Reflexive representability and stable metrics

Itaï Ben Yaacov (1) , Alexander Berenstein (1, 2) , Stefano Ferri (2)

1 ICJ - Institut Camille Jordan
2 UNIANDES - Departamento de Matematicas [Univ de los Andes Colombia]

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.

Mots clés: Weakly almost periodic function; weakly almost periodic compactification; reflexively representable group; stable Banach space

Domaines: Logique [math.LO]

Itaï Ben Yaacov, Alexander Berenstein, Stefano Ferri. Reflexive representability and stable metrics. 2009. ⟨hal-00351012v1⟩