The positive element in a C*-algebra

Question

The following is a theorem of Conway's Functional Analysis:

for the proof ($c\to a$), I think we can say: for $\lambda\in \sigma(a)\subset \Bbb R$, there is a character $h:C(\sigma(a))\to\Bbb C$ such that $\lambda=h(a)=|h(x)|^2$, so $\lambda\geq 0$ and $\sigma(a)\subset [0,\infty)$.

I think it's easier than Conway's way, but I'm not sure that it's correct or not. Please check it. Thanks in advance.

Answer 1

No, it isn't correct, because you are trying to use "$|h(x)|$" without it having been defined.