Geodesic
Topological invariants of elliptic operators.~I. $K$-homology
Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 751-792

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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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     author = {G. G. Kasparov},
     title = {Topological invariants of elliptic {operators.~I.} $K$-homology},
     journal = {Izvestiya. Mathematics },
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     number = {4},
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     language = {en},
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}
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G. G. Kasparov. Topological invariants of elliptic operators.~I. $K$-homology. Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 751-792. http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a3/