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!



About $C(K)$-spaces containing no copy of $l_{1}$
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 $C(K)$.


