Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T07:30:26.717Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-K-exact uniform Roe C*-algebras

Published online by Cambridge University Press:  03 August 2010

Ján Špakula
Ján Špakula*
Affiliation:
Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, jan.spakula@uni-muenster.de
Get access

Abstract

We prove that uniform Roe C*-algebras C*uX associated to some expander graphs X coming from discrete groups with property (τ) are not K-exact. In particular, we show that this is the case for the expander obtained as Cayley graphs of a sequence of alternating groups (with appropriately chosen generating sets).

Keywords

uniform Roe C*-algebrasK-exactnessexpanders
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 1 , August 2012 , pp. 191 - 201
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Anantharaman-Delaroche, C. and Renault, J., Amenable groupoids, vol. 36 of Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], L'Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain.Google Scholar
2.Chen, X. and Wang, Q., Ideal structure of uniform Roe algebras of coarse spaces, J. Funct. Anal. 216 (2004), 191–211.CrossRefGoogle Scholar
3.de la Harpe, P., Robertson, A., and Valette, A., On the spectrum of the sum of generators for a finitely generated group, Israel J. Math. 81 (1993), 65–96.Google Scholar
4.Gromov, M., Spaces and questions, Geom. Funct. Anal., (2000), 118–161. GAFA 2000 (Tel Aviv, 1999).Google Scholar
5.Guentner, E. and Kaminker, J., Exactness and the Novikov conjecture, Topology 41 (2002), 411–418.Google Scholar
6.Higson, N., Lafforgue, V., and Skandalis, G., Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), 330–354.Google Scholar
7.Kassabov, M., Symmetric groups and expander graphs, Invent. Math. 170 (2007), 327–354.CrossRefGoogle Scholar
8.Kassabov, M. and Nikolov, N., Cartesian products as profinite completions, Int. Math. Res. Not., (2006), pp. Art. ID 72947, 17.CrossRefGoogle Scholar
9.Lubotzky, A., Discrete groups, expanding graphs and invariant measures, Progress in Mathematics 125, Birkhäuser Verlag, Basel, 1994. With an appendix by Jonathan D.Google Scholar
10.Lubotzky, A. and Żuk, A., On property (τ), in preparation, 2003.Google Scholar
11.Manuilov, V. and Thomsen, K., On the lack of inverses to C*-extensions related to property T groups, Canad. Math. Bull. 50 (2007), 268–283.Google Scholar
12.Ozawa, N., Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 691–695.CrossRefGoogle Scholar
13.Ozawa, N., An application of expanders to, J. Funct. Anal. 198 (2003), 499–510.Google Scholar
14.Roe, J., Lectures on coarse geometry, University Lecture Series 31, American Mathematical Society, Providence, RI, 2003.Google Scholar
15.Skandalis, G., Tu, J. L., and Yu, G., The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), 807–834.Google Scholar
16.Tu, J. L., La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory 17 (1999), 215–264.Google Scholar
17.Tu, J. L., The Baum-Connes conjecture for groupoids, in C*-algebras (Münster, 1999), Springer, Berlin, 2000, 227–242.Google Scholar
18.Ülgen Yildirim, S., K-exact group C*-algebras, PhD thesis, Purdue University, 2005.Google Scholar
19.Ülgen Yildirim, S., K-exact groups and coarsely embeddable groups, Proc. AMS 137 (2009), 2393–2402.Google Scholar
20.Wang, P. S., On isolated points in the dual spaces of locally compact groups, Math. Ann. 218 (1975), 19–34.Google Scholar
21.Wang, Q., Remarks on ghost projections and ideals in the Roe algebras of expander sequences, Arch. Math. (Basel) 89 (2007), 459–465.Google Scholar
22.Wassermann, S., C*-algebras associated with groups with Kazhdan's property T, Ann. of Math. (2), 134 (1991), 423–431.Google Scholar
23.Yu, G., The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), 201–240.CrossRefGoogle Scholar