If $C(K)$ is isomorphic to $C(H)$, are $K$ and $H$ homeomorphic?

Question

Let $K,H$ be compact Hausdorff spaces. The Banach Stone theorem says that $K$ and $H$ are homeomorphic $\iff$ $C(K)$ and $C(H)$ are isometrically isomorphic. Is it true that if $C(K)$ and $C(H)$ are just isomoprhic, that $K$ is homeomorphic to $H$?

Answer 1

If $K$ and $H$ are both uncountable, then by Miljutin theorem (see theorem 4.4.8 in Topics in Banach space theory by Fernando Albiac, Nigel J. Kalton) $C(K)$ and $C(H)$ are isomorphic.

Thus there is an abundance of counterexamples: take any two uncountable topological spaces that are not homeomorphic, say $K=[0,1]$, $H=[0,1]\cup\{2\}$.

As for countable compacts, see S. V. Kislyakov. Classification of spaces of continuous functions of ordinals. Siberian Mathematical Journal, 1975, Volume 16, Number 2, Page 226

Answer 2

If K is the unit circle, and H is the unit interval, I think you can construct an explicit isomorphism between C(K) and C(H) by sending trigonometric polynomials to ordinary polynomials and vice versa. By the Stone-Weierstrass theorem, the above rule defines an isomorphism.