TITLE: About $C(K)$-spaces containing no copy of $l_{1}$
QUESTION [7 upvotes]: Let $K$ be a compact Hausdorff space. I wonder whether there are characterizations of $K$ such that $C(K)$ contains no copy of $l_{1}$. There are some compact Hausdorff spaces $K$ such that $C(K)$ contains no copy of $l_{1}$, for example, if $K$ is a countable compact metric space and has finite Cantor-Bendixson index. Are there more compact Hausdorff spaces $K$ such that $C(K)$ contains no copy of $l_{1}$?
Thank you!

REPLY [7 votes]: Yes, there is such characterisation. $C(K)$ contains no isomorphic copy of $\ell_1$ if and only if $K$ is scattered. Indeed, if $K$ is scattered then $C(K)^*$ is isometric to $\ell_1(K)$, so $C(K)$ cannot contain $\ell_1$, as then $C(K)^*$ would have contained a copy of $L_1$. Conversely, if $K$ is not scattered, then you may find a copy of $C[0,1]$ in $C(K)$.