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Stability of ternary homomorphisms via generalized Jensen equation. (English) Zbl 1114.39010

A ternary algebra \(\bigl({\mathcal A},[\;]\bigr)\) is a linear space \({\mathcal A}\) over \({\mathbb R}\) or \({\mathbb C}\) with a ternary product \([\;]\) s.t. \( \bigl[[abc]de\bigr]=\bigl[a[bcd]e\bigr]=\bigl[ab[cde]\bigr] \), \(a,b,c,d,e\in{\mathcal A}\). If \({\mathcal A}\) is a Banach space then \({\mathcal A}\) is a Banach ternary algebra if \(\bigl\| [abc]\bigr\| \leq\| a\| \,\| b\| \,\| c\| \), \(a,b,c\in{\mathcal A}\). A mapping \(T:{\mathcal A}\to{\mathcal B}\) between ternary algebras is a ternary homomorphism if \(T\bigl([abc]\bigr)=\bigl[T(a)T(b)T(c)\bigr]\), \(a,b,c\in{\mathcal A}\). The stability of ternary homomorphisms in the sense of Hyers, Ulam and Rassias is considered. Auxiliarily the stability of the generalized Jensen’s equation
\[ rf\Bigl(\frac{sx+ty}{r}\Bigr)=sf(x)+tf(y) \]
is established. Using this result the following theorem is proved.
Theorem: Let \({\mathcal A}\) be a ternary algebra, \({\mathcal B}\) be a Banach ternary algebra and \(f:{\mathcal A}\to{\mathcal B}\) be a mapping s.t. \(f(0)=0\). Assume that there exists a function \(\varphi:{\mathcal A}^5\to[0,\infty)\) s.t.
\[ \widetilde{\varphi}(x,y,u,v,w) :=\frac{1}{r}\sum_{j=0}^{\infty} \Bigl(\frac{r}{s}\Bigr)^{-j}\varphi\biggl( \Bigl(\frac{r}{s}\Bigr)^{j}x, \Bigl(\frac{r}{s}\Bigr)^{j}y, \Bigl(\frac{r}{s}\Bigr)^{j}u, \Bigl(\frac{r}{s}\Bigr)^{j}v, \Bigl(\frac{r}{s}\Bigr)^{j}w \biggr)<\infty \] and
\[ \biggl\| rf\biggl(\frac{\mu sx+\mu ty+[uvw]}{r}\biggr) -\mu sf(x)+\mu tf(y)-\bigl[f(u)f(v)f(w)\bigr] \biggr\| \leq\varphi(x,y,u,v,w) \]
for all \(\mu\in\bigl\{z\in{\mathbb C}:| z| =1\bigr\}\) and all \(x,y,u,v,w\in{\mathcal A}\). Then there exists a unique ternary homomorphism \(T:{\mathcal A}\to{\mathcal B}\) s.t. \( \bigl\| f(x)-T(x)\bigr\| \leq\widetilde{\varphi}(x,x,0,0,0) \) for all \(x\in{\mathcal A}\).
Some related results are also proved.

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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