Group equivariant Radon-Nikod\'ym property and its characterizations

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.