  We prove index formulas for elliptic operators acting between sections of
C*-vector bundles on a closed manifold. The formulas involve Karoubi's Chern
character from K-theory of a C*-algebra to de Rham homology of smooth
subalgebras. We show how they apply to the higher index theorem for coverings
and to flat foliated bundles, and prove an index theorem for C*-dynamical
systems associated to actions of compact Lie groups. In an Appendix we relate
the pairing of odd K-theory and KK-theory to the noncommutative spectral flow
and prove the regularity of elliptic pseudodifferential operators over
C*-algebras.
