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.
Reviewer: Luigi Rodino (Torino)
MSC:
58J40 | Pseudodifferential and Fourier integral operators on manifolds |
47A53 | (Semi-) Fredholm operators; index theories |
35S05 | Pseudodifferential operators as generalizations of partial differential operators |
Citations:
Zbl 0647.46053References:
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[2] | Sternin, B. Yu, No article title, Cent. Europ. J. Math., 9, 814-832 (2011) · Zbl 1241.58012 · doi:10.2478/s11533-011-0045-8 |
[3] | A. Savin and B. Sternin, J. Noncommutative Geom. 7 (2013), ArXiv:1106.4195. |
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[5] | Kasparov, G., No article title, Inv. Math., 91, 147-201 (1988) · Zbl 0647.46053 · doi:10.1007/BF01404917 |
[6] | G. K. Pedersen, C*-Algebras and Their Automorphism Groups (Academic, London, 1979). · Zbl 0416.46043 |
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[8] | V. M. Manuilov and E. V. Troitskii, Hilbert C*-Modules (Faktorial, Moscow, 2001; AMS, Providence, RI, 2005). |
[9] | Mishchenko, A. S., No article title, Izv. Akad. Nauk, Ser. Mat., 38, 85-111 (1974) |
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