Separating vector and representations

Question

Let $\mathcal A$ be a von Neumann algebra acting on the separable Hilbert space H (which has not necessarily a separating vector).

Can we find a faithful representation $\,\pi$ of $\,\mathcal{A}\,$ on some separable Hilbert space $\,K$ in which the von Neumann algebra $\,\pi(\mathcal{A})$ has a separating vector?

Thank you in advance.

Answer 1

Yes. Because $H$ is separable, $\mathcal A$ is $\sigma$-finite. Then you can apply Lemma 2.8 in Haagerup's 1975 Standard Form paper, that says that the standard form of $\mathcal A$ has a cyclic and separating vector.