University of MaraghehSahand Communications in Mathematical Analysis2322-5807Articles in Press20230312Dynamical Systems Implemented by Isomorphic Groups of Unitaries70319410.22130/scma.2023.559758.1158ENMaysamMosadeqDepartment of Mathematics, Larestan Branch, Islamic Azad University, Larestan, Iran.0000-0003-0456-3145Journal Article20220806Let $\varphi:A\to B$ be an isomorphism of $C^*$-algebras and $I$ be an ideal of $A.$ Introducing the concepts of unitary equivalent and the implemented Finsler modules, we show that the $\frac{A}{I}$-module $\frac{E}{E_{I}}$ and the implemented $\frac{B}{\varphi(I)}$-module $\frac{F}{F_{\varphi(I)}}$ are unitary equivalent. We also, establish a one to one correspondence between the groups $U(E)$ and $U(F)$ of unitaries on full Finsler modules $E$ and $F,$ respectively. Finally, we explain regularized dynamical systems and apply the aforementioned one to one correspondenec to prove that each regularized dynamical system in $U(E)$ implements a regularized dynamical system in $U(F).$