Given $C^{\ast}$-algebras $A_1$, $A_2$ and $B$, if $\pi_1 : A_1 \to B$ and $\pi_2: A_2 \to B$ are homomorphism with commuting ranges, then there exists unique homomorphism $\pi: A_1 \otimes^{\text{max}}A_2 \to B$ satisfying $$\pi(a \otimes b) = \pi_1(a) \pi_2(b) $$
Suppose $\pi_1$ and $\pi_2$ are faithful. Can we say that $\pi$ is faithful?
No. Take $A_1=A_2=B=C[0,1]$ and $\pi_1=\pi_2=\text{id}$. Now take $f,g\in C[0,1]$ continuous non-zero functions with $f$ supported inside $[0,1/3]$ and $g$ supported inside $[2/3,1]$. Since both are non-zero, the elementary tensor $f\otimes g$ is non-zero. However, $\pi(f\otimes g)=f\cdot g=0$.