If I have a commutative $C^*$-algebra $\mathfrak{A}=C(X)$ and $\Phi : C(X) \rightarrow C(X)$ a *-endomorphism, what can I say about its structure? I was reading that every *-endomorphism between commutative $C^*$-algebra is a composition operator (in particular it is the pull-back of a map acting on its spectrum $X$), but I was searching for a clear explanation of this, also with a contextualized definition of pullback.
Thanks
Given any $x\in X$, you can define a character $C_x$ given by $C_xf=(\Phi f)(x)$. As characters are point evaluations, there exists $h(x)\in X$ such that $(\Phi f)(x)=f(h(x))$. You do this for every $x\in X$, and you show that $h$ is continuous. So $$\Phi f=f\circ h.$$