2025/10/27 by Sheldon Dantas, Dantas, Sheldon, Michal Doucha +5
Mathematics · #math.FA
paper · pdf · doi:10.48550/arxiv.2510.23100
We introduce and study equivariant versions of the Radon-Nikodým property for Banach spaces, together with the closely related notions such as dentability, the Bishop-Phelps and Krein-Milman properties, and Lindenstrauss' property A, all considered in the presence of a continuous group action by linear isometries. While in the classical setting the Radon-Nikodým property, the Bishop-Phelps property and dentability are equivalent, the equivariant situation turns out to depend essentially on the acting group and requires non-trivial tools from abstract harmonic analysis and representation theory. We establish several implications among the equivariant counterparts of these properties. Namely, given a compact group G, the G-Bishop-Phelps property implies strong G-dentability, which in turn implies the G-Krein-Milman property and the classical Bishop-Phelps property, for any G-Banach space. Moreover, given a locally compact and second countable group G, the G-Radon-Nikodým property is equivalent to the classical Radon-Nikodým property, for any G-Banach space.