Approximate Identity: Dense Ideal

Question

Given a C*-algebra without identity $1\notin A$.
Consider its minimal unitization $A\rtimes\mathbb{C}$.
Regard a dense twosided ideal $\overline{I}=A$.

Construct the element: $$I\owns i\geq0:\quad\frac{i}{\varepsilon+i}\in I\quad(\varepsilon>0)$$

Why does it belong to the ideal?

For closed ideals this is clear: $$\frac{1}{\varepsilon+i}\in A\rtimes \mathbb{C}\implies\frac{\sqrt{i}}{\varepsilon+i}\in A\implies\frac{i}{\varepsilon+i}\in I$$

But what about dense ideals?

Reference: Dixmier, C*-algebras, Proof to Proposition 1.7.2

Answer 1

$(i+ \epsilon)^{-1}$ lives in the unitization of $A$, which contains $I$ as an ideal. You don't need that $I$ is dense.