Let $X$ be a compact countable Hausdorff space. By Sierpinski-Mazurkiewicz Theorem we know that $X$ is a compact countable ordinal, i.e.
$$
X \simeq \omega ^{\alpha} \cdot n + 1
$$
where $\alpha$ is countable and $n \ge 1$ an integer.


My question is: What do we know about $\mathcal{C}(X)$?


Obviously it is a commutative unital C\*-Algebra. But can we say anything more? Ideal of course would be a statement of the form: "$\mathcal{C}(X) $ has property $P$ (or is of the form $F$) iff $X$ is countable cpct Hsdff."



What can be said about the algebra of continuous functions on compact countable ordinals?
Let $X$ be a compact Hausdorff space. Then $C(X)$ has separable dual if and only if $X$ is countable. In this case, the dual is isomorphic to $\ell\_1$.


In the class of compact metric space, $C(X)$ has countable Szlenk index if and only if $X$ is countable.


