A question about maximal and minimal tensor product

Question

Let $A, B$ be two C*-algebras and $\pi: A\otimes_{\max} B\rightarrow M_{n}(\mathbb{C})$ be a representations, then this $\pi$ can factor through the minimal tensor product $A\otimes_{\min} B$ ? (That is, do there exist two representations $\phi:A\otimes_{\max} B\rightarrow A\otimes_{min}B$ and $\psi: A\otimes_{\min} B\rightarrow M_{n}(\mathbb{C})$ such that $\pi=\psi\circ\phi$.)

Answer 1

I try to answer my own question. And I think this question is not hard to answer if we notice the fact that $\pi=\pi_{A}\times\pi_{B}$, where $\pi_{A}: A\rightarrow M_{n}(\mathbb{C})$ and $\pi_{B}: B\rightarrow M_{n}(\mathbb{C})$ are two $*$-representations with commuting ranges.

Now, from universality, we have $$A\otimes_{max} B \xrightarrow{} A\otimes_{min} B\xrightarrow{\pi_{A}\otimes\pi_{B}}\pi_{A}(A)\otimes_{min}\pi_{B}(B)=\pi_{A}(A)\otimes_{max}\pi_{B}(B)\xrightarrow{I\times I} M_{n}(\mathbb{C}).$$ by $$a\otimes b\mapsto a\otimes b \mapsto\pi_{A}(a)\otimes \pi_{B}(b)\mapsto \pi_{A}(a)\pi_{B}(b).$$

where $I$ is the identity on $M_{n}(\mathbb{C})$.

Hence, the $\phi$, in the question, is the natural surjective *-homomorphisms from $A\otimes_{max}B\rightarrow A\otimes_{min} B$, while the $\psi=(I\times I)\circ {(\pi_{A}\otimes\pi_{B})}.$