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."