Dynamical Systems on Hilbert C*-Modules

Abstract

We investigate the generalized derivations and show that every generalized derivation on a simple Hilbert $C^{\ast }$-module either is closable or has a dense range. We also describe dynamical systems on a full Hilbert $C^{\ast }$-module $\mathcal{M}$ over a $C^{\ast }$-algebra $\mathcal{A}$ as a one-parameter group of unitaries on $\mathcal{M}$ and prove that if $\alpha :\mathbb{R}\to U(\mathcal{M})$ is a dynamical system, where $U(\mathcal{M})$ denotes the set of all unitary operator on $\mathcal{M}$, then we can correspond a $C^{\ast }$-dynamical system ${\alpha }^{{}^{\prime }}$ on $\mathcal{A}$ such that if $\delta$ and $d$ are the infinitesimal generators of $\alpha$ and ${\alpha }^{{}^{\prime }}$ respectively, then $\delta$ is a $d$-derivation.

7 pages, minor changes, to appear in Bull. Iranian Math. Soc

Cited by in corpus (1)

• Generators of Dynamical Systems on Hilbert Modules