Dynamical Systems on Hilbert Modules Over Locally $C^*$-Algebras

L. Naranjani, M. Hassani, M. Amyari

Let $\mathcal{A}$ be a locally $C^*$-algebra and $S(\mathcal{A})$ be the family of continuous $C^*$-seminorms and let $\mathcal{E}$ be a Hilbert $\mathcal{A}$-module. We prove that every dynamical system of unitary operators on $\mathcal{E}$ defines a dynamical system of automorphisms on the compact operators on $\mathcal{E}$ and show that under certain conditions, the converse is true. We define a generalized derivation on $\mathcal{E}$ and prove that if $\mathcal{E}$ is a full Hilbert $\mathcal{A}$-module and $\delta : \mathcal{E} \to \mathcal{E}$ is a bounded generalized derivation, then $\delta_p: \mathcal{E}_p \to \mathcal{E}_p$ is a generalized derivation on the Hilbert module $\mathcal{E}_p$ over the $C^*$-algebra $\mathcal{A}_p$ for each $p \in S(\mathcal{A})$.