Homomorphism between compact Hausdorff spaces.

Question

Suppose that X and Y are compact Hausdorff spaces and $p:C(X)\to C(Y)$ is a unital * homomorphism. Prove that there exists a continuous function $h: Y \to X $ such that $p(f)=f\circ h $ for all f in $C(X) $

I have managed to prove the other 3 parts ( show the statement false if $p$ is not unital as well as finding conditions on $h$ for $p$ to injective and surjective.

I have found a function that does what I want but I can't prove it's continuous.

Any help much appreciated.

Answer 1

The idea is this: For each $y\in Y$, the map $\varphi_y:C(X)\to\mathbb C$ given by $\varphi_y(f)=(pf)(y)$ is a $*$-homomorphism. Thus there is some $h(y)\in X$ such that $(pf)(y)=f(h(y))$ for all $f\in C(X)$ (because $\ker(\varphi_y)$ is a maximal ideal in $C(X)$, and maximal ideals correspond to elements of $X$). To show continuity, assume $\{y_\gamma\}$ is a net in $Y$ convergent to some $y\in Y$. Then for each $f\in C(X)$ we have $$f(h(y))=(pf)(y)=\lim_\gamma\ (pf)(y_\gamma)=\lim_\gamma\ f(h(y_\gamma))$$ and therefore $h(y)=\lim_\gamma h(y_\gamma)$.