Let $A$ be a $C^{\ast}$-algebra. Let $I_1$ and $I_2$ be closed ideals in $A$, then it is known that $I_1+I_2$ is closed in $A$. This motivated me to ask the following:
Let $\{I_{\alpha}\}_ {\alpha \in I}$ be a collection of closed ideals in $A$, then is it true that $\sum_{\alpha \in I} I_{\alpha}= \{ \sum_ { \alpha \in I'} x_{\alpha}, x_\alpha \in I_{\alpha} , I' \subseteq I, I'\hspace{0.1mm} \text{is finite}\}$ is closed in $A$.
Any references?
Such a thing already fails for families of subspaces (you always need to take the closure). In any case, here is a specific example in your situation.
Let $A=\ell^\infty(\mathbb N)$, $I=\mathbb N$, $I_\alpha=\mathbb C\,e_n$. Then what you called $\sum_\alpha I_\alpha$ is $c_{00}$, the set of functions that are eventually zero. The closure of $c_{00}$ is $c_0$