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Index of nonlocal elliptic operators over \(C^*\)-algebras. (English. Russian original) Zbl 1186.58017
Dokl. Math. 79, No. 3, 369-372 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 426, No. 3, 314-317 (2009).
The elliptic theory of pseudo-differential operators over \(C^*\)-algebras was constructed in [A. S. Mishchenko and A. T. Fomenko, Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831–859 (1979; Zbl 0416.46052)]. In this case, the index of an elliptic operator is an element of the \(K\)-group \(K_0(A)\), where \(A\) denotes the \(C^*\)-algebra over which operators are considered. Pseudo-differential operators are local operators in the sense that their commutators with multiplication operators by functions are compact.

The authors consider nonlocal elliptic operators over \(C^*\)-algebras associated with a discrete diffeomorphism group of a manifold, and give a formula expressing the analytical numerical invariants of these operators in topological terms. As a consequence of the formula, a simple proof of higher index formulas for nonlocal elliptic operators, which were given in [V. E. Nazaikinskii, A. Yu. Savin and B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry. Operator Theory: Advances and Applications 183. Advances in Partial Differential Equations. Basel: Birkhäuser. (2008; Zbl 1158.58013)], is given.
MSC:
58J42 Noncommutative global analysis, noncommutative residues
58J05 Elliptic equations on manifolds, general theory
58J40 Pseudodifferential and Fourier integral operators on manifolds
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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References:
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[2] V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry (Birkhäuser, Basel, 2008).
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