Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a family of copies of $c_0(\omega_1)$ in $C(K)$ such that

$Y_\alpha\cap Y_\beta=\{0\}$ for $\alpha\neq \beta$

Can we describe somehow the closure $Y$ of the subspace

$\sum_{\alpha<\omega_1}Y_\alpha$?

May it be isomorphic to $c_0(\omega_1)$ in some particular cases?