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

Artur Michalak. "Notes on binary trees of elements in $C(K)$ spaces with an application to a proof of a theorem of H. P. Rosenthal." Commentationes Mathematicae 46.2 (2006): null. <http://eudml.org/doc/292236>.

@article{ArturMichalak2006,
abstract = {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) $c \max _\{2^n\le k2^\{n+1\}\} |a_k| \le \Vert \sum _\{k=2^n\}^\{2^\{n+1\}-1\} a_k e_k \le C\max _\{2^n\le k2^\{n+1\}\} |a_k|$ for some constants $0c \le C$ 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$S\colon E \rightarrow X$ 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.},
author = {Artur Michalak},
journal = {Commentationes Mathematicae},
keywords = {$C(K)$-spaces},
language = {eng},
number = {2},
pages = {null},
title = {Notes on binary trees of elements in $C(K)$ spaces with an application to a proof of a theorem of H. P. Rosenthal},
url = {http://eudml.org/doc/292236},
volume = {46},
year = {2006},
}

TY - JOUR
AU - Artur Michalak
TI - Notes on binary trees of elements in $C(K)$ spaces with an application to a proof of a theorem of H. P. Rosenthal
JO - Commentationes Mathematicae
PY - 2006
VL - 46
IS - 2
SP - null
AB - 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) $c \max _{2^n\le k2^{n+1}} |a_k| \le \Vert \sum _{k=2^n}^{2^{n+1}-1} a_k e_k \le C\max _{2^n\le k2^{n+1}} |a_k|$ for some constants $0c \le C$ 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$S\colon E \rightarrow X$ 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.
LA - eng
KW - $C(K)$-spaces
UR - http://eudml.org/doc/292236
ER -