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1. CJM 2009 (vol 62 pp. 133)
Some Applications of the Perturbation Determinant in Finite von Neumann Algebras 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 Categories:47A55, 47C15, 47A53 |
2. CJM 2004 (vol 56 pp. 742)
Similarity Classification of Cowen-Douglas Operators Let $\cal H$ be a complex separable Hilbert space
and ${\cal L}({\cal H})$ denote the collection of
bounded linear operators on ${\cal H}$.
An operator $A$ in ${\cal L}({\cal H})$
is said to be strongly irreducible, if
${\cal A}^{\prime}(T)$, the commutant of $A$, has no non-trivial idempotent.
An operator $A$ in ${\cal L}({\cal H})$ is said to a Cowen-Douglas
operator, if there exists $\Omega$, a connected open subset of
$C$, and $n$, a positive integer, such that
(a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$
(b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$;
(c) $\bigvee_{z{\in}{\Omega}}$\ker$(A-z)={\cal H}$ and
(d) $\dim \ker(A-z)=n$ for $z$ in $\Omega$.
In the paper, we give a similarity classification of strongly
irreducible Cowen-Douglas operators by using the $K_0$-group of
the commutant algebra as an invariant.
Categories:47A15, 47C15, 13E05, 13F05 |