Geodesic
The index of elliptic operators over $C^*$-algebras
Izvestiya. Mathematics , Tome 15 (1980) no. 1, pp. 87-112

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In this paper natural generalizations are developed of the theory of elliptic operators invariant under the action of a $C^*$-algebra. The theory of compact and Fredholm operators acting in spaces of the type of a Hilbert space over a $C^*$-algebra is developed. A formula of the Atiyah–Singer type for elliptic operators over a $C^*$-algebra is developed. Bibliography: 16 titles. 
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     author = {A. S. Mishchenko and A. T. Fomenko},
     title = {The index of elliptic operators over $C^*$-algebras},
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A. S. Mishchenko; A. T. Fomenko. The index of elliptic operators over $C^*$-algebras. Izvestiya. Mathematics , Tome 15 (1980) no. 1, pp. 87-112. http://geodesic.mathdoc.fr/item/IM2_1980_15_1_a3/