Let $A,B$ be $C^*$-algebras and assume that at least one of the two is nuclear, so $A\odot B$ admits a unique $C^*$-norm. Let $J\triangleleft A\otimes B$ be a closed, two-sided ideal in $A\otimes B$.
What I am trying to prove is this: for every $x\in J$ and any $\varepsilon>0$ we can find $z\in A\odot B$, say $z=\sum_{j=1}^na_i\otimes b_i$ such that $\|x-z\|<\varepsilon$ and every elementary tensor $a_i\otimes b_i$ belongs to $J$.
The thing is that I have no idea where to get started with this. For example, if $A,B$ are unital, then $A\otimes B$ contains canonical copies of $A$ and $B$ ($A\cong A\otimes\mathbb{C}1_B$, $B\cong\mathbb{C}1_A\otimes B$). The ideal $J$ then induces ideals $J_A\triangleleft A$ and $J_B\triangleleft B$ (through the above identifications) and these satisfy $J_A\otimes J_B\subset J$. The question is the same as showing the reverse inclusion. I believe something similar can be done for the non-unital case, probably by passing to the unitization. On the other hand, given any pair of ideals $I_1\triangleleft A$ and $I_2\triangleleft B$, we obtain an ideal $I_1\otimes I_2\triangleleft A\otimes B$.
This argument shows that our question is more or less the same as asking whether the ideals of $A\otimes B$ correspond to pairs of ideals in $A$ and $B$.
I think this is implied in a proof I'm studying but I have no reference. Does anyone know how to prove this or know where I should be looking for a proof?