why is $f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$ injective? If $f:A\to B$, $g:C\to D$ are injective $\ast$-homomorphisms between $C^*$-algebras $A, B, C, D$, is the induced map on the spatial tensor product $$f\otimes g:A\otimes_{\min}C\to  B \otimes_{\min} D$$ injective as well?
I tried to prove it, but I'm stuck.
The spartial tensor norm on $A\otimes B$ is $$\|\sum\limits_{i=1}^na_i\otimes b_i\|_{\min}=\sup\{\|\pi_A\otimes \pi_B)(\sum\limits_{i=1}^na_i\otimes b_i)\|: \pi_a,\pi_B\;\ast-\text{respresentations}\}.$$
Maybe the fact $\pi_A$ and $\pi_B$ injective $=>$ $\pi_A\otimes \pi_B$ injective will be helpful. But my problem is that  it shouln't be enough to check injectivity of $f\otimes g$ on sums $z=\sum\limits_{i=1}^na_i\otimes c_i$. 
Could anyone give some help?

Tensor products of c.p. faithful maps with respect to the minimal norm are faithful:

D. Avitzour, Free products of C*-algebras, Trans. Amer. Math. Soc. 271 (1982), 423–435.

Certainly, *-homomorphisms are c.p.
