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Perturbation of the Wigner equation in inner product $C^*$-modules. (English) Zbl 1153.81342
Editorial remark: Let $\germ A$ be a $C^*$-algebra and $\germ B$ be a von Neumann algebra that both act on a Hilbert space $\cal H$. Let $\cal M$ and $\cal N$ be inner product modules over $\germ A$ and $\germ B$, respectively. Under certain assumptions we show that for each mapping $f: \cal M\to\cal N$ satisfying $$\Vert\,\vert\langle f(x)f(y)\rangle\vert-\vert\langle xy\rangle\vert \Vert\leq\phi(x,y)\qquad (x,y\in\cal M),$$ where $\phi$ is a control function, there exists a solution $I: \cal M\to\cal N$ of the Wigner equation $$\vert\langle I(x)I(y)\rangle\vert=\vert\langle xy\rangle\vert\qquad (x, y \in\cal M)$$ such that $$\Vert f(x)-I(x)\Vert \le \sqrt{\phi(x,x)}\qquad (x\in\cal M).$$

39B72Systems of functional equations and inequalities
47B48Operators on Banach algebras
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