Unitary equivalence between two representations

Here is a quotation of a book:

Let $\phi$ and $\psi$ be the faithful states on $A$ and $B$ respectively, and let $||.||_{\alpha}$ be any C*-norm on $A\odot B$ (algebraic tensor product). As we know, we can extend $\phi\odot \psi$ (on $A\odot B$) to a state $\phi \otimes_{\alpha} B$ on $A\otimes_{\alpha} B$. By uniqueness of GNS representations it follows that the representations $\pi_{\phi\otimes_{\alpha} \psi}|_{A\odot B}$ and $\pi_{\phi} \odot \pi_{\psi}$ are unitarily equivalent. (Here, the $\pi_{\phi\otimes_{\alpha} \psi}$, $\pi_{\phi}$ and $\pi_{\psi}$ denote the GNS representations of $\phi\otimes_{\alpha} \psi$, $\phi$ and $\psi$ respectively.)

I do not know how to verify the unitarily equivalent here. Could someone give me some hints?

Solution 1:

You need to convince yourself that $\pi_\phi\otimes\pi_\psi$ is a GNS representation for $\phi\otimes_\alpha\psi$.

You have \begin{align*} \langle \pi_\phi\otimes\pi_\psi(\sum a_j\otimes b_j)\,v_\phi\otimes v_\psi,v_\phi\otimes v_\psi\rangle &=\sum_j \langle \pi_\phi(a_j)v_\phi,v_\phi\rangle\,\langle\pi_\psi(b_j)v_\psi,v_\psi\rangle\\ &=\sum_j\phi(a_j)\psi(b_j)\\ &=\phi\otimes\psi(\sum_ja_j\otimes b_j). \end{align*} And $$ \pi_\phi\otimes\pi_\psi(\sum_ja_j\otimes b_j)(v_\psi\otimes v_\psi)=\sum \pi_\phi(a_j)v_\phi\otimes \pi_\psi(b_j)v_\psi $$ shows that $H=\overline{\pi_\phi\otimes\pi_\psi(A\odot B)v_\phi\otimes v_\psi}=H_\phi\otimes H_\psi$. Then the uniqueness of the GNS representation (a particular case of the uniqueness in Stinespring) shows that existence of your unitary.