Essentially Invertible Measurable Operators Affiliated to a Semifinite von Neumann Algebra and Commutators

Siberian Mathematical Journal - Suppose that a von Neumann operator algebra  $ {\mathcal{M}} $ acts on a Hilbert space  $ {\mathcal{H}} $ and $ \tau $...     

冯·诺依曼代数的可测算子的性质

本文研究了冯·诺依曼代数的可测算子的基本性质,定义了阶梯算子,证明了任意一个正可测算子可以由阶梯算子在定义域内按照强算子拓扑逼近,从而证明了任意一个可测算子可以由投影在定义域内按照强算子拓扑逼近.此外,还讨论了可测算子与有界算子的复合算子的可测性.     

Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

Let M be a type I von Neumann algebra with the center Z, and a faithful normal semi-finite trace τ. Consider the algebra L(M, τ) of all τ-measurable operators with respect to M and let S ₀(M, τ) be the subalgebra of τ-compact operators in L(M, τ). We prove that any Z-linear derivation of S ₀(M, τ) is spatial and generated by an element from L(M, τ).      

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Lifting of Kadec-Klee Properties to Symmetric Spaces of Measurable Operators

We show that if is a separable symmetric Banach function space on the positive half-line, then has the Kadec-Klee property (respectively, uniform Kadec-Klee property) for local convergence in measure if and only if, for every semifinite von Neumann algebra , the associated space of -measurable operators has the same property.

     

Generalized Wave Operators in Von Neumann Algebras

Let M?B(H)be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weight r where B(H)is the set of all bound...     

Almost-Everywhere Convergence and (o)-Convergence in Rings of Measurable Operators Associated with a Finite von Neumann Algebra

We study the relationship between (o)-convergence and almost-everywhere convergence in the Hermite part of the ring of unbounded measurable operators associated with a finite von Neumann algebra. In particular, we prove a theorem according to which (o)-convergence and almost-everywhere convergence are equivalent if and only if the von Neumann algebra is of the type I.     

Prediction Operators in Banach Ideal Spaces

A necessary and sufficient condition for an operator in a Banach ideal space to be a prediction operator is given.     

The Distance to Upper Triangular Operator Spaces of Von Neumann Algebras

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Operator Space Embedding of Schatten p-Classes Into Von Neumann Algebra Preduals

Let X₁ and X₂ be subspaces of quotients of R OH and C OH respectively. We use new free probability techniques to construct a completely isomorphic embedding of the Haagerup tensor product into the predual of a sufficiently large QWEP von Neumann algebra. As an immediate application, given any 1 < q ≤ 2, our result produces a completely isomorphic embedding of (equipped with its natural operator space structure) into with a QWEP von Neumann algebra. Received: June 2006, Revision: June 2007, Accepted: September 2007     

Spaces of Ideal Convergent Sequences of Bounded Linear Operators

Abstract

The notation of I–convergence was introduced and studied by Kostyrko, Macaj, Salat, and Wilczynski. Recently, the concept of I–convergent for a sequence of bounded linear operators has been studied by Khan and Shafiq. This has motivated us to introduce and study some new spaces of double sequences of bounded linear operators and their basic topological and algebraic properties of these spaces. And we study some of their basic topological and algebraic properties of these spaces. We prove some inclusion relations on these spaces.     

Metrics on Projections of the Von Neumann Algebra Associated with Tracial Functionals

Siberian Mathematical Journal - Let φ be a positive functional on a von Neumann algebra $$mathscr{A}$$ and let $$mathscr{A}^{rm{pr}}$$ be the projection lattice in $$mathscr{A}$$. Given...     

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H₀, and let {T_t, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {T_t} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans [3] have shown that Lhas the form [formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H₀ K is a bounded operator satisfying the ‘minimality’condition that the set {(RX–(X)R)u, uH₀, XA} is totalin K, and K₀ is a fixed element of A. The unitality of {T_t}implies that L(1) = 0, and consequently K₀=iH–R^*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensen–Evans (CE) parameters which determine theCE generator L of the semigroup {T_t}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25.     

Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra

In this paper we study the metric geometry of the space ofpositive invertible elements of a von Neumann algebra A witha finite, normal and faithful tracial state . The trace inducesan incomplete Riemannian metric x,y_a = (ya^–1xa^–1),and, though the techniques involved are quite different, thesituation here resembles in many relevant aspects that of then x n matrices when they are regarded as a symmetric space.For instance, we prove that geodesics are the shortest pathsfor the metric induced, and that the geodesic distance is aconvex function; we give an intrinsic (algebraic) characterizationof the geodesically convex submanifolds M of ; and under a suitablehypothesis we prove a factorization theorem for elements inthe algebra that resembles the Iwasawa decomposition for matrices.This factorization is obtained via a nonlinear orthogonal projection_M : M, a map which turns out to be contractive for the geodesicdistance.     

Gamma-Compactification of Measurable Spaces

We define the gamma-compactification of an arbitrary measurable space and study its structure and properties in the general and topological cases. We introduce and study the notion of gamma-extension of a singleton in a topological space. We consider the procedure of extension of finitely additive measures from the original space to regular countably additive measures on the gamma-compactification of the space.     

严格循环的算子代数

设X是半自共轭的Banach空间,而A是B(X)的严格循环的交换子代数,考虑算子代数A的一些性质,这些性质与算子代数的自反性有关,而且这些性质为讨论A的自反性提供了理论基础.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

A Wide Class of Ultrabornological Spaces of Measurable Functions

Our main result states that a bornological locally convex space having a suitable Boolean algebra of projections is ultrabornological. This general theorem, whose proof is a variation of the sliding-hump techniques used in [Díaz et al., Arch. Math. (Basel)60 (1993), 73-78; Díaz et al., Resultate Math.23 (1993), 242-250; Drewnowski el al., Proc. Amer. Math. Sec.114 (1992), 687-694; Drewnowski et al., Atti. Sem. Mat. Fis. Univ. Modena41 (1993), 317-329], is applied to prove that some non-complete normed spaces such as the spaces of Dunford, Pettis, or McShane integrable functions, as well as other interesting spaces of weakly or strongly measurable functions, are ultrabornological. We also give applications to vector-valued sequence spaces; in particular, we prove that ℓ^p{X} (1 ≤ p < ∞) is an ultrabornological DF-space when X is.     

Hardy空间到Zygmund型空间的Riemann-Stieltjes算子

利用Hardy空间中函数的高阶导数的估计,通过构造一些新的检验函数,运用解析函数的性质与算子理论,给出了从Hardy空间到Zygmund型空间的Riemann—Stieltjes算子的有界性和紧性的特征,获得了若干个充要条件.