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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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##### References:
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