If $X$ is compact Hausdorff, is every isomorphicm $\mathcal{C}(X) \to \mathcal{C}(X)$ continuous?

Question

If $X$ is a compact Hausdorff space, is then every isomorphism from ${\mathcal C}(X)$ onto ${\mathcal C}(X)$ is continuous?

Answer 1

You should provide more information.

If you mean that we view $C(X)$ as a $C^*$-algebra and isomorphism means unital $^*$-isomorphism, then the answer is yes. In fact, any unital $^*$-homomorphism between two unital $C^*$-algebras is continuous.

Even weaker, every unital algebra automorphism of $C(X)$ is continuous.

Answer 2

Even more, in this case every ring isomorphism $C(X)\to C(Y)$ is automatically norm-continuous. This is part of the Gelfand-Kolmogorov theorem.