Let $||.||_{\alpha}$ be a C*-norm on $A\odot B$, $A\otimes_{\alpha} B$ be the completion and $\xi$ be a state on $A\otimes_{\alpha} B$. Let ($\pi_{\xi}, H_{\xi}, v_{\xi}$) be the GNS triplet and $\pi_{\xi, A}$ and $\pi_{\xi, B}$ be the restriction homomorphisms on $A$ and $B$ respectively.
My question is: How to verify $\pi_{\xi}(A\otimes_{\alpha} B)$ is contained in the $C^{*}$-algebra generated by $\pi_{\xi, A}$ and $\pi_{\xi, B}$ ?
You have $$ \pi_\xi(a\otimes b)\left(\sum c_j\otimes d_j\right)v_\xi=\left(\sum ac_j\otimes bd_j\right)v_\xi=\pi_{\xi,A}(a)\left(\sum c_j\otimes bd_j\right)v_\xi =\pi_{\xi,A}(a)\pi_{\xi,B}(b)\left(\sum c_j\otimes d_j\right)v_\xi. $$ Then $$ \pi_\xi(\sum a_j\otimes b_j)=\sum \pi_{\xi,A}(a_j)\pi_{\xi,B}(b_j)\in C^*(\pi_{\xi\,A}(A),\pi_{\xi,B}(B)). $$ By continuity, $\pi_\xi(A\otimes_\alpha B)\subset C^*(\pi_{\xi\,A}(A),\pi_{\xi,B}(B)).$