Let A be a C$^*$-Algebra, $a\in A$. Why is $a\ge 0$ (a is called "positive") iff $\forall \varphi\in S(A): \varphi\ge0$? S(A) is the set of linear positive functional $\eta:A\to\mathbb{C}$ with $\|\eta\|=1$.
Maybe i have to use Gelfand-Naimark/continuous functional-calculus
Is this Theorem 3.4.3 in Murphy's book"C*-Algebras and Operator Theory"? Why $\|\eta\|=1$ is important?
If $\varphi(a)\geq0$ for all states $\varphi$, you can do the following:
Note that a state is selfadjoint, i.e. it maps selfadjoints to real numbers. This, because any selfadjoint is a difference of two positives.
Write $a=b+ic$ with $b,c$ selfadjoint. Then $\varphi(b)$ and $\varphi(c)$ are real. So $\varphi(a)=\varphi(b)+i\varphi(c)$ is positive, which then shows that $\varphi(c)=0$. As $\varphi$ can be any state, we conclude that $c=0$. So $a$ is selfadjoint.
Now you can use the Gelfand representation to conclude that $a\geq0$.