Crossed product for covariant systems of finite type

Crossed product is a powerful tool in generating von Neumann algebras from covariant systems. In classical cases, the action space of a von Neumann algebra from a crossed product is extremely abstract. In order to make the action space simple, a covariant system of finite type was defined. In the system a concise characterization of the crossed product was given by constructing a new von Neumann algebra, which is isomorphic to the algebra from the classical case.