Please help me with this problem. I want to prove that: "An element $x$ of a C*-algebra $A$ is positive if and only if for every state $\tau$ of $A$, we have $\tau(x) \geq 0$ ."
If $x$ is positive, it is easy to show that $\tau(x) \geq 0$. But my problem is that, if $\tau(x) \geq 0$ how can we deduce that $x \geq 0$ ?