Dynamical Systems on Hilbert C*-Modules

Abstract

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