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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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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
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\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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A. S. Mishchenko, A. T. Fomenko, “The index of elliptic operators over $C^*$-algebras”, Math. USSR-Izv., 15:1 (1980), 87–112
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A. S. Mishchenko, “Banach algebras, pseudodifferential operators, and their application to $K$-theory”, Russian Math. Surveys, 34:6 (1979), 77–91
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G. G. Kasparov, “The $K$-functor in the theory of extensions of $C^*$-algebras”, Funct. Anal. Appl., 13:4 (1979), 296–297
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A. S. Mishchenko, Yu. P. Solov'ev, “Representations of Banach algebras and formulas of Hirzebruch type”, Math. USSR-Sb., 39:2 (1981), 189–205
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G. G. Kasparov, “The operator $K$-functor and extensions of $C^*$-algebras”, Math. USSR-Izv., 16:3 (1981), 513–572
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Nicolae Teleman, “The index of signature operators on Lipschitz manifolds”, Publ Math Inst Hautes Étud Sci, 58:1 (1983), 39
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Nicolae Teleman, “The index theorem for topological manifolds”, Acta Math, 153:1 (1984), 117
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E. V. Troitskii, “The equivariant index of $C^*$-elliptic operators”, Math. USSR-Izv., 29:1 (1987), 207–224
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V. M. Manuilov, “$K$-homology of $C^*$-algebras”, Math. USSR-Sb., 59:2 (1988), 533–540
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V. M. Manuilov, “Diagonalization of compact operators in Hilbert modules over finite $W^*$-algebras”, Russian Math. Surveys, 49:2 (1994), 166–167
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