The Daugavet property in spaces of vector-valued Lipschitz functions

TL;DR

This paper demonstrates that certain metric spaces with the finite CEP property induce the Daugavet property in associated vector-valued Lipschitz function spaces, expanding understanding of geometric properties in Banach space theory.

Contribution

It establishes that metric spaces with the finite CEP property lead to the Daugavet property in Lipschitz function spaces, generalizing previous results to a broader class of spaces.

Findings

Spaces with finite CEP have the Daugavet property in their Lipschitz function spaces.

Injective Banach spaces and convex subsets of Hilbert spaces also exhibit the Daugavet property.

The results apply to spaces where the dual is an $L_1(u)$ space.

Abstract

We prove that if a metric space $M$ has the finite CEP then $\mathcal F(M)\widehat{\otimes}_{\pi} X$ has the Daugavet property for every non-zero Banach space $X$. This applies, for instance, if $M$ is a Banach space whose dual is isometrically an $L_1(\mu)$ space. If $M$ has the CEP then $L(\mathcal F(M),X)=\Lip(M,X)$ has the Daugavet property for every non-zero Banach space $X$, showing that this is the case when $M$ is an injective Banach space or a convex subset of a Hilbert space.