The HI extension of the standard HI spaces

TL;DR

This paper introduces a method to construct Hereditarily Indecomposable (HI) extensions of standard HI Banach spaces, including notable examples like the Gowers-Maurey space, addressing a longstanding open problem.

Contribution

It provides a new technique for defining HI-extensions of standard HI spaces, expanding the class of known HI extensions and advancing understanding of their structure.

Findings

Developed a method to construct HI-extensions of standard HI spaces

Included key examples like Gowers-Maurey space in the class of HI-extensions

Addressed the open problem of existence of HI-extensions for all HI spaces

Abstract

A Hereditarily Indecomposable (HI) Banach space $X$ admits an HI extension if there exists an HI space $Z$ such that $X$ is isomorphic to a subspace $Y$ of $Z$ and $Z/Y$ is of infinite dimension. The problem whether or not every HI space admits an HI extension is attributed to A. Pelczynski. In this paper we present a method to define HI-extensions of the standard HI spaces, a class which includes the Gowers-Maurey space, asymptotic $\ell_{p}$-HI spaces and others.