Quotients of Banach spaces with the Daugavet property

TL;DR

This paper generalizes the Daugavet property in Banach spaces, introduces related concepts like narrow operators, and demonstrates that certain quotients of $L_1[0,1]$ can lack this property, answering a longstanding question.

Contribution

It extends the Daugavet property framework to a broader setting and provides a counterexample regarding quotients of $L_1[0,1]$, which was previously unresolved.

Findings

A quotient of $L_1[0,1]$ over an $\\ell_1$-subspace can fail the Daugavet property.

The generalized concept encompasses both the usual and weak$^*$ Daugavet properties.

New analogues for narrow operators and rich subspaces are introduced and studied.

Abstract

We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative.