Geodesic
Some Applications of the Perturbation Determinant in Finite von Neumann Algebras
Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 133-156

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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.
DOI : 10.4153/CJM-2010-008-x
Mots-clés : perturbation determinant, trace formulae, von Neumann algebras 
Makarov, Konstantin A.; Skripka, Anna. Some Applications of the Perturbation Determinant in Finite von Neumann Algebras. Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 133-156. doi: 10.4153/CJM-2010-008-x
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     title = {Some {Applications} of the {Perturbation} {Determinant} in {Finite} von {Neumann} {Algebras}},
     journal = {Canadian journal of mathematics},
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     year = {2010},
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     doi = {10.4153/CJM-2010-008-x},
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