# Tensor product of faithful normal states is faithful

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.

This is true. Consider the GNS representations $$(\pi_\omega, H_\omega)$$ and $$(\pi_\varpi, H_\varpi)$$. Both are normal faithful representations, so $$A \otimes_{vN} B$$ can be regarded as acting on $$H_\omega \otimes H_\varpi$$. Since $$\omega$$ is faithful, $$\hat{1} \in H_\omega$$ is separating for $$A$$, whence it is cyclic for $$A’$$. Similarly, $$\hat{1} \in H_\varpi$$ is cyclic for $$B’$$. Hence $$\hat{1} \otimes \hat{1} \in H_\omega \otimes H_\varpi$$ is cyclic for $$(A \otimes_{vN} B)’ = A’ \otimes_{vN} B’$$, so it is separating for $$A \otimes_{vN} B$$. As $$\hat{1} \otimes \hat{1} \in H_\omega \otimes H_\varpi$$ implements the state $$\omega \otimes \varpi$$, this is the same as saying $$\omega \otimes \varpi$$ is faithful.