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

TL;DR

This paper investigates the stability of the Wigner equation within inner product modules over $C^*$- and von Neumann algebras, demonstrating that approximate solutions are close to exact solutions under certain conditions.

Contribution

It establishes the existence of exact solutions near approximate ones for the Wigner equation in the setting of inner product modules over operator algebras.

Findings

Existence of a solution $I$ close to the approximate solution $f$.

Bound on the deviation between $f$ and $I$ depending on the control function.

Extension of stability results to modules over $C^*$- and von Neumann algebras.

Abstract

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 $$\|f(x)-I(x)\|\leq\sqrt{\phi(x,x)} \qquad (x\in {\mathcal M}).$$