Quotients of Banach Spaces with the Daugavet Property
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 of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ by an $\ell_1$-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.