I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in verifying a statement given in last paragraph of Page $112$.
Let $A$ and $B$ be $C^{\ast}$-algebras and $\epsilon: A \otimes B \to A\otimes B$ be identity map. Since Haagerup norm dominates the min norm, so $\epsilon$ extends to a contraction from $ A \otimes^h B$ to $ A \otimes^{\text{min}} B$. It is known that $\epsilon$ is injective.
Let $I$ be an ideal of $ A \otimes^h B$ then $\overline{\epsilon(I) }$ is an ideal of $ A \otimes^{\text{min}} B$.
It is easy to see that if we take an elementary tensor from $\overline{\epsilon(I) }$ and multiply with an element of $A\otimes^{\text{min}} B$, then it belongs to $\overline{\epsilon(I) }$ but I'm having trouble in verifying in general case. It must be easy but I'm unable to see it somehow. Please help.
By construction, the range of $\epsilon $ is dense.
Given $y\in A\otimes^\min B$, there exists $\{z_n\}\subset A\otimes B$ with $y=\lim_n \epsilon(z_n)$. For any $x\in J$, $$ y\epsilon(x)=\lim_n\epsilon(z_n)\epsilon(x)=\lim_n\epsilon(z_nx)\in\overline{\epsilon(J)}. $$ So if $w\in\overline{\epsilon(J)}$ there exists $\{x_n\}\subset J$ such that $w=\lim_n\epsilon(x_n)$. Thus by the above $$ yw=\lim_k y\epsilon(x_k)\in\overline{\epsilon(I)}. $$