all $$x,y,z \in A$$. Then there exists a unique $$C^{*}$$-ternary quadratic mapping $$Q: A \to B$$ such that
$\|f(x) - Q(x) \| \leq \widetilde{\varphi} \bigg(\frac{x}{2},\frac{x}{2}, 0\bigg)$ for all $$x \in A$$, where
$\widetilde{\varphi}(x,y,z) = \sum_{j = 0}^{\infty} 4^{j} \varphi \bigg(\frac{x}{2^{j}}, \frac{y}{2^{j}}, \frac{z}{2^{j}}\bigg)$ for all $$x,y,z \in A$$.
Let $$f: A \to B$$ be a mapping for which there exists a function $$\varphi: A^{3} \to [0,\infty)$$ satisfying (1) and (2) such that
$\widetilde{\varphi}(x,y,z) = \sum_{j = 0}^{\infty} \frac{1}{4^{j}} \varphi(2^{j}x, 2^{j}y, 2^{j}z) < \infty$ for all $$x,y,z \in A$$. Then there exists a unique $$C^{*}$$-ternary quadratic mapping $$Q: A \to B$$ such that
$\|f(x) - Q(x) \| \leq \frac{1}{4} \widetilde{\varphi}(x, x, 0)$ for all $$x \in A$$.

##### 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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