On the non-embedding of ${\ell }_1$ in the James Tree Space

Abstract

James Tree Space ($\mathcal{J}\mathcal{T}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing ${\ell }^1$. James actually proved that every infinite dimensional subspace of $\mathcal{J}\mathcal{T}$ contains a Hilbert space, which implies the ${\ell }^1$ non-embedding. In this expository article, we present a direct proof of the ${\ell }^1$ non-embedding, using Rosenthal's ${\ell }^1$- Theorem and some measure theoretic arguments, namely Riesz's Representation Theorem.

arXiv commentAccepted in Expositiones Mathematicae-Elsevier