A group action approach to the Daugavet property

Mathematics > Functional Analysis [Submitted on 18 Jun 2026] Title:A group action approach to the Daugavet property View PDF HTML (experimental)Abstract:We introduce the $G$-Daugavet property ($G$-DPr, for short) for Banach spaces endowed with an action of a group $G$ by surjective linear isometries. This notion provides a common framework for the classical Daugavet property and the alternative Daugavet property, which correspond respectively to the trivial action and to the scalar action of $S_{\mathbb{K}}$. We establish several characterizations of the $G$-DPr in terms of $G$-slices and closed convex $G$-invariant hulls, recovering the usual slice descriptions of the DPr and the aDPr as particular cases. We show that the presence of a group action leads to new behavior in Daugavet theory. In particular, the $G$-DPr may hold on classical reflexive spaces in sharp contrast with the classical Daugavet property. We relate this phenomenon to convex transitivity, almost transitivity and finite-dimensional rotation problems. We also prove group-action versions of the classical characterizations for $L^1(\mu, X)$- and $C(K,X)$-spaces. The paper also studies group separable determination, $G$-versions of numerical radius and numerical index, and connections between the $G$-DPr and strong Radon-Nikodým and SCD operators. Finally, we introduce a parameter which measures how far the $G$-DPr is from the classical DPr in a quantitative manner. As a consequence of these results, we obtain conditions under which the $G$-DPr recovers several classical implications, including the failure of the RNP for both $X$ and $X^*$, the presence of copies of $\ell_1$ and the failure of the unit ball to be an SCD set. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.