On metric characterizations of the Radon-Nikod\'ym and related properties of Banach spaces

TL;DR

This paper characterizes metric structures that cannot be embedded into Banach spaces with the Radon-Nikodým property, using martingale techniques and providing new insights into the geometric properties of Banach spaces.

Contribution

It introduces a class of metric structures that characterize the absence of the Radon-Nikodým property in Banach spaces without relying on metric differentiation.

Findings

Infinite diamond and Laakso spaces do not embed into RNP Banach spaces.

A dual Banach space lacks RNP iff it contains a bilipschitz image of the infinite diamond.

Characterizations of reflexivity and the infinite tree property are provided.

Abstract

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod\'ym property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond. The paper also contains related characterizations of reflexivity and the infinite tree property.