Geodesic
Noncommutative elliptic theory. Examples
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 204-223.

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We study differential operators with coefficients in noncommutative algebras. As an algebra of coefficients, we consider crossed products corresponding to the action of a discrete group on a smooth manifold. We give index formulas for the Euler, signature, and Dirac operators twisted by projections over the crossed product. The index of Connes operators on the noncommutative torus is computed. 
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A. Yu. Savin; B. Yu. Sternin. Noncommutative elliptic theory. Examples. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 204-223. http://geodesic.mathdoc.fr/item/TM_2010_271_a14/

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