  We relate the spectral flow to the index for paths of selfadjoint
Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra,
generalizing results of Robbin-Salamon and Pushnitski. Then we prove the
vanishing of the von Neumann spectral flow for the tangential signature
operator of a foliated manifold when the metric is varied. We conclude that the
tangential signature of a foliated manifold with boundary does not depend on
the metric. In the Appendix we reconsider integral formulas for the spectral
flow of paths of bounded operators.
