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 $x\in A$, $x^*x$ 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!
Let $x\in A$ be selfadjoint with $\pi(x)=k$. Write $x=x^+-x^-$, with $x^+,x^-$ positive and $x^+x^-=0$. Then $$ k=\pi(x^+)-\pi(x^-). $$ The uniqueness of the plus-minus decomposition implies that $\pi(x^-)=0$. Explicitly, \begin{align} 0&\leq\pi(x^-)^{1/2}k\pi(x^-)^{1/2}=\pi((x^-)^{1/2}x^+(x^-)^{1/2}-(x^-)^2)\\[0.3cm] &=-\pi(x^-)^2, \end{align} implying that $\pi(x^-)=0$.