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-exact groups and coarsely embeddable groups
Author:
Semail Ülgen Yildirim
Journal:
Proc. Amer. Math. Soc. 137 (2009), 2393-2402
MSC (2000):
Primary 46L80; Secondary 19K35, 46L06
Posted:
February 17, 2009
MathSciNet review:
2495274
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Abstract: In this paper we introduce the notion of -exact -algebras, in particular -exact groups. We prove that -exactness is stable under direct limits and that coarsely embeddable groups are -exact groups under a technical condition.
- 1.
A.
Connes, J.
Cuntz, E.
Guentner, N.
Higson, J.
Kaminker, and J.
E. Roberts, Noncommutative geometry, Lecture Notes in
Mathematics, vol. 1831, Springer-Verlag, Berlin, 2004. Lectures given
at the C.I.M.E. Summer School held in Martina Franca, September 3–9,
2000; Edited by S. Doplicher and R. Longo; Fondazione C.I.M.E.. [C.I.M.E.
Foundation]. MR
2067646 (2005b:00021)
- 2.
M.
Rørdam, F.
Larsen, and N.
Laustsen, An introduction to 𝐾-theory for
𝐶*-algebras, London Mathematical Society Student Texts,
vol. 49, Cambridge University Press, Cambridge, 2000. MR 1783408
(2001g:46001)
- 3.
N. Higson, J. Roe, Coarse Baum-Connes conjecture and groupoids, Topology, 41 (2002), 807-834.
- 4.
Nigel
Higson and John
Roe, On the coarse Baum-Connes conjecture, (Oberwolfach,
1993) London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ.
Press, Cambridge, 1995, pp. 227–254. MR 1388312
(97f:58127), http://dx.doi.org/10.1017/CBO9780511629365.008
- 5.
Erik
Guentner, Nigel
Higson, and Shmuel
Weinberger, The Novikov conjecture for linear groups, Publ.
Math. Inst. Hautes Études Sci. 101 (2005),
243–268. MR 2217050
(2007c:19007), http://dx.doi.org/10.1007/s10240-005-0030-5
- 6.
Eberhard
Kirchberg and Simon
Wassermann, Permanence properties of 𝐶*-exact groups,
Doc. Math. 4 (1999), 513–558 (electronic). MR 1725812
(2001i:46089)
- 7.
Erik
Guentner and Jerome
Kaminker, Exactness and uniform embeddability of discrete
groups, J. London Math. Soc. (2) 70 (2004),
no. 3, 703–718. MR 2160829
(2006i:43006), http://dx.doi.org/10.1112/S0024610704005897
- 8.
Erik
Guentner and Jerome
Kaminker, Exactness and the Novikov conjecture, Topology
41 (2002), no. 2, 411–418. MR 1876896
(2003e:46097a), http://dx.doi.org/10.1016/S0040-9383(00)00036-7
- 9.
Marius
Dadarlat and Erik
Guentner, Constructions preserving Hilbert space
uniform embeddability of discrete groups, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3253–3275 (electronic). MR 1974686
(2004e:20070), http://dx.doi.org/10.1090/S0002-9947-03-03284-7
- 10.
Jean-Louis
Tu, La conjecture de Baum-Connes pour les feuilletages
moyennables, 𝐾-Theory 17 (1999), no. 3,
215–264 (French, with English and French summaries). MR 1703305
(2000g:19004), http://dx.doi.org/10.1023/A:1007744304422
- 11.
Jean-Louis
Tu, The Baum-Connes conjecture for groupoids,
𝐶*-algebras (Münster, 1999) Springer, Berlin, 2000,
pp. 227–242. MR 1798599
(2001j:46109)
- 12.
S. Ülgen,
-Exactness of Group -algebras, Ph.D. Thesis, Department of Mathematics, Purdue University, West Lafayette, IN, 2005.
- 13.
Narutaka
Ozawa, Amenable actions and exactness for discrete groups, C.
R. Acad. Sci. Paris Sér. I Math. 330 (2000),
no. 8, 691–695 (English, with English and French summaries). MR 1763912
(2001g:22007), http://dx.doi.org/10.1016/S0764-4442(00)00248-2
- 14.
Piotr
W. Nowak, Coarsely embeddable metric spaces without Property
A, J. Funct. Anal. 252 (2007), no. 1,
126–136. MR 2357352
(2008i:54026), http://dx.doi.org/10.1016/j.jfa.2007.06.014
- 15.
Joachim
Cuntz, 𝐾-theoretic amenability for discrete groups, J.
Reine Angew. Math. 344 (1983), 180–195. MR 716254
(86e:46064), http://dx.doi.org/10.1515/crll.1983.344.180
- 16.
Georges
Skandalis, Une notion de nucléarité en
𝐾-théorie (d’après J.\
Cuntz),
𝐾-Theory 1 (1988), no. 6, 549–573
(French, with English summary). MR 953916
(90b:46131), http://dx.doi.org/10.1007/BF00533786
- 17.
Guoliang
Yu, The coarse Baum-Connes conjecture for spaces which admit a
uniform embedding into Hilbert space, Invent. Math.
139 (2000), no. 1, 201–240. MR 1728880
(2000j:19005), http://dx.doi.org/10.1007/s002229900032
- 18.
M.
Gromov, Asymptotic invariants of infinite groups, Geometric
group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser.,
vol. 182, Cambridge Univ. Press, Cambridge, 1993,
pp. 1–295. MR 1253544
(95m:20041)
- 19.
M.
Gromov, Hyperbolic groups, Essays in group theory, Math. Sci.
Res. Inst. Publ., vol. 8, Springer, New York, 1987,
pp. 75–263. MR 919829
(89e:20070), http://dx.doi.org/10.1007/978-1-4613-9586-7_3
- 20.
Erik
Guentner, Exactness of the one relator
groups, Proc. Amer. Math. Soc.
130 (2002), no. 4,
1087–1093. MR 1873783
(2002m:46084), http://dx.doi.org/10.1090/S0002-9939-01-06195-0
- 21.
Bruce
Blackadar, 𝐾-theory for operator algebras, 2nd ed.,
Mathematical Sciences Research Institute Publications, vol. 5,
Cambridge University Press, Cambridge, 1998. MR 1656031
(99g:46104)
- 22.
Peter
A. Fillmore, A user’s guide to operator algebras,
Canadian Mathematical Society Series of Monographs and Advanced Texts, John
Wiley & Sons Inc., New York, 1996. A Wiley-Interscience Publication. MR 1385461
(97i:46094)
- 23.
Simon
Wassermann, Exact 𝐶*-algebras and related topics,
Lecture Notes Series, vol. 19, Seoul National University Research
Institute of Mathematics Global Analysis Research Center, Seoul, 1994. MR 1271145
(95b:46081)
- 1.
- A. Connes, J. Cuntz, E. Guentner, N. Higson, J. Kaminker, J.E. Roberts, Noncommutative Geometry, Lecture Notes in Math., vol. 1831, Springer-Verlag, Berlin, 2004.MR 2067646 (2005b:00021)
- 2.
- M. Rørdam, F. Larsen, N. J. Laustsen, An Introduction to
-theory for -algebras, London Math. Soc. Student Texts, 49, Cambridge Univ. Press, Cambridge, 2000, 1-239. MR 1783408 (2001g:46001)
- 3.
- N. Higson, J. Roe, Coarse Baum-Connes conjecture and groupoids, Topology, 41 (2002), 807-834.
- 4.
- N. Higson, J. Roe, On the coarse Baum-Connes conjecture, Novikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., 227, Cambridge Univ. Press, Cambridge, 1995, 227-254. MR 1388312 (97f:58127)
- 5.
- E. Guentner, N. Higson, S. Weinberger, The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci., 101 (2005), 243-268. MR 2217050 (2007c:19007)
- 6.
- E. Kirchberg, S. Wassermann, Permanence properties of
-exact groups, Doc. Math, 4 (1999), 513-558 (electronic). MR 1725812 (2001i:46089)
- 7.
- E. Guentner, J. Kaminker, Exactness and uniform embeddability of discrete groups, J. Lond. Math. Soc., 70 (2004), 703-718. MR 2160829 (2006i:43006)
- 8.
- E. Guentner, J. Kaminker, Exactness and the Novikov conjecture, Topology, 41 (2002), no. 2, 411-418. MR 1876896 (2003e:46097a)
- 9.
- 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)
- 10.
- J. L. Tu, La conjecture de Baum-Connes pour les feuilletages moyennables,
-theory, 17 (1999), 215-264. MR 1703305 (2000g:19004)
- 11.
- J. L. Tu, The Baum-Connes conjecture for groupoids, s, Springer, Berlin, 2000, 227-242. MR 1798599 (2001j:46109)
- 12.
- S. Ülgen,
-Exactness of Group -algebras, Ph.D. Thesis, Department of Mathematics, Purdue University, West Lafayette, IN, 2005.
- 13.
- N. Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), no. 8, 691-695. MR 1763912 (2001g:22007)
- 14.
- P. Nowak, Coarsely embeddable metric spaces without Property A, J. Funct. Anal., 252 (2007), 126-136. MR 2357352 (2008i:54026)
- 15.
- J. Cuntz,
-theoretic amenability for discrete groups, J. Reine Angew. Math., 344 (1983), 180-195. MR 716254 (86e:46064)
- 16.
- G. Skandalis, Une notion de nucl
arit en -théorie (d'après J. Cuntz), -theory, 1 (1998), 549-573. MR 953916 (90b:46131)
- 17.
- G. L. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., 139 (2000), 201-240. MR 1728880 (2000j:19005)
- 18.
- M. L. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory (A. Niblo and M. Roller, eds.), London Math. Soc. Lecture Note Ser., 182, Cambridge University Press, Cambridge, 1993, 1-295, MR 1253544 (95m:20041)
- 19.
- M. L. Gromov, Hyberbolic groups, in Essays in Group Theory, edited by S.M. Gersten, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. MR 919829 (89e:20070)
- 20.
- E. Guentner, Exactness of the one relator groups, Proc. Amer. Math. Soc., 130 (2002), 1087-1093. MR 1873783 (2002m:46084)
- 21.
- B. Blackadar, K-theory for Operator Algebras, Cambridge Univ. Press, Cambridge, 1998. MR 1656031 (99g:46104)
- 22.
- P. A. Fillmore, A User's Guide to Operator Algebras, John Wiley and Sons, Inc., New York, 1996. MR 1385461 (97i:46094)
- 23.
- S. Wassermann, Exact
-algebras and Related Topics, Lecture Notes Series, 19, Global Analysis Research Center, Seoul National University, 1994. MR 1271145 (95b:46081)
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Additional Information
Semail Ülgen Yildirim
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Lunt Hall 223, Evanston, Illinois 60208-2370
Email:
sulgen@math.northwestern.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09870-0
PII:
S 0002-9939(09)09870-0
Received by editor(s):
October 2, 2008
Posted:
February 17, 2009
Communicated by:
Marius Junge
Article copyright:
© Copyright 2009 American Mathematical Society
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