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Some Applications of the Perturbation Determinant in Finite von Neumann Algebras

Published online by Cambridge University Press:  20 November 2018

Konstantin A. Makarov
Affiliation:
Konstantin A. Makarov, Department of Mathematics, University of Missouri, Columbia, MO 65211, USA, e-mail: [email protected]
Anna Skripka
Affiliation:
Anna Skripka, Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, e-mail: [email protected]
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Abstract

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In the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements ${{H}_{0}}$ and $H$ in the algebra and relate it to the concept of the de la Harpe–Skandalis homotopy invariant determinant associated with piecewise ${{C}^{1}}$-paths of operators joining ${{H}_{0}}$ and $H$. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman–Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier–Fuglede–Kadison differentiation formula.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

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