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Zbl 1221.39034
Abbaspour, Gh.; Ramezanpour, M.
A fixed point approach to the stability of $\varphi$-morphisms on Hilbert $C^*$-modules.
(English)
[J] Ann. Funct. Anal. AFA 1, No. 1, 44-50, electronic only (2010). ISSN 2008-8752/e

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 \lq\lq control\rq\rq\ 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 $$\|U(x)-T(x)\|\leq {\sqrt{c}(2^p+2)\over \vert 2-2^p\vert}\|x\|^p\quad\text{for }x\in E.$$ If $p<0$ both inequalites (in the assumption and in the assertion) are postulated for $x,y\in E\setminus\{ 0\}$.
[Tomasz Kochanek (Katowice)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
46L08 C*-modules
39B52 Functional equations for functions with more general domains

Keywords: Hyers-Ulam-Rassias stability; Hilbert $C^\ast$-modules; $\varphi$-morphism

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