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Uniformization of nonlocal elliptic operators and \(KK\)-theory. (English. Russian original) Zbl 1284.58013
Dokl. Math. 87, No. 1, 20-22 (2013); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 448, No. 1, 27-29 (2013).
The authors consider operators of the form \[ P= \sum_g P_g T_g, \] where \(P_g\) are pseudo-differential operators of order zero on a compact manifold \(M\) and \(T_g\) is the action of an element \(g\) belonging to a discrete finitely generated group acting on \(M\). The sum defining \(P\) is assumed to be finite. A notion of symbol of \(P\) is introduced, so that ellipticity of the symbol grants the Fredholm property for \(P: L^2(M)\to L^2(M)\). By using the KK-theory of G. Kasparov [Invent. Math. 91, No. 1, 147–201 (1988; Zbl 0647.46053)], the index of \(P\) is then computed.

58J40 Pseudodifferential and Fourier integral operators on manifolds
47A53 (Semi-) Fredholm operators; index theories
35S05 Pseudodifferential operators as generalizations of partial differential operators
Full Text: DOI
[1] Savin, A Yu; Sternin, B Yu, No article title, Dokl. Math., 81, 258-261, (2010) · Zbl 1213.47012
[2] Sternin, B Yu, No article title, Cent. Europ. J. Math., 9, 814-832, (2011) · Zbl 1241.58012
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[4] Savin, A Yu; Sternin, B Yu; Schrohe, E, No article title, Dokl. Math., 84, 846-849, (2011) · Zbl 1248.47046
[5] Kasparov, G, No article title, Inv. Math., 91, 147-201, (1988) · Zbl 0647.46053
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