positive elements in $C^*$ algebras and states I have problems to prove that an element $a $ is a $C^*$-algebra is positive if and only if $f(a) \geq 0$ for all states $f$.
The definitions I use: -$f:A\to\mathbb{C}$ linear functional on a C*-algebra A is called positive, if $f(a^*a)\ge 0$ for every a in A. 
-a state f is a positive functional with norm 1. 
-$a\ge0$, this means a is selfadjoint and $\sigma(a)\subseteq[0,\|a\|]$. 
The direction => is clear. But i stuck on <=.
If i follow the hint, i use the representation theorem: I have a universal representation $(H,g)$ of A and is $x\in A$, a positive linear functional $f:A\to\mathbb{C}\; f(b)=\langle g(b)x,x\rangle $ such that $f(a)\ge 0$. This is similar as you can see in Murphys book, Theorem 3.4.3. But now, i can't say that $g(a)$ is selfadjoint, because a is not selfadjoint in general. How can i continue? Regards 
An other hint is: First consider $A\cong C_0(X)$ commutative. But i dont know to prove this with this hint.  Maybe you could prove it without the representiation theorem.

Asyou remarked, one direction is trivial from definitions. 
For the other direction you can use the following well known facts:
1) An element is positive if and only if its spectrum is contained in $[0,\infty )$
2) If $\lambda$ belongs to spectrum of $a$ then there exists a state $f$ such that $f(a)=\lambda$.
