Some Applications of the Perturbation Determinant in Finite von Neumann Algebras

摘要

In the finite von Neumann algebra setting, we introduce the conceptof a perturbation determinant associated with a pair of self-adjointelements $H_0$ and $H$ in the algebra and relate it to the concept ofthe de la Harpe--Skandalis homotopy invariant determinant associatedwith piecewise $C^1$-paths of operators joining $H_0$ and $H$. Weobtain an analog of Krein's formula that relates the perturbationdeterminant and the spectral shift function and, based on thisrelation, we derive subsequently (i) the Birman--Solomyak formula fora general non-linear perturbation, (ii) a universality of a spectralaveraging, and (iii) a generalization of theDixmier--Fuglede--Kadison differentiation formula.