# functional analysis – about positive elements in a \$C^*\$- algebra

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!