Let $X$ and $Y$ be compact Hausdorff spaces, and let $F$ be a continuous function from $X$ to $Y$.
Define a function $\Phi_F$ from $C(Y)$ to $C(X)$ by $\Phi_F(f)=f\circ F.$
I have shown $\Phi_F$ is a unital $*$-homomorphism from $C(Y)$ to $C(X)$, where $*$ is the usual complex conjugation of complex-valued functions and it is naturally an involution. (i.e. I proved that $\Phi_F$ is an algebra homomorphism that carries identity element to identity element, and respects complex conjugation)
Now the problem is to prove that the mapping $F\mapsto \Phi_F$ is a bijection between the set of continuous functions from $X$ to $Y$ and the set of unital $*$-homomorphism from $C(Y)$ to $C(X)$.
The fact that this mapping is injective is trivial. But I can not figure out the surjective part. Can someone help me? Any help will be appreciated.
Thanks!