This might be a basic question but I'm not seen it through. Let $\mathcal A$ be a C$^*$-algebra and $I$ a two sided closed ideal, we define the projection $\pi:\mathcal A \to I$ given by $$\pi(a)=1_I\cdot a$$ where $1_I$ is the unit of $I$. This should be a well defined continuous projection.
If we consider the case where we have no unit in $I$ could we take an approximate unit of $I$ and just define the projection with it? I mean something like $\tilde{\pi}:\mathcal A\to I $ such that $$\tilde{\pi}(a)=\lim_\lambda e_\lambda a$$ where $\{e_\lambda\}_{\lambda}$ is an approximation unit in $I$, I'm not even sure this limit exists. Any help would be appreciated!