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) \(c \max_{2^n\leq k\tl 2^{n+1}} |a_k| \leq \|\sum_{k=2^n}^{2^{n+1}-1} a_k e_k \leq C\max_{2^n\leq k\lt 2^{n+1}} |a_k|\) for some constants \(0\lt c \leq C\) 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\(S\colon E \to X\) is a continuous linear operator such that \(S^{∗}\) 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.