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Group equivariant Radon-Nikodým property and its characterizations

Sheldon Dantas, Michal Doucha, Mingu Jung, Tomáš Raunig

TL;DR

This work develops a cohesive theory of group-equivariant versions of the Radon-Nikodým property and related geometric notions for Banach spaces under continuous group actions by linear isometries. The authors define $G$-RNP, $G$-BP, $G$-dentability, $G$-Krein-Milman property, and $G$-Lindenstrauss property A, and establish a structured web of implications that depend on the acting group $G$ (notably when $G$ is compact or locally compact, σ-compact, and separable). Key contributions include: (i) $G$-BP implying strong $G$-dentability for compact $G$; (ii) strong $G$-dentability implying $G$-KMP; (iii) an equivalence between weak $G$-dentability and $G$-RNP under separability and under certain group-structure hypotheses; (iv) a detailed martingale-based approach linking $G$-RNP to weak dentability; and (v) an in-depth study of norm-attaining $G$-equivariant operators, including the $G$-property A for $oldsymbol{ extell}_1$ and related renorming phenomena. The results illuminate how combination of geometry of Banach spaces with representation theory of $G$ drives a richer, group-sensitive landscape than in the classical theory, with implications for equivariant operator theory and harmonic analysis on groups.

Abstract

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 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, $σ$-compact, separable group $G$, weak $G$-dentability is equivalent to the $G$-Radon-Nikodým property.

Group equivariant Radon-Nikodým property and its characterizations

TL;DR

This work develops a cohesive theory of group-equivariant versions of the Radon-Nikodým property and related geometric notions for Banach spaces under continuous group actions by linear isometries. The authors define -RNP, -BP, -dentability, -Krein-Milman property, and -Lindenstrauss property A, and establish a structured web of implications that depend on the acting group (notably when is compact or locally compact, σ-compact, and separable). Key contributions include: (i) -BP implying strong -dentability for compact ; (ii) strong -dentability implying -KMP; (iii) an equivalence between weak -dentability and -RNP under separability and under certain group-structure hypotheses; (iv) a detailed martingale-based approach linking -RNP to weak dentability; and (v) an in-depth study of norm-attaining -equivariant operators, including the -property A for and related renorming phenomena. The results illuminate how combination of geometry of Banach spaces with representation theory of drives a richer, group-sensitive landscape than in the classical theory, with implications for equivariant operator theory and harmonic analysis on groups.

Abstract

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 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 , the -Bishop-Phelps property implies strong -dentability, which in turn implies the -Krein-Milman property. Moreover, for a locally compact, -compact, separable group , weak -dentability is equivalent to the -Radon-Nikodým property.
Paper Structure (14 sections, 26 theorems, 93 equations, 2 figures)

This paper contains 14 sections, 26 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions, basic properties and examples
  4. $G$-Bishop-Phelps property
  5. $G$-dentability and weaker notions
  6. $G$-Krein-Milman property
  7. $G$-Radon-Nikodým property
  8. Consequences of the $G$-Bishop-Phelps property
  9. Consequences of $G$-dentability
  10. Strong $G$-dentability implies $G$-KMP
  11. Weak $G$-dentability implies the $G$-RNP
  12. Consequences of the $G$-Radon-Nikodým property
  13. Norm-attaining $G$-equivariant operators
  14. Summary of the results

Key Result

Lemma 3.4

Let $G$ be a topological group and let $X$ be a $G$-Banach space. Let $C \subseteq X$ be a $G$-invariant subset. The following conditions are equivalent.

Figures (2)

  • Figure 1: $G$-dentability concepts.
  • Figure 2: This diagram summarizes all the implications among the properties we have considered in the paper. Dotted arrows denote cases where it remains unclear whether the implication holds, whereas crossed arrows mark implications that generally fail to hold; counterexamples are provided throughout the text.

Theorems & Definitions (68)

  • Definition 3.1: $G$-Bishop-Phelps property
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5: $G$-dentabilities for Banach spaces
  • Remark 3.6
  • proof
  • proof
  • Corollary 3.9
  • ...and 58 more