I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful.

I also believe I can prove that if $A,B$ equal the von Neumann algebras $B(\mathcal{H}), B(\mathcal{K})$ for some Hilbert spaces $\mathcal{H},\mathcal{K}$, then if $\omega, \varpi$ are additionally normal, the tensor product state $\omega\otimes\varpi$ is faithful (and normal) on the standard tensor product $A\otimes_{\text{vN}}B=B(\mathcal{H}\otimes\mathcal{K})$, essentially by representing $\omega,\varpi$ by trace-class operators and then reasoning in an appropriate eigenbasis of $\mathcal{H}\otimes\mathcal{K}$.

This sets up the following question: suppose $A,B$ are any von Neumann algebras and $\omega,\varpi$ faithful normal states. Is it true that $\omega\otimes\varpi$ is faithful on $A\otimes_{\text{vN}}B$? Unfortunately $\otimes_{\text{vN}}\neq\otimes_{\text{min}}$ for von Neumann algebas, so I cannot apply the first result I cited.