I quote this excerpt from Conway:
"A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ where $g \in C(X)_+$; hence $\rho(f)=\rho(g)^2=\rho(g)^\ast\rho(g)\geq0$. So $\rho$ is a positive map. One might expect, by analogy with the Riesz Representation Theorem, that $\rho(f)=\int f dE$..."
There are some statements that are not clear to me and I ask for a clarification.
First of all, $C(X)_+$ is never defined before. I can think it's natural that is the set of continue positive real functions on $X$, but Conway always speaks about complex functions before. It's the set of continue positive real functions on $X$, right?
After that, it's not clear why $\rho$ is a positive map. I know when an operator is positive and when a linear functional is positive, but $\rho$ is neither an operator nor a functional.
And lastly it's not clear why the positiveness condition would help.
Thank you in advance for your answers.
Yes, $C(X)_+$ is the set of continuous functions on $X$ such that $f(x)\in[0,\infty)$ for all $x\in X$.
A positive map can be defined whenever you have a positive cone in the domain and a positive cone in the codomain. In this case, $\rho$ maps positive functions to positive operators, and so it is customary to call it "positive".
In the case of a $*$-homomorphism positiveness occurs by default (as Conway argument shows) so it is not particularly useful. In the quoted paragraph the point is that there seems to an analogy with characters.