Perturbation of the Wigner equation in inner product C*-modules

Details Perturbation of the Wigner equation in inner product C*-modules Perturbation of the Wigner equation in inner product C*-modules

2007-12-29

arxiv.org/abs/0801.0085v2

J. Math. Phys. 49 (2008), no. 3, 033519, 8 pp.

Mathematics

Operator Algebras

12 Pages, To appaer in J. Math. Phys. (won an ISFE medal in the 45th International Symposium on Functional Equations, Poland,

Scientific paper

10.1063/1.2898486

Let $\A$ be a $C^*$-algebra and $\B$ be a von Neumann algebra that both act on a Hilbert space $\Ha$. Let $\M$ and $\N$ be inner product modules over $\A$ and $\B$, respectively. Under certain assumptions we show that for each mapping $f\colon{\mathcal M} \to {\mathcal N}$ satisfying $$\||\ip{f(x)}{f(y)}|-|\ip{x}{y}| \|\leq\phi(x,y)\qquad (x,y\in{\mathcal M}),$$ where $\phi$ is a control function, there exists a solution $I\colon{\mathcal M} \to {\mathcal N}$ of the Wigner equation $$|\ip{I(x)}{I(y)}|=|\ip{x}{y}|\qquad (x, y \in {\mathcal M})$$ such that $$\|f(x)-I(x)\|\leq\sqrt{\phi(x,x)} \qquad (x\in {\mathcal M}).$$

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