Reflexive representability and stable metrics

TL;DR

This paper establishes an equivalence between a metrisable group's reflexive representability and the existence of a uniformly equivalent stable metric, linking topological group properties with Banach space geometry.

Contribution

It proves that for metrisable groups, reflexive representability is characterized by stable metrics, connecting topological and geometric properties.

Findings

Equivalence between reflexive representability and stable metrics in metrisable groups

Partial negative answer to Megrelishvili's problem

Insight into the structure of topological groups via Banach space theory

Abstract

It is well-known that a topological group can be represented as a group of isometries of a reflexive Banach space if and only if its topology is induced by weakly almost periodic functions (see \cite{Shtern:CompactSemitopologicalSemigroups}, \cite{Megrelishvili:OperatorTopologies} and \cite{Megrelishvili:TopologicalTransformations}). We show that for a metrisable group this is equivalent to the property that its metric is uniformly equivalent to a stable metric in the sense of Krivine and Maurey (see \cite{Krivine-Maurey:EspacesDeBanachStables}). This result is used to give a partial negative answer to a problem of Megrelishvili.