A $c_0$ saturated Banach space with tight structure

TL;DR

This paper introduces a new $c_0$ saturated Banach space, $rak{X}_0$, with a tight structure, not embeddable into spaces with unconditional bases, and with a unique subspace classification.

Contribution

It constructs a novel $c_0$ saturated space with a rigid structure and detailed subspace and operator properties, extending the HI space methods.

Findings

The space $rak{X}_0$ is not embeddable into spaces with unconditional bases.

Complemented subspaces are either isomorphic to $rak{X}_0$ or to subspaces of $c_0$.

Any analytic decomposition splits into one subspace of each type.

Abstract

It is shown that variants of the HI methods could yield objects closely connected to the classical Banach spaces. Thus we present a new $c_0$ saturated space, denoted as $\mathfrak{X}_0$, with rather tight structure. The space $\mathfrak{X}_0$ is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure. Everyone is either of type I, namely, contains an isomorph of $\mathfrak{X}_0$ itself or else is isomorphic to a subspace of $c_0$ (type II). Furthermore for any analytic decomposition of $\mathfrak{X}_0$ into two subspaces one is of type I and the other is of type II. The operators of $\mathfrak{X}_0$ share common features with those of HI spaces.