Invariant subspaces and representations of certain von Neumann algebras (Q2750843)

Let \(\sigma\) be a unitary representation of a locally compact ordered group \(G\) on a Hilbert space \(H\) and let \(N\) be a von Neumann algebra acting on \(H\) such that \(\sigma(g)N\sigma(g)^*=N\) for each \(g\in G\). Denote by \(M\) (resp. \(B\)) the von Neumann algebra generated by \(N\) and by all \(\sigma(g)\), where \(g\in G\) (resp. \(g\in G_+\)). In the case, when \(G\) is either \(\mathbb Z\) or \(\mathbb R\), it is proved that, if there exists a pure full \(B\)-invariant subspace of \(H\), then \(M=N\rtimes_\sigma G\) and \(B=N\rtimes_\sigma G_+\).