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)



Coarse embeddings of metric spaces into Banach spaces

Author: Piotr W. Nowak
Journal: Proc. Amer. Math. Soc. 133 (2005), 2589-2596
MSC (2000): Primary 46C05; Secondary 46T99
Published electronically: April 19, 2005
MathSciNet review: 2146202
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There are several characterizations of coarse embeddability of locally finite metric spaces into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces $L_p(\mu)$, we get their coarse embeddability into a Hilbert space for $0<p<2$. This together with a theorem by Banach and Mazur yields that coarse embeddability into $\ell_2$ and into $L_p(0,1)$ are equivalent when $1 \le p<2$. A theorem by G.Yu and the above allow us to extend to $L_p(\mu)$, $0<p\le 2$, the range of spaces, coarse embeddings into which is guaranteed for a finitely generated group $\Gamma$ to satisfy the Novikov Conjecture.

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

  • [Ah] I. AHARONI, Every separable metric space is Lipschitz equivalent to a subset of $c_0$, Isr. J. Math. 19 (1974) 284-291. MR 0511661 (58:23471a)
  • [AMM] I. AHARONI, B. MAUREY, B. S. MITYAGIN, Uniform embeddings of metric spaces and of Banach spaces into Hilbert Spaces, Isr. J. Math., 52 (1985), 251-265. MR 0815815 (87b:46011)
  • [Ba] S. BANACH, Theory of Linear Operations, North-Holland Mathematical Library, Volume 38 (1987). MR 0880204 (88a:01065)
  • [BL] Y. BENYAMINI, J. LINDENSTRAUSS, Geometric nonlinear functional anlysis, Volume 48 of Colloquium Publications. American Mathematical Society, Providence, R.I., 2000. MR 1727673 (2001b:46001)
  • [DG] M. DADARLAT, E. GUENTNER, Constructions preserving Hilbert space uniform embeddability of discrete groups, Trans. Amer. Math. Soc. 355 (2003), 3253-3275. MR 1974686 (2004e:20070)
  • [DGLY] A.N. DRANISHNIKOV, G. GONG, V. LAFFORGUE, G. YU, Uniform Embeddings into Hilbert Space and a Question of Gromov, Canad. Math. Bull., Vol.45(1), 2002, 60-70. MR 1884134 (2003a:57043)
  • [En] P. ENFLO, On a problem of Smirnov, Ark. Math. 8 (1969), 107-109. MR 0415576 (54:3661)
  • [Gr] M. GROMOV, Asymptotic invariants of infinite groups, London Mathematical Society Lecture Notes, no.182, s. 1-295, Cambridge University Press, 1993. MR 1253544 (95m:20041)
  • [LT] J. LINDENSTRAUSS, L. TZAFRIRI, Classical Banach spaces, Springer-Verlag Lecture Notes in Mathematics 338 (1973). MR 0415253 (54:3344)
  • [Mo] E.H. MOORE, On properly positive Hermitian matrices, Bull. Am. Math. Soc. 23 (1916), 59, 66-67.
  • [Roe] J. ROE, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, AMS 1996.MR 1399087 (97h:58155)
  • [Sch$_1$] I.J. SCHOENBERG, On certain metric spaces arising from euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. Math. 38 (1937), 787-793. MR 1503370
  • [Sch$_2$] I.J. SCHOENBERG, Metric spaces and positive definite functions,Trans. Am. Math. Soc. 44 (1938), 522-536. MR 1501980
  • [Yu] G. YU, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. (1) 139 (2000), 201-240.MR 1728880 (2000j:19005)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46C05, 46T99

Retrieve articles in all journals with MSC (2000): 46C05, 46T99

Additional Information

Piotr W. Nowak
Affiliation: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland – and – Department of Mathematics, Tulane University, 6823 St. Charles Avenue, New Orleans, Louisiana 70118
Address at time of publication: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240

Keywords: Coarse embeddings, metric spaces, Novikov Conjecture
Received by editor(s): October 5, 2003
Published electronically: April 19, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society