Exercise 3.4.1. Let $\pi: A \otimes B \rightarrow C$ be a $*$-homomorphism which is injective when restricted to $A\odot B$. Show that $\pi$ must be injective on all of $A\otimes B$. Is this still true if one replaces $\|\cdot\|_{\text{min}}$ by $\|\cdot\|_{\text{max}}$?
Here, the $A\otimes B$ denote the completion of $A\odot B$ with respect to $\|\cdot\|_{\text{min}}$.
Could someone give me some hints of this exercise? Many thanks.
You can define a C*-norm $\|\,.\|_\pi$ on $A\odot B$ by $\|\sum a_i\otimes b_i\|_\pi = \|\pi(\sum a_i\otimes b_i)\|$. Since $\|\,.\|_{\text{min}}$ is the minimal C*-norm, it must be true that $\|\sum a_i\otimes b_i\|_\pi \geqslant \|\sum a_i\otimes b_i\|_{\text{min}}$. Deduce that $\pi$ must be an isometry from $A\otimes B$ to $C$ and is therefore injective.
To see whether this holds with $\|\,.\|_{\text{max}}$ replacing $\|\,.\|_{\text{min}}$, think about what happens if $C$ is $A\otimes B$ with the $\|\,.\|_{\text{min}}$-norm and $\pi$ is the canonical quotient map.