We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. We prove also a $C_p$-version of Rosenthal's theorem by showing that the space $C_p(X)$ contains a closed copy of $c_{0}(\Gamma)$ (with the pointwise topology) for some uncountable set $\Gamma$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$.

The talk is based on a joint work with Jerzy Kakol and Wieslaw Sliwa.