Notes on binary trees of elements in $C(K)$ spaces with an application to a proof of a theorem of H. P. Rosenthal

A Banach space $X$ contains an isomorphic copy of $C([0,1])$, if it contains a binary tree $(e_n)$ with the following properties (1) $e_n=e_{2n}+e_{2n+1}$ and (2) $c\underset{2^n\le k\text{tl}{2}^{n+1}}{max}|a_k|\le \Vert \sum _{k=2^n}^{2^{n+1}-1}{a}_k{e}_k\le C\underset{2^n\le k<2^{n+1}}{max}|a_k|$ for some constants $0<c\le C$ and every $n$ and any scalars $a_{2^n},\dots ,a_{2^{n+1}-1}$. We present a proof of the following generalization of a Rosenthal result: if $E$ is a closed subspace of a separable $C(K)$ space with separable annihilator and$S:E\to X$ is a continuous linear operator such that $S^{\ast }$ has nonseparable range, then there exists a subspace $Y$ of $E$ isomorphic to $C([0,1])$ such that $S{|}_Y$ is an isomorphism, based on the fact.