  We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by
C*-vector bundles. We use it to derive a general product formula for eta-forms
and to define and study new rho-invariants generalizing Lott's higher rho-form.
The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be
recovered by applying the theorem to Dirac operators twisted by the
Mishenko-Fomenko bundle associated to the reduced C*-algebra of the fundamental
group.
