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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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