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# Some Applications of the Perturbation Determinant in Finite von Neumann Algebras

Published:2009-12-04
Printed: Feb 2010
• Konstantin A. Makarov,
University of Missouri, USA
• Anna Skripka,
Texas A & M University, USA
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## Abstract

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.
 Keywords: perturbation determinant, trace formulae, von Neumann algebras
 MSC Classifications: 47A55 - Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47C15 - Operators in $C^*$- or von Neumann algebras 47A53 - (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

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