Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:29:34Z http://mathoverflow.net/feeds/question/108254 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108254/reference-request-tensor-products-of-states-separate-the-points-of-tensor-produc Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras Huichi Huang 2012-09-27T15:01:58Z 2012-09-28T02:34:37Z <p>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:</p> <p>is there any reference for us to cite this result?</p> http://mathoverflow.net/questions/108254/reference-request-tensor-products-of-states-separate-the-points-of-tensor-produc/108264#108264 Answer by Alain Valette for Reference request: tensor products of states separate the points of tensor product of $C^*$-alagebras Alain Valette 2012-09-27T16:57:27Z 2012-09-27T16:57:27Z <p>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 $0$.</p>