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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."

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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.

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  • $\begingroup$ I see, thanks Tomek! I was not familiar with Szlenk indices. Is there anything known about the concrete form of the algebras $C(X)$? How to construct them? $\endgroup$ – Niki Jan 11 '17 at 13:56

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