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!

subspaceisomorphic/isometric to $\ell^1$? $\endgroup$ – YCor Jun 10 '16 at 23:06