Invariant subspaces and representations of certain Von Neumann algebras

Let (N,alpha ,G) be a covariant system and let (pi ,U) be a covariant repre sentation of (N,alpha ,G) on a Hilbert space H. In this note, we investigat e the representation of the covariance algebra M and the sigma -weakly clos ed sub-algebra U generated by pi (N) and {U-g}(g greater than or equal to0) in the case of G = Z or R when there exists a pure, full, U-invariant subs pace of H.