In this paper an equivariant generalization of the Mishchenko-Fomenko theorem on the index of an elliptic operator over a C*-algebra is obtained. Β§ 0. Introduction
The Atiyah-Singer index theorem proved in the early sixties has found numerous applications during the last 20 years being the center of a field of mathematics developing intensively. Among the results we can find the general Lefschetz formula, formulas for various characteristic classes, a variety of results in the theory of elliptic problems and in theoretical physics.
Such a fruitfulness of using the index theorem caused Singer in 1971 (cf. [25]) to formulate a programme of the further development of the field. In particular, the problem of involving von Neumann factors as coefficients was raised.
In the work of Mishchenko and Fomenko (cf. [17]) an index theorem has been obtained for a wider class of Banach algebras, i.e. C*-algebras. The formula of [17] was successfully applied to investigate elliptic operators with operator-valued symbols (cf. [5]) and symbols with random coefficients (cf. [18]), to prove the Novikov conjecture in some particular cases (cf. [17] and [23]), and to study the higher ).
Simultaneously, Kasparov developed another approach based on Atiyah's ideas on the analytic construction of K-homology. Such a K-homology was defined in [11]. The further development led Kasparov to the creation of the operator K-bifunctor. The results and methods of these works have had numerous applications, too (cf., e.g. [14D.
In the case that the base space is a G-manifold it seems to be interesting and useful to have a more delicate C*-index formula (compared to the one of [17]) taking into account the G-structure. This work is devoted mainly to a solution of this problem for a compact Lie group G.
Acknowledgement. I am indebted to Yu. P. Soloviev for having suggested this problem and for stimulating discussions.