Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra

Benjamin Küter

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On approximation of functions by certain operators preserving $x^{2}$

Lucyna Rempulska, Karolina Tomczak (2008)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving $e_{k} (x) = x^{k}$ , $k = 0, 2$ . Using a modification of certain operators $L_{n}$ preserving $e_{0}$ and $e_{1}$ , we introduce operators $L_{n}^{*}$ which preserve $e_{0}$ and $e_{2}$ and next we define operators $L_{n; r}^{*}$ for $r$ -times differentiable functions. We show that $L_{n}^{*}$ and $L_{n; r}^{*}$ have better approximation properties than $L_{n}$ and $L_{n; r}$ .

Subadditive measures on projectors of a von Neumann algebra

Leszek J. Ciach

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CONTENTSIntroduction...............................................................................................................................................5§0. Fundamental definitions and notations...............................................................................................7§1. Subadditive measure on projectors of a von Neumann algebra.........................................................8§2. m-measurable operators. Convergence in measure.........................................................................10§3....

On shape and multiplicity of solutions for a singularly perturbed Neumann problem

J. Chabrowski, Peter J. Watson, Jianfu Yang (2001)

Annales Polonici Mathematici

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We investigate the effect of the topology of the boundary ∂Ω and of the graph topology of the coefficient Q on the number of solutions of the nonlinear Neumann problem $(1_{d})$ .

Dual algebras generated by von Neumann n-tuples over strictly pseudoconvex sets

Michael Didas

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Let D ⋐ X denote a relatively compact strictly pseudoconvex open subset of a Stein submanifold X ⊂ ℂⁿ and let H be a separable complex Hilbert space. By a von Neumann n-tuple of class over D we mean a commuting n-tuple of operators T ∈ L(H)ⁿ possessing an isometric and weak* continuous $H^{\infty} (D)$ -functional calculus as well as a ∂D-unitary dilation. The aim of this paper is to present an introduction to the structure theory of von Neumann n-tuples of class over D including the necessary function-...

The von Neumann algebra associated with an infinite number of t-free noncommutative gaussian random variables

Janusz Wysoczański (2006)

Banach Center Publications

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We show that the von Neumann algebras generated by an infinite number of t-deformed free gaussian operators are factors of type $I_{\infty}$ .

The order topology for a von Neumann algebra

Emmanuel Chetcuti, Jan Hamhalter, Hans Weber (2015)

Studia Mathematica

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The order topology $τ_{o} (P)$ (resp. the sequential order topology $τ_{o s} (P)$ ) on a poset P is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra M we consider the following three posets: the self-adjoint part $M_{s a}$ , the self-adjoint part of the unit ball $M ¹_{s a}$ , and the projection lattice P(M). We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology...

Spectral subspaces and non-commutative Hilbert transforms

Narcisse Randrianantoanina (2002)

Colloquium Mathematicae

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Let G be a locally compact abelian group and ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. We study Hilbert transforms associated with G-flows on ℳ and closed semigroups Σ of Ĝ satisfying the condition Σ ∪ (-Σ) = Ĝ. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from L¹(ℳ,τ) into $L^{1, \infty} (ℳ, τ)$ . As an application, we obtain a Matsaev-type result for p = 1: if x is a quasi-nilpotent compact...

Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty}$ calculus, $H^{\infty} (T)$ , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_{p}$ .

On the Neumann-Poincaré operator

Josef Král, Dagmar Medková (1998)

Czechoslovak Mathematical Journal

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Let $Γ$ be a rectifiable Jordan curve in the finite complex plane $ℂ$ which is regular in the sense of Ahlfors and David. Denote by $L_{C}^{2} (Γ)$ the space of all complex-valued functions on $Γ$ which are square integrable w.r. to the arc-length on $Γ$ . Let $L^{2} (Γ)$ stand for the space of all real-valued functions in $L_{C}^{2} (Γ)$ and put $L_{0}^{2} (Γ) = {h \in L^{2} (Γ) \int_{Γ} h (ζ) | d ζ | = 0} .$ Since the Cauchy singular operator is bounded on $L_{C}^{2} (Γ)$ , the Neumann-Poincaré operator $C_{1}^{Γ}$ sending each $h \in L^{2} (Γ)$ into $C_{1}^{Γ} h (ζ_{0}) : = ℜ {(π i)}^{- 1} P . V . \int_{Γ} \frac{h (ζ)}{ζ - ζ_{0}} d ζ, ζ_{0} \in Γ,$ is bounded on $L^{2} (Γ)$ . We show that the inclusion $C_{1}^{Γ} (L_{0}^{2} (Γ)) \subset L_{0}^{2} (Γ)$ characterizes the circle in the...

On completely bounded bimodule maps over W*-algebras

Bojan Magajna (2003)

Studia Mathematica

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It is proved that for a von Neumann algebra A ⊆ B(ℋ ) the subspace of normal maps is dense in the space of all completely bounded A-bimodule homomorphisms of B(ℋ ) in the point norm topology if and only if the same holds for the corresponding unit balls, which is the case if and only if A is atomic with no central summands of type $I_{\infty, \infty}$ . Then a duality result for normal operator modules is presented and applied to the following problem. Given an operator space X and a von Neumann algebra...