Are two compact Hausdorff spaces homeomorphic if their algebras of continuous functions are isomorphic?

Question

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!

Answer 1

This is the so-called Gelfand-Kolmogorov theorem. It says:

Let $X$ and $Y$ be compact, Hausdorff spaces. Suppose that there exists a ring isomorphism $T\colon C(X)\to C(Y)$. Then there exists a homeomorphism $h\colon Y\to X$ such that $$Tf = f\circ h\text{ for all }f\in C(X).$$ In particular, $T$ is a continuous algebra isomorphism.

Click here for some references. Actually there is a more general result due to Milgram which asserts that two compact Hausdorff spaces $X$ and $Y$ are homeomorphic if and only if there exists a multiplicative bijection between $C(X)$ and $C(Y)$.

Answer 2

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.