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A fixed point approach to the stability of $$\varphi$$-morphisms on Hilbert $$C^*$$-modules. (English) Zbl 1221.39034
Let $$E$$, $$F$$ be Hilbert modules over $$C^\ast$$-algebras $$A$$, $$B$$, respectively, and let $$\varphi: A\to B$$ be a map. We say that $$U: E\to F$$ is a $$\varphi$$-morphism if
$\langle U(x),U(y)\rangle=\varphi(\langle x,y\rangle)\quad\text{for }x,y\in E.$
Using a fixed point approach the authors prove stability results for $$\varphi$$-morphisms under the assumption of the form
$\|\langle U(x),U(y)\rangle-\varphi(\langle x,y\rangle)\|\leq\rho(x,y)\quad\text{for }x,y\in E,$
where $$\rho(x,y)$$ is a \\ control\'\' function satisfying some technical conditions. In particular, for $$\rho(x,y)=c\|x\|^p\|y\|^p$$ (with some $$c,p\geq 0$$ and $$p\not=1$$), there exists a unique $$\varphi$$-morphism $$T: E\to F$$ such that
$\|U(x)-T(x)\|\leq {\sqrt{c}(2^p+2)\over | 2-2^p|}\|x\|^p\quad\text{for }x\in E.$
If $$p<0$$ both inequalites (in the assumption and in the assertion) are postulated for $$x,y\in E\setminus\{ 0\}$$.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 46L08 $$C^*$$-modules 39B52 Functional equations for functions with more general domains and/or ranges
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