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

Abstract

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

12 Pages, To appaer in J. Math. Phys. (won an ISFE medal in the 45th International Symposium on Functional Equations, Poland, 2007)

References in corpus (1)

• Stability of Derivations on Hilbert $C^{\ast }$-Modules

Cited by in corpus (2)

• Angle Preserving Mappings
• Relationship between the Hyers-Ulam stability and the Moore-Penrose inverse