Let $A$ be a $C^*$-Algebra and $a\in A$.
I'm stuck in the proof of:
$a\ge 0\iff $ it is $\varphi(a)\ge 0$ for all states (=positive linear functionals with norm 1) $\varphi:A\to\mathbb{C}$.
Proof:
$\Rightarrow$ is no problem, because if $a\ge 0$, there is a $b\in A$ such that $a=b^*b$. Therefore it is $\varphi(a)=\varphi(b^*b)\ge 0$, because $\varphi$ is positive.
The direction $\Leftarrow$ seems to be more complicated. For $a\in A$ you can write $a=Re(a)+iIm(a)$. Then: For all states $\varphi$ of A it is $\varphi(a)=\varphi(Re(a)^)+i\varphi(Im(a))\ge 0, \Rightarrow \varphi(Im(a))=0$ for all states $\varphi$ of A. But I don't know how to continue. It is to Show that a is self adjoint and $\sigma(a)\subseteq [0,\infty )$.
By Gelfand transform you can show that $\sigma(a)=\{\varphi(a):\varphi \text{ is a multiplicative linear fuctional } \}$. Since every multiplicative linear fuctional is positive, then you obtain $\Leftarrow$.