Invariant subspaces and representations of certain von Neumann algebras

Let $(N,\alpha ,G)$ be a covariant system and let $(\pi,U)$be a covariant representation of $(N,\alpha,G)$ on a Hilbert space $\mathcal{H}$. In this note, we investigate the representation of the covariance algebra $M$ and the $\sigma$-weakly closed subalgebra $\mathfrak{A}$ generated by $\pi (N)$ and $\{U_{g}\}_{g \geq 0}$ in the case of $G= \mathbb{Z}$ or $\mathbb{R}$ when there exists a pure, full, $\mathfrak{A}$-invariant subspace of $\mathcal{H}$.