This is an announcement for the paper "Connections between metric characterizations of superreflexivity and Radon-Nikod'ym property for dual Banach spaces" by Mikhail I. Ostrovskii.

Abstract: Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikod'ym property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample.

Archive classification: math.FA math.MG

Mathematics Subject Classification: 46B85 (primary), 46B07, 46B22 (secondary)

Submitted from: ostrovsm@stjohns.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1406.0904

or

http://arXiv.org/abs/1406.0904