Suppose $A\otimes B$ is the minimal tensor product of two unital $C^\*$ algebras $A$ and $B$.
We know that the set of states, $\{\phi\otimes\psi|\phi\in S(A),\psi\in S(B) \}$ on $A\otimes B$ separates the points of $A\otimes B$. Here $S(A)$ and $S(B)$ are the state spaces of $A$ and $B$ respectively.
 My question is:


is there any reference for us to cite this result?



Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras 
Let $\pi\_A$ and $\pi\_B$ be faithful rep's of $A$ and $B$. Then $\pi=:\pi\_A\otimes\pi\_B$ is a faithful rep of $A\otimes\_{min} B$ (see Thm 4.9(iii) in Chapter 4 in M. Takesaki, Theory of Operator algebras I, Springer-Verlag, 1979). For $x$ a non-zero element in $A\otimes\_{min} B$, use then a vector state associated with some elementary tensor $\xi\_A\otimes\xi\_B$ to separate $\pi(x)$ from $0$.


