Let $M$ and $N$ be two von Neumann algebras. Suppose $\omega_1$ and $\omega_2$ are two normal states of $M$ and $N$ respectively.
We consider the tensor product von Neumann algebra $M\otimes N$. If we assume that $(M\otimes N)_{\omega_1\otimes \omega_2}=M_{\omega_1}\otimes N_{\omega_2}$, where $(M\otimes N)_{\omega_1\otimes \omega_2}$ is the centralizer of $M\otimes N$.
Can we have the following equality:
$$(M\otimes N)_{\omega_1\otimes \omega_2}' \cap (M\otimes N) =(M_{\omega_1}'\otimes N_{\omega_2}')\cap (M\otimes N) = (M_{\omega_1}'\cap M) \otimes (N_{\omega_2}'\cap N).$$
It is generally true that, when $A \subset M$ and $B \subset N$ are von Neumann subalgebras, then $(A \otimes B)’ \cap (M \otimes N) = (A’ \otimes B’) \cap (M \otimes N) = (A’ \cap M) \otimes (B’ \otimes N)$.