Non-separable tree-like Banach spaces and Rosenthal's $\ell_1$-theorem

Abstract

We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we can not achieve a satisfactory extension of Rosenthal's $\ell_1$-theorem to spaces of the type $\ell_1(\kappa)$, for $\kappa$ an uncountable cardinal.