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

TL;DR

This paper explores how classical geometric properties of Banach spaces, like the Radon-Nikodým property, behave under group actions, revealing new relationships and dependencies on the acting group.

Contribution

It introduces equivariant versions of key Banach space properties and establishes their implications, highlighting the role of group structure in these properties.

Findings

For compact groups, the G-Bishop-Phelps property implies strong G-dentability.

Strong G-dentability implies the G-Krein-Milman property.

For certain groups, weak G-dentability is equivalent to the G-Radon-Nikodým property.

Abstract

We introduce and study equivariant versions of the Radon-Nikod\'ym 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-Nikod\'ym property, the Bishop-Phelps property and dentability are equivalent, the equivariant situation turns out to depend essentially on the acting group and requires nontrivial tools from abstract harmonic analysis and representation theory. We establish several implications among the equivariant counterparts of these properties. Namely, for a compact group $G$, the $G$-Bishop-Phelps property implies strong $G$-dentability, which in turn implies the $G$-Krein-Milman property. Moreover, for a locally compact, $\sigma$-compact, separable group $G$, weak $G$-dentability is equivalent to the $G$-Radon-Nikod\'ym property.