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:
| [1] | Savin, A. Yu; Sternin, B. Yu, No article title, Dokl. Math., 81, 258-261 (2010) · Zbl 1213.47012 · doi:10.1134/S1064562410020262 |
| [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. |
| [4] | Savin, A. Yu; Sternin, B. Yu; Schrohe, E., No article title, Dokl. Math., 84, 846-849 (2011) · Zbl 1248.47046 · doi:10.1134/S1064562411070350 |
| [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 |
| [7] | Antonevich, A. B.; Lebedev, A. V., No article title, Proc. St. Petersburg Math. Soc., 6, 25-116 (2000) |
| [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) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
