Given two C*-algebras $A$ and $B$ and let $A_1$ and $B_1$ be their C*-subalgebras. Can we conclude that $A_1 \otimes_\min B_1$ is a subalgebra of $A \otimes_\min B$?
I think that this is not true, however I don't know how to construct a counter-example.
It is true for the min-norm, because you can construct the minimal tensor product explicitly by using two faithful representations. Namely, if you fix two faithful representations $\pi:A\to B(H)$ and $\mu:B\to B(K)$, then you have $$ A\otimes_\min B=\overline{\pi(A)\otimes\mu(B)}\subset B(H\otimes K). $$ Now you can use the restrictions $\pi|_{A_1}$ and $\mu|_{B_1}$ to construct $A_1\otimes_\min B_1$, which then sits naturally in $A\otimes_\min B$.