We introduce the concept of a perturbation determinant associated with a pair (H_0,H_1) of self-adjoint elements in a finite von Neumann algebra and relate it to the one of the de la Harpe-Skandalis homotopy invariant determinant associated with piecewise C^1-paths of operators joining H_0 and H_1. We represent Krein's spectral shift function (which is the difference of the spectrum distribution functions for H_0 and H_1 in the case of a finite von Neumann algebra) in terms of the perturbation determinant and, based on this representation, we obtain a version of the Newton-Leibniz formula for operator-valued functions. This formula naturally extends the Birman-Solomyak spectral averaging formula and suggests an extension of Krein's trace formula to a certain class of non-differentiable functions.

The talk is based on joint work with K. A. Makarov.