Variants of the James Tree space

TL;DR

This paper introduces variants of the James Tree space to explore the existence of $ ext{ell}_2$-saturated, hereditarily indecomposable Banach spaces, addressing a question about their structure and properties.

Contribution

The paper defines and studies two new variants of the James Tree space, providing a framework for future spaces that could be $ ext{ell}_2$-saturated and hereditarily indecomposable.

Findings

Defined two variants of the James Tree space, $JT_{2,p}$ and $JT_G$.

Provided structural analysis of these variants.

Laid groundwork for constructing $ ext{ell}_2$-saturated hereditarily indecomposable spaces.

Abstract

Recently, W. Cuellar Carrera, N. de Rancourt, and V. Ferenczi introduced the notion of $d_2$-hereditarily indecomposable Banach spaces, i.e., non-Hilbertian spaces that do not contain the direct sum of any two non-Hilbertian subspaces. They posed the question of the existence of such spaces that are $\ell_2$-saturated. Motivated by this question, we define and study two variants $JT_{2,p}$ and $JT_G$ of the James Tree space $JT$. They are meant to be classical analogues of a future space that will affirmatively answer the aforementioned question.