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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
\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
\jour Math. USSR-Izv.
\yr 1975
\vol 9
\issue 4
\pages 751--792

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    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  mathnet  crossref  mathscinet  zmath  isi
    2. A. S. Mishchenko, “Banach algebras, pseudodifferential operators, and their application to $K$-theory”, Russian Math. Surveys, 34:6 (1979), 77–91  mathnet  crossref  mathscinet  zmath
    3. G. G. Kasparov, “The $K$-functor in the theory of extensions of $C^*$-algebras”, Funct. Anal. Appl., 13:4 (1979), 296–297  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi
    5. G. G. Kasparov, “The operator $K$-functor and extensions of $C^*$-algebras”, Math. USSR-Izv., 16:3 (1981), 513–572  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. Nicolae Teleman, “The index of signature operators on Lipschitz manifolds”, Publ Math Inst Hautes Étud Sci, 58:1 (1983), 39  crossref  mathscinet  zmath
    7. Nicolae Teleman, “The index theorem for topological manifolds”, Acta Math, 153:1 (1984), 117  crossref  mathscinet  zmath  isi
    8. E. V. Troitskii, “The equivariant index of $C^*$-elliptic operators”, Math. USSR-Izv., 29:1 (1987), 207–224  mathnet  crossref  mathscinet  zmath
    9. V. M. Manuilov, “$K$-homology of $C^*$-algebras”, Math. USSR-Sb., 59:2 (1988), 533–540  mathnet  crossref  mathscinet  zmath
    10. Nigel Higson, “Algebraic K-theory of stable -algebras”, Advances in Mathematics, 67:1 (1988), 1  crossref
    11. Richard B Melrose, Paolo Piazza, “Analytic K-theory on manifolds with corners”, Advances in Mathematics, 92:1 (1992), 1  crossref
    12. H. Moscovici, F. -B. Wu, “Localization of topological pontryagin classes via finite propagation speed”, GAFA Geom funct anal, 4:1 (1994), 52  crossref  mathscinet  zmath
    13. V. M. Manuilov, “Diagonalization of compact operators in Hilbert modules over finite $W^*$-algebras”, Russian Math. Surveys, 49:2 (1994), 166–167  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. John Roe, “Kasparov Products and Dual Algebras”, Journal of Functional Analysis, 155:1 (1998), 286  crossref
    16. Dorin Dumitraşcu, “Asymptotic morphisms, K-homology and Dirac operators”, Pacific J Math, 218:1 (2005), 53  crossref
    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  crossref
    18. Richard Melrose, Frédéric Rochon, “Index in K-Theory for Families of Fibred Cusp Operators”, K-Theory, 37:1-2 (2006), 25  crossref  mathscinet  zmath  isi
    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  crossref
    20. Fabio Cipriani, Jean-Luc Sauvageot, “Fredholm Modules on P.C.F. Self-Similar Fractals and Their Conformal Geometry”, Comm Math Phys, 2008  crossref  mathscinet  isi
    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  mathnet  crossref  mathscinet  elib
    22. Yu. A. Kordyukov, “Index theory and non-commutative geometry on foliated manifolds”, Russian Math. Surveys, 64:2 (2009), 273–391  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. Paul Baum, Hervé Oyono-Oyono, Thomas Schick, Michael Walter, “Equivariant geometric K-homology for compact Lie group actions”, Abh Math Sem Hamburg, 2010  crossref
    24. A. Yu. Savin, B. Yu. Sternin, “On the index of noncommutative elliptic operators over $C^*$-algebras”, Sb. Math., 201:3 (2010), 377–417  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    25. Robin J. Deeley, “Geometric K-homology with coefficients I: ℤ/kℤ-cycles and Bockstein sequence”, J. K-Theory, 2011, 1  crossref
    26. Rufus Willett, Guoliang Yu, “Higher index theory for certain expanders and Gromov monster groups, I”, Advances in Mathematics, 229:3 (2012), 1380  crossref
    27. R.J.. Deeley, “Geometric K-homology with coefficients II: The Analytic Theory and Isomorphism”, J. K-Theory, 2013, 1  crossref
    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  crossref
    29. Fabio Cipriani, Daniele Guido, Tommaso Isola, Jean-Luc Sauvageot, “Spectral triples for the Sierpinski gasket”, Journal of Functional Analysis, 2014  crossref
    30. Xiaoman Chen, Qin Wang, Zhijie Wang, “Fibred coarse embedding into non-positively curved manifolds and higher index problem”, Journal of Functional Analysis, 2014  crossref
    31. Xiaoman Chen, Qin Wang, Guoliang Yu, “The coarse Novikov conjecture and Banach spaces with Property (H)”, Journal of Functional Analysis, 2015  crossref
    32. Jens Kaad, “On the Unbounded Picture of $KK$-Theory”, SIGMA, 16 (2020), 082, 21 pp.  mathnet  crossref
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