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 Izv. Akad. Nauk SSSR Ser. Mat., 1975, Volume 39, Issue 4, Pages 796–838 (Mi izv2053)

This article is cited in 32 scientific papers (total in 32 papers)

Topological invariants of elliptic operators. I. $K$-homology

G. G. Kasparov

Abstract: In this paper the homological $K$-functor is defined on the category of involutory Banach algebras, and Bott periodicity is proved, along with a series of theorems corresponding to the Eilenberg–Steenrod axioms. As an application, a generalization of the Atiyah–Singer index theorem is obtained, and some problems connected with representation rings of discrete groups and higher signatures of smooth manifolds are discussed.
Bibliography: 16 items.

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English version:
Mathematics of the USSR-Izvestiya, 1975, 9:4, 751–792

Bibliographic databases:

UDC: 513.8
MSC: Primary 58G10; Secondary 55F50, 55B20
Received: 31.07.1974

Citation: G. G. Kasparov, “Topological invariants of elliptic operators. I. $K$-homology”, Izv. Akad. Nauk SSSR Ser. Mat., 39:4 (1975), 796–838; Math. USSR-Izv., 9:4 (1975), 751–792

Citation in format AMSBIB
\Bibitem{Kas75} \by G.~G.~Kasparov \paper Topological invariants of elliptic operators.~I. $K$-homology \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1975 \vol 39 \issue 4 \pages 796--838 \mathnet{http://mi.mathnet.ru/izv2053} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=488027} \zmath{https://zbmath.org/?q=an:0328.58016} \transl \jour Math. USSR-Izv. \yr 1975 \vol 9 \issue 4 \pages 751--792 \crossref{https://doi.org/10.1070/IM1975v009n04ABEH001497} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. S. Mishchenko, A. T. Fomenko, “The index of elliptic operators over $C^*$-algebras”, Math. USSR-Izv., 15:1 (1980), 87–112
2. A. S. Mishchenko, “Banach algebras, pseudodifferential operators, and their application to $K$-theory”, Russian Math. Surveys, 34:6 (1979), 77–91
3. G. G. Kasparov, “The $K$-functor in the theory of extensions of $C^*$-algebras”, Funct. Anal. Appl., 13:4 (1979), 296–297
4. A. S. Mishchenko, Yu. P. Solov'ev, “Representations of Banach algebras and formulas of Hirzebruch type”, Math. USSR-Sb., 39:2 (1981), 189–205
5. G. G. Kasparov, “The operator $K$-functor and extensions of $C^*$-algebras”, Math. USSR-Izv., 16:3 (1981), 513–572
6. Nicolae Teleman, “The index of signature operators on Lipschitz manifolds”, Publ Math Inst Hautes Étud Sci, 58:1 (1983), 39
7. Nicolae Teleman, “The index theorem for topological manifolds”, Acta Math, 153:1 (1984), 117
8. E. V. Troitskii, “The equivariant index of $C^*$-elliptic operators”, Math. USSR-Izv., 29:1 (1987), 207–224
9. V. M. Manuilov, “$K$-homology of $C^*$-algebras”, Math. USSR-Sb., 59:2 (1988), 533–540
10. Nigel Higson, “Algebraic K-theory of stable -algebras”, Advances in Mathematics, 67:1 (1988), 1
11. Richard B Melrose, Paolo Piazza, “Analytic K-theory on manifolds with corners”, Advances in Mathematics, 92:1 (1992), 1
12. H. Moscovici, F. -B. Wu, “Localization of topological pontryagin classes via finite propagation speed”, GAFA Geom funct anal, 4:1 (1994), 52
13. V. M. Manuilov, “Diagonalization of compact operators in Hilbert modules over finite $W^*$-algebras”, Russian Math. Surveys, 49:2 (1994), 166–167
14. V. M. Manuilov, “Diagonalization of compact operators on Hilbert modules over $C^*$-algebras of real rank zero”, Math. Notes, 62:6 (1997), 726–730
15. John Roe, “Kasparov Products and Dual Algebras”, Journal of Functional Analysis, 155:1 (1998), 286
16. Dorin Dumitraşcu, “Asymptotic morphisms, K-homology and Dirac operators”, Pacific J Math, 218:1 (2005), 53
17. Hela Bettaieb, Michel Matthey, Alain Valette, “Unbounded symmetric operators in K-homology and the Baum–Connes conjecture”, Journal of Functional Analysis, 229:1 (2005), 184
18. Richard Melrose, Frédéric Rochon, “Index in K-Theory for Families of Fibred Cusp Operators”, K-Theory, 37:1-2 (2006), 25
19. Lin Shan, Qin Wang, “The coarse geometric Novikov conjecture for subspaces of non-positively curved manifolds”, Journal of Functional Analysis, 248:2 (2007), 448
20. Fabio Cipriani, Jean-Luc Sauvageot, “Fredholm Modules on P.C.F. Self-Similar Fractals and Their Conformal Geometry”, Comm Math Phys, 2008
21. V. E. Nazaikinskii, A. Yu. Savin, B. Yu. Sternin, “Noncommutative geometry and classification of elliptic operators”, Journal of Mathematical Sciences, 164:4 (2010), 603–636
22. Yu. A. Kordyukov, “Index theory and non-commutative geometry on foliated manifolds”, Russian Math. Surveys, 64:2 (2009), 273–391
23. Paul Baum, Hervé Oyono-Oyono, Thomas Schick, Michael Walter, “Equivariant geometric K-homology for compact Lie group actions”, Abh Math Sem Hamburg, 2010
24. A. Yu. Savin, B. Yu. Sternin, “On the index of noncommutative elliptic operators over $C^*$-algebras”, Sb. Math., 201:3 (2010), 377–417
25. Robin J. Deeley, “Geometric K-homology with coefficients I: ℤ/kℤ-cycles and Bockstein sequence”, J. K-Theory, 2011, 1
26. Rufus Willett, Guoliang Yu, “Higher index theory for certain expanders and Gromov monster groups, I”, Advances in Mathematics, 229:3 (2012), 1380
27. R.J.. Deeley, “Geometric K-homology with coefficients II: The Analytic Theory and Isomorphism”, J. K-Theory, 2013, 1
28. Xiaoman Chen, Qin Wang, Guoliang Yu, “The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space”, Advances in Mathematics, 249 (2013), 88
29. Fabio Cipriani, Daniele Guido, Tommaso Isola, Jean-Luc Sauvageot, “Spectral triples for the Sierpinski gasket”, Journal of Functional Analysis, 2014
30. Xiaoman Chen, Qin Wang, Zhijie Wang, “Fibred coarse embedding into non-positively curved manifolds and higher index problem”, Journal of Functional Analysis, 2014
31. Xiaoman Chen, Qin Wang, Guoliang Yu, “The coarse Novikov conjecture and Banach spaces with Property (H)”, Journal of Functional Analysis, 2015
32. Jens Kaad, “On the Unbounded Picture of $KK$-Theory”, SIGMA, 16 (2020), 082, 21 pp.
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