Self-adjoint lift of idempotent in quotient

Question

Let $A$ be a $C^*$-algebra and $I\subseteq A$ a closed ideal. Let $a\in A$ be such that $a^2-a\in I$.

Question: Can one construct an element $b\in A$ in terms of $a$ such that $b$ is a self-adjoint lift of $[a]$?

Remark: By a lift I mean $[b]=[a]\in A/I$.

Answer 1

If $a\in A$ is such that $[a]$ is self-adjoint, then $b=\frac{a+a^*}{2}$ is self-adjoint and satisfies $[b]=\frac{[a]+[a]^*}{2}=[a]$. Conversely, if such a $b$ exists, then clearly $[a]=[b]$ must be self-adjoint. So assuming that $[a]$ is idempotent is not relevant; instead, such a $b$ exists iff $[a]$ is self-adjoint.