Homological index formulas for elliptic operators over C*-algebras (Q1043770)

This paper with remarkable contents starts with a concise, readable introduction. Recalled and slightly extended from \textit{M. Karoubi} [``Cyclic homology and K-theory'', Astérisque 149 (1987; Zbl 0648.18008)] the definition of Karoubi's Chern character (from \(C^*\)-algebra K-theory to the de Rham homology of smooth subalgebras) and its properties that are relevant to index theory, and also the Chern character of \textit{A. S. Mishchenko} and \textit{A. T. Fomenko} [Math. USSR, Izv. 15, 87--112 (1980; Zbl 0448.46039)] for \(C^*\)-algebra bundles based on the Künneth formula. A formulation of the Mishchenko-Fomenko index theorem is given, adapted to applications. As applications, deduced is the higher index theorem for coverings of \textit{A. Connes} and \textit{H. Moscovici} [Topology 29, No.~3, 345--388 (1990; Zbl 0759.58047)] and that for flat foliated bundles motivated from \textit{X. Jiang} [K-Theory 12, No.~4, 319--359 (1997; Zbl 0913.58054)], and also an index theorem is proved for Toeplitz operators associated with a \(C^*\)-dynamical system by a compact Lie group, with respect to trace (of Breuer), previously proved in \textit{M. Lesch} [J. Oper. Theory 26, No.~1, 73--92 (1991; Zbl 0784.46041)]. In the Appendix, it is explained how pairing of K-theory with KK-theory is related to index theory via spectral flow, and some useful facts (and beyond) about pseudodifferential operators over \(C^*\)-algebras are collected.