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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?

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Thanks. It seems that this result (tensor states separate points) is well-known for experts, but I could not find it in any textbook or papers. – Huichi Huang Sep 28 '12 at 2:33
up vote 4 down vote accepted

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 $0$.

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