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Some Applications of the Perturbation Determinant in Finite von Neumann Algebras |
Canad. J. Math. 62(2010), 133-156
Published:2009-12-04
Printed: Feb 2010
Printed: Feb 2010
- Konstantin A. Makarov,
- Anna Skripka,
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] |