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Search: MSC category 47C15 ( Operators in $C^$- or von Neumann algebras *$- or von Neumann algebras * )

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1. CJM Online first

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang
Strict comparison of positive elements in multiplier algebras
Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if ``finitely many extremal traces" is replaced by ``quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Keywords:strict comparison, bi-diagonal form, positive combinations
Categories:46L05, 46L35, 46L45, 47C15

2. CJM 2009 (vol 62 pp. 133)

Makarov, Konstantin A.; Skripka, Anna
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

3. CJM 2004 (vol 56 pp. 742)

Jiang, Chunlan
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

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