It seems that given positive elements in a $C^*$ algebra A, we can give an ordering on its elements. Namely given elements $a,b\in A$ is positive iff $a\geq 0$ and $b\leq 0$ iff $-b\geq 0$.
My question is that in a proof of $c^*c\geq 0$ in A the author took a contradiction approach, and I don't see why we can just assume $c^*c< 0$ because I don't see why $\sigma(c^*c)\subseteq \mathbb{R}_{<0}$. Can anyone explain this small nuance to me.
You cannot do the proof by contradiction assuming $c^*c<0.$
The book by Davidson "C*-algebras by example" contains the standard proof that $c^*c\geq 0.$ The author writes $c^*c=b_+-b_-,$ where $b_+,b_-\geq 0,\ b_+b_-=0$ and shows that $b_-=0.$