Let $A$ be a $C^*$-algebra.
Let $pi: A rightarrow A/I$ be the canonical *- homomorphism, where $I$ is a closed ideal of A.
Show that if $k$ is positive in $A/I$, then there exists a positive element $a$ in $A$ such that $pi(a)=k$.
I know that since $k$ is self-adjoint, we can find a self-adjoint $x in A$ such that $pi(x)= k $.
Also, we know that for any $xin A$, $x^*x$ is positive.
So $pi(x^*x) = pi(x^*) pi (x) = kk$. But here I’m not sure how to use the positivity of $k$ to show $x$ is positive.
Any help will be appreciated!