Actions of compact groups on algebras, the \(C^*\)-index theorem and families (Q1848086)

\textit{A. S. Mishchenko} and \textit{A. T. Fomenko} [Math. USSR, Izv. 15, 87-112 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831-859 (1979; Zbl 0416.46052)] proved an index theorem for operators elliptic over a \(C^{*}\)-algebra. The author [Math. USSR, Izv. 29, 207-224 (1987); translation from Izv.Akad. Nauk SSSR, Ser. Mat. 50, 849-865 (1986; Zbl 0641.46047)] generalized this theorem to the \(G\)-equivariant case, where the compact Lie group \(G\) acts on all structures, but the action on the \(C^{*}\)-algebra is trivial. The paper under review extends this result to permit a nontrivial action of \(G\) on the \(C^{*}\)-algebra. \textit{G. G. Kasparov} [Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 987-1000 (1984; Zbl 0571.46047)] stated a more general theorem and outlined a proof whose correctness is not questioned. There are some differences between the proof outlined by Kasparov and the proof in the paper under review. By focusing on compact \(G\), the author can use imbedding in a \(G\)-representation and an equivariant Thom isomorphism to define the topological index.