engleski

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

Let $\A$ be a $C^*$-subalgebra of a von Neumann algebra $\B$. Let $\M$ and $\N$ be inner product modules over $\A$ and $\B$, respectively. Under some assumptions we show that for each mapping $f\colon{; ; ; \mathcal M}; ; ; \to {; ; ; \mathcal N}; ; ; $ satisfying $$\|\, |\ip{; ; ; f(x)}; ; ; {; ; ; f(y)}; ; ; |-|\ip{; ; ; x}; ; ; {; ; ; y}; ; ; |\, \|\leq\varphi(x, y)\qquad (x, y\in{; ; ; \mathcal M}; ; ; ), $$ where $\varphi$ 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 $$ \|f(x)-I(x)\|\leq\sqrt{; ; ; \varphi(x, x)}; ; ; \qquad (x\in {; ; ; \mathcal M}; ; ; ). $$

Wigner equation ; C*-modules

nije evidentirano