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Embeddability of locally finite metric spaces into Banach spaces is finitely determined


Author: M. I. Ostrovskii
Journal: Proc. Amer. Math. Soc. 140 (2012), 2721-2730
MSC (2010): Primary 46B85; Secondary 05C12, 46B08, 46B20, 54E35
DOI: https://doi.org/10.1090/S0002-9939-2011-11272-3
Published electronically: November 28, 2011
MathSciNet review: 2910760
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Abstract: The main purpose of the paper is to prove the following results:

  • Let $ A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $ X$. Then $ A$ admits a bilipschitz embedding into $ X$.

  • Let $ A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $ X$. Then $ A$ admits a coarse embedding into $ X$.

These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009).

One of the main steps in the proof is: each locally finite subset of an ultraproduct $ X^\mathcal {U}$ admits a bilipschitz embedding into $ X$. We explain how this result can be used to prove analogues of the main results for other classes of embeddings.


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Additional Information

M. I. Ostrovskii
Affiliation: Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, New York 11439
Email: ostrovsm@stjohns.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11272-3
Keywords: Banach space, bilipschitz embedding, coarse embedding, locally finite metric space
Received by editor(s): March 3, 2011
Published electronically: November 28, 2011
Dedicated: This paper is dedicated to the memory of Nigel J. Kalton
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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