I'm currently reading proof of Proposition $5.2$ from the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras. Let $A$ be a simple $C^{\ast}$-algebra and $B$ be any $C^{\ast}$-algebra. Let $I$ be a closed ideal of $A\otimes^h B$. If $I_1$ and $I_2$ are ideals of $ B$ satisfying $M\otimes^h I_i \subseteq I$ for $i=1,2$. Then $$M\otimes^h(I_1+I_2)= M\otimes^h I_1+ M\otimes^h I_2 \subseteq I. $$ Thus
There is a largest closed ideal $J$ of $B$ such that $A \otimes^h J \subseteq I$
I'm not able to follow this step. By applying Zorn's lemma I can see that there is a maximal element $J$ but i cannot see the existence of largest ideal. Can someone please explain.
Let $\mathcal F$ be the set of ideals $I'\subset B$ such that $A\otimes^h I'\subset I$. Let $$ J=\overline{\Big\{\sum_{\alpha\in F}x_\alpha:\ F\subset\mathcal F,\ \text{ finite}, x_\alpha\in\alpha\Big\}}\subset B. $$ Then $J$ is an ideal of $B$ and by definition it contains all ideals $I'$ in $B$ such that $A\otimes^hI'\subset I$.
So all we need to do is to show that $A\otimes^h J\subset I$. But this follows easily from the fact that $$ A\otimes^h J=\overline{A\otimes\Big\{\sum_{\alpha\in F}x_\alpha:\ F\subset\mathcal F,\ \text{ finite}, x_\alpha\in\alpha\Big\}}\subset I. $$