Let $\rho$ be a hermitian functional on $C(X)$ where $X$ is a compact and Hausdorff space. Suppose $\|\rho\|=\rho(1)$. Then show that $\rho$ is a positive linear functional on $C(X)$. Part of the problem was to specifically use a theorem that shows that $\rho$ can be written as a difference of linear functionals $\rho^+$ and $\rho^-$. The second part which is the one that I am stuck on asks us to do it without this theorem. This is problem 3.5.45 of Fundamentals of the Theory of Operator Algebras. Volume I: Elementary Theory Richard V. Kadison and John R. Ringrose.
I figured that an easy way to show this would be to show that if $f\geq 0$. Then $\rho(f)=|\rho(f)|$. However, when I tried this, I required the fact that $\rho$ was multiplicative. I'm not sure if there is enough information to show that $\rho$ is multiplicative. Any suggestions on different approaches or showing that $\rho$ is multiplicative?
For those interested here is what I did:
Assume $f\geq 0$. Then
$|\rho(f)|^2=\rho(f(x))\overline{\rho(f(x))}=\rho(f(x))\rho(\overline{f(x)})=\rho(f(x)\overline{f(x)})$ if $\rho$ is multiplicative.
$=\rho(|f(x)|^2)=\rho(f(x)^2)=\rho(f(x))^2$ again assuming $\rho$ is multiplicative.
I'm not sure where the fact that $\|\rho\|=\rho(1)$ comes in. Any help is appreciated.