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# Connections Between Metric Characterizations of Superreflexivity and the Radon-Nikodý Property for Dual Banach Spaces

Published:2014-11-03
Printed: Mar 2015
• Mikhail I. Ostrovskii,
Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway, Queens, NY 11439, USA
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## Abstract

Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikodým 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.
 Keywords: Banach space, diamond graph, finite representability, metric characterization, Radon-Nikodým property, superreflexivity
 MSC Classifications: 46B85 - Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science [See also 05C12, 68Rxx] 46B07 - Local theory of Banach spaces 46B22 - Radon-Nikod{y}m, Kreiin-Milman and related properties [See also 46G10]

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