Subspaces of almost Daugavet spaces

TL;DR

This paper investigates the structure of subspaces within almost Daugavet spaces, establishing conditions under which subspaces inherit the almost Daugavet property, especially focusing on quotients lacking an $ ext{ell}_1$ subspace.

Contribution

It proves that subspaces with quotients free of $ ext{ell}_1$ copies in separable almost Daugavet spaces also possess the almost Daugavet property.

Findings

Subspaces inherit the almost Daugavet property under certain conditions.

Quotients containing no $ ext{ell}_1$ copies ensure subspace properties.

Main result applies to separable Banach spaces with the almost Daugavet property.

Abstract

We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\ell_1$, then $Z$ has the almost Daugavet property, too.