Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 
 

 

Characterization of quasi-Banach spaces which coarsely embed into a Hilbert space


Author: N. Lovasoa Randrianarivony
Journal: Proc. Amer. Math. Soc. 134 (2006), 1315-1317
MSC (2000): Primary 46B20; Secondary 51F99
DOI: https://doi.org/10.1090/S0002-9939-05-08416-9
Published electronically: October 25, 2005
MathSciNet review: 2199174
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a linear subspace of $ L_0(\mu)$ for some probability space $ (\Omega, \mathcal{B}, \mu)$.


References [Enhancements On Off] (What's this?)

  • [A] T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588-594. MR 0014182 (7:250d)
  • [AMM] I. Aharoni, B. Maurey, and B. S. Mityagin, Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces, Israel J. Math. 52 (1985), no. 3, 251-265. MR 0815815 (87b:46011)
  • [BDK] J. Bretagnolle, D. Dacunha Castelle, J. L. Krivine, Lois stables et espaces $ L^p$, Ann. Inst. H. Poincaré Sect. B (N. S.) 2 (1965/1966), 231-259. MR 0203757 (34:3605)
  • [BL] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Coll. Pub. 48, Amer. Math. Soc., Providence, RI, 2000. MR 1727673 (2001b:46001)
  • [G] M. Gromov, Asymptotic invariants for infinite groups, Geometric Group Theory, G. A. Niblo and M. A. Roller, eds., Cambridge University Press, Cambridge, 1993. MR 1253544 (95m:20041)
  • [GD] S. Guerre-Delabrière, Types et suites symétriques dans $ L^p, 1\leq p <+\infty, p\neq 2$, Israel J. Math. 53 (1986), no. 2, 191-208. MR 0845871 (87m:46047)
  • [JR] W. B. Johnson and N. L. Randrianarivony, $ \ell_p$ $ (p>2)$ does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. (to appear).
  • [MN] M. Mendel, A. Naor, Euclidian quotients of finite metric spaces, Adv. Math. 189 (2004), 451-494. MR 2101227 (2005h:54031)
  • [Ni] E. M. Nikišin, Resonance theorems and expansions in eigenfunctions of the Laplace operator (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 795-813, English translation: Math. USSR-Izv. 6 (1972), 788-806.
  • [No] P. Nowak, Coarse embeddings of metric spaces into Banach spaces, Proc. Amer. Math. Soc. 133 (2005), 2589-2596. MR 2146202
  • [R] S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astrono. Phys. 5 (1957), 471-473. MR 0088682 (19:562d)
  • [S] I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522-536. MR 1501980

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B20, 51F99

Retrieve articles in all journals with MSC (2000): 46B20, 51F99


Additional Information

N. Lovasoa Randrianarivony
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Address at time of publication: Department of Mathematics, University of Missouri-Columbia, Mathematical Sciences Building, Columbia, Missouri 65211-4100
Email: nirina@math.tamu.edu, lova@math.missouri.edu

DOI: https://doi.org/10.1090/S0002-9939-05-08416-9
Keywords: Coarse embedding, uniform embedding
Received by editor(s): November 17, 2004
Published electronically: October 25, 2005
Additional Notes: The author was supported in part by NSF 0200690 and Texas Advanced Research Program 010366-0033-20013.
This paper represents a portion of the author’s dissertation being prepared at Texas A&M University under the direction of William B. Johnson.
Communicated by: David Preiss
Article copyright: © Copyright 2005 by the author

American Mathematical Society