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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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