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.