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Nonlocal elliptic operators for compact Lie groups. (English. Russian original) Zbl 1213.47012
Dokl. Math. 81, No. 2, 258-261 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 431, No. 4, 457-460 (2010).
The paper considers the class of nonlocal operators corresponding to shifts along orbits under the action of a compact Lie group on a smooth manifold. Suppose that a compact Lie group \(G\) acts on a smooth closed manifold \(M\). Let \(T^*_{G}M:=\{(x, \xi) \in T^*M \mid \langle\xi, X_{M}(x) = 0\;\forall X \in \mathcal(G)\}\) and \(S^*_{G}M\) be the subspace consisting of covectors with unit length in \(T^*_{G}M\). Then \(G\) acts on \(T^*_{G}M\) and \(S^*_{G}M\) by differentials. Denote the crossed product of the algebra \(C(S^*_{G}M)\) and \(G\) corresponding to the action of \(G\) by \(C(S^*_{G}M) \rtimes G\).
The paper introduces a definition of ellipticity for nonlocal operators. Namely, a nonlocal operator on \(L^{2}(M, {\mathbb{C}}^{N})\) is called elliptic if and only if its principal symbol is invertible in the algebra Mat\(_{N}( C(S^*_{G}M) \rtimes G )^{+}\). The main results of the paper are Theorems 1,2 which assert that elliptic nonlocal operators are Fredholm and give an index formula for these operators in topological terms.

47A53 (Semi-) Fredholm operators; index theories
Full Text: DOI
[1] M. F. Atiyah and G. B. Segal, Ann. Math. 87, 531–545 (1968). · Zbl 0164.24201 · doi:10.2307/1970716
[2] M. F. Atiyah, Elliptic Operators and Compact Groups (Springer-Verlag, Berlin, 1974). · Zbl 0297.58009
[3] B. Blackadar, K-Theory for Operator Algebras (Cambridge Univ. Press, Cambridge, 1998). · Zbl 0913.46054
[4] T. Kawasaki, Nagoya Math. J. 84, 135–157 (1981). · Zbl 0437.58020 · doi:10.1017/S0027763000019589
[5] J. L. Koszul, in Colloques Internationaux Du Centre National De La Recherche Scientifique, Strasbourg 1953 (Centre Nat. Rech. Sci., Paris, 1953), pp. 137–141.
[6] G. Luke, J. Different Equations 12, 566–589 (1972). · Zbl 0238.35077 · doi:10.1016/0022-0396(72)90026-5
[7] M. Vergne, Duke Math. J. 12, 637–652 (1996). · Zbl 0874.57029 · doi:10.1215/S0012-7094-96-08226-5
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