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Cyclic cohomology after the excision theorem of Cuntz and Quillen

Published online by Cambridge University Press:  17 May 2013

Jacek Brodzki
Jacek Brodzki*
Affiliation:
School of Mathametics, University of Southampton, Highfield, Southampton, SO17 1BJ, EnglandJ.Brodzki@soton.ac.uk
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Abstract

The excision theorem of Cuntz and Quillen established the existence of a six term exact sequence in the bivariant periodic cyclic cohomology HP*(–,–) associated with an arbitrary algebra extension 0 → S → P → Q → 0. This remarkable result enabled far reaching developments in the purely algebraic periodic cyclic cohomology. It also provided a new formalism that led to the creation of new versions of this theory for topological and bornological algebras. In this article we outline some of the developments that resulted from this breakthrough.

Keywords

Excision theoremperiodic cyclic cohomologyChern characterbornological algebrasPrimary 19D55Secondary 48L8058B3446A17
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 3: The Legacy of Daniel Quillen , June 2013 , pp. 575 - 598
Copyright
Copyright © ISOPP 2013 

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References

1.Baum, P., Higson, N., Plymen, R. J., Representation theory of p-adic groups: a view from operator algebras, in: Proc. Sympos. Pure. Math. 68, American Mathematical Society, Providence RI, 2000, 111–149.Google Scholar
2.Brodzki, J., Plymen, R., Entire cyclic cohomology of Schatten ideals. Homology, Homotopy Appl. 7 (2005), no. 3, 37–52.CrossRefGoogle Scholar
3.Connes, A., Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360.Google Scholar
4.Connes, A., Moscovici, H., Cyclic Cohomology, the Novikov Conjecture and Hyperbolic Groups, Topology 29 (1990), 345–388.Google Scholar
5.Cuntz, J., A new look at KK -theory, K -Theory 1, 31–51 (1987).Google Scholar
6.Cuntz, J., Excision in periodic cyclic theory for topological algebras. Cyclic coho-mology and noncommutative geometry (Waterloo, ON, 1995), 43–53, Fields Inst. Commun. 17, Amer. Math. Soc., Providence, RI, 1997.Google Scholar
7.Cuntz, J., Skandalis, G., Tsygan, B., Cyclic Homology in Noncommutative Geometry, Encyclopaedia of Mathematical Sciences 121, Springer Verlag, 2004.Google Scholar
8.Cuntz, J., Bivariante K-Theorie für localkonvexe Algebren und der Chern-Connes-Charakter, Doc. Math. 2 (1997), 139–182.Google Scholar
9.Cuntz, J., Quillen, D., Excision in bivariant periodic cyclic cohomology, Invent. math. 127 (1997), 67–98.CrossRefGoogle Scholar
10.Cuntz, J., Quillen, D., Cyclic homology and nonsingularity, Journal Amer. Math. Soc. 8 (1995) 373–442.CrossRefGoogle Scholar
11.Cuntz, J., Quillen's work on the foundations of cyclic cohomology; this issue.Google Scholar
12.Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires, AMS Memoir 16 (1966).Google Scholar
13.Kasparov, G., Skandalis, G., Groupes boliques et conjecture de Novikov, C.R.A.S. 319 (1994), 815–820.Google Scholar
14.Kassel, C., Caractère de Chern bivariant, K-Theory 3 (1989), 367–400.Google Scholar
15.Lafforgue, V., K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math. 149 (2002), 1–95.Google Scholar
16.Mathai, V., Stevenson, D., On a generalized Connes-Hochschild-Kostant-Rosenberg theorem. Adv. Math. 200 (2006), no. 2, 303–335.CrossRefGoogle Scholar
17.Mathai, V., Stevenson, D., Entire cyclic homology of stable continuous trace algebras. Bull. Lond. Math. Soc. 39 (2007), no. 1, 71–75.Google Scholar
18.Meyer, R., Cyclic cohomology theories and nilpotent extensions, Münster University thesis, Münster 1999, math.KT/9906205.Google Scholar
19.Meyer, R., Local and Analytic Cyclic Homology. European Mathematical Society Publishing House, EMS Tracts in Mathematics 3, August 2007, 368 pages.Google Scholar
20.Meyer, R., Excision in entire cyclic cohomology. J. Eur. Math. Soc. (JEMS) 3 (2001), no. 3, 269–286.Google Scholar
21.Mineyev, I., Yu, G., The Baum-Connes conjecture for hyperbolic groups, Invent. Math. 149 (2002), 97–122.Google Scholar
22.Nistor, V., A bivariant Chern-Connes character. Ann. of Math. (2) 138 (1993), no. 3, 555–590.Google Scholar
23.Nistor, V., On the Cuntz-Quillen boundary map. C. R. Math. Rep. Acad. Sci. Canada 16 (1994), no. 5, 203–208.Google Scholar
24.Nistor, V., Higher index theorems and the boundary map in cyclic cohomology. Doc. Math. 2 (1997), 263–295.Google Scholar
25.Puschnigg, M., Excision in cyclic homology theories. Invent. Math. 143 (2001), no. 2, 249–323.Google Scholar
26.Puschnigg, M., The Kadison-Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153–194.Google Scholar
27.Puschnigg, M., Asymptotic cyclic cohomology. Lecture Notes in Mathematics 1642. Springer-Verlag, Berlin, 1996.Google Scholar
28.Quillen, D., Cyclic cohomology and algebra extensions. K-Theory 3 (1989), no. 3, 205–246.Google Scholar
29.Solleveld, M., Some Fréchet algebras for which the Chern character is an isomorphism. K-Theory 36 (2005), no. 3–4, 275–290.Google Scholar
30.Voigt, C.. Equivariant local cyclic homology and the equivariant Chern-Connes character. Doc. Math. 12 (2007), 313–359 (electronic).Google Scholar
31.Voigt, C.. Equivariant periodic cyclic homology. J. Inst. Math. Jussieu 6(4): 689–763, 2007.CrossRefGoogle Scholar
32.Wodzicki, M., Excision in cyclic homology and in rational algebraic K-theory, Ann. of Math. 129 (1989), 591–639.Google Scholar