Are two compact Hausdorff spaces homeomorphic if their algebras of continuous functions are isomorphic? Suppose that $X$ and $Y$ are two compact Hausdorff spaces and $F\colon C(X) → C(Y)$ is a continuous isomorphism of algebras. Can I say $X$ and $Y$ are homeomorphic?
The key words always lead me to other questions. Can anyone give me some reference? Thanks a lot!

An algebra isomorphism of $C(X)$ to $C(Y)$ is automatically continuous in the norm, because $\|f\| = \sup \{|\lambda|: \lambda \in \sigma(f)\}$.  $F$ induces a homeomorphism of the maximal ideal spaces, and these are in turn homeomorphic to $X$ and $Y$ respectively.
