Embeddability of locally finite metric spaces into Banach spaces is finitely determined
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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.
References
- Florent Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Arch. Math. (Basel) 89 (2007), no. 5, 419–429. MR 2363693, DOI 10.1007/s00013-007-2108-4
- F. Baudier and G. Lancien, Embeddings of locally finite metric spaces into Banach spaces, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1029–1033. MR 2361878, DOI 10.1090/S0002-9939-07-09109-5
- Nathanial Brown and Erik Guentner, Uniform embeddings of bounded geometry spaces into reflexive Banach space, Proc. Amer. Math. Soc. 133 (2005), no. 7, 2045–2050. MR 2137870, DOI 10.1090/S0002-9939-05-07721-X
- D. Dacunha-Castelle and J. L. Krivine, Applications des ultraproduits à l’étude des espaces et des algèbres de Banach, Studia Math. 41 (1972), 315–334 (French). MR 305035, DOI 10.4064/sm-41-3-315-334
- Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
- Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
- Maurice Fréchet, Les dimensions d’un ensemble abstrait, Math. Ann. 68 (1910), no. 2, 145–168 (French). MR 1511557, DOI 10.1007/BF01474158
- W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
- M. Ĭ. Kadec′ and A. Pelčin′ski, Basic sequences, bi-orthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297–323 (Russian). MR 181886
- Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR 1899299, DOI 10.1007/978-1-4613-0039-7
- Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90
- M. I. Ostrovskii, Coarse embeddings of locally finite metric spaces into Banach spaces without cotype, C. R. Acad. Bulgare Sci. 59 (2006), no. 11, 1113–1116. MR 2293922
- M. I. Ostrovskii, On comparison of the coarse embeddability into a Hilbert space and into other Banach spaces, unpublished manuscript, 2006, available at http:// facpub.stjohns.edu/ostrovsm
- M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proc. 33 (2009), 163–183. MR 2471569
Additional Information
- M. I. Ostrovskii
- Affiliation: Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, New York 11439
- MR Author ID: 211545
- Email: ostrovsm@stjohns.edu
- Received by editor(s): March 3, 2011
- Published electronically: November 28, 2011
- Communicated by: Thomas Schlumprecht
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2910760
Dedicated: This paper is dedicated to the memory of Nigel J. Kalton