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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.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
47A53 (Semi-) Fredholm operators; index theories
35S05 Pseudodifferential operators as generalizations of partial differential operators
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