Tensor product of separating vectors in von Neumann algebra.

Question

Let $\mathcal{M}_1\subseteq \mathcal{B}(\mathcal{H}_1)$ and $\mathcal{M}_2\subseteq\mathcal{B}(\mathcal{H}_2)$ be von Neumann algebras with separating vectors $\xi_1\in \mathcal{H}_1$ and $\xi_2\in\mathcal{H}_2$ respectively. Then how can we prove that $\xi_1 \otimes \xi_2\in\mathcal{H}_1\overline\otimes\mathcal{H}_2$ is a separating vector of the von Neumann algebra $\mathcal{M}_1\overline\otimes\mathcal{M}_2$?

Answer 1

As $\xi_1\otimes\xi_2$ is cyclic for $\mathcal{M}_1'\overline\otimes\mathcal{M}_2'$ and we have $\mathcal{M}_1'\overline\otimes\mathcal{M}_2'\subseteq (\mathcal{M}_1\overline\otimes\mathcal{M}_2)'$, therefore $\xi_1\otimes\xi_2$ is cyclic for $(\mathcal{M}_1\overline\otimes\mathcal{M}_2)'$. Hence $\xi_1\otimes\xi_2$ is separating for $\mathcal{M}_1\overline\otimes\mathcal{M}_2$.