Arens spaces associated with von Neumann algebras and normal states

In the present paper we introduce the notion of Arens space associated with a finite von Neumann algebra and a faithful normal finite state. Relations between these spaces and Arens algebras with respect to traces are investigated.     

Convergences in JW-algebras and in their enveloping von Neumann algebras

We study properties of different convergences in JW-algebras with a faithful normal state. The relationship between these convergences and similar convergences in enveloping von Neumann algebras is established. Based on this, ergodic theorems are proved.     

Derivations of noncommutative Arens algebras

The present paper deals with derivations of noncommutative Arens algebras. We prove that every derivation of an Arens algebra associated with a von Neumann algebra and a faithful normal finite trace is inner. In particular, each derivation on such algebras is automatically continuous in the natural topology, and in the commutative case, even for semi-finite traces, all derivations are identically zero. At the same time, the existence of noninner derivations is proved for noncommutative Arens algebras with a semi-finite but nonfinite trace.     

The(**)-Haagerup Property for C*-Algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the(**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki's question(2013), and then obtain several results of(**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the(**)-Haagerup property. Some heredity results concerning the(**)-Haagerup property are also proved.     

**-Haagerup Property for C*-Algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the $(**)$-Haagerup property for $C^{*}$-algebras in this paper. They first give an answer to Suzuki''s question (2013), and then obtain several results of $(**)$-Haagerup property parallel to those of Haagerup property for $C^{*}$-algebras. It is proved that a nuclear unital $C^{*}$-algebra with a faithful tracial state always has the $(**)$-Haagerup property. Some heredity results concerning the $(**)$-Haagerup property are also proved.     

关于强奇异极大交换子代数

设M_1和M_2是有限的冯·诺依曼代数,τ_1和τ_2是M_1和M_2的正规的,忠实的,正规化的迹.假设A_1和A_2分别是M_1和M_2的极大交换子代数,E_(Ai)是由M_i到A_i 的保迹的条件期望(i=1,2).若E_(A1)和E_(A2)是渐近同态条件期望,则A_1■A_2是M_1■M_2的强奇异极大交换子代数.另外,我们证明了若A是没有原子的有限冯·诺依曼代数M_1的强奇异极大交换子代数,M_2是有限冯·诺依曼代数,则A是M_1和M_2的约化自由积M_1*M_2 的强奇异极大交换子代数.     

Normal state spaces of Jordan and von Neumann algebras

Normal state spaces of Jordan and von Neumann algebras are characterized among convex sets. The normal state spaces of JBW-algebras are precisely those which are spectral and elliptic; along these the normal state spaces of von Neumann algebras are distinguished by the global 3-ball property.     

Isometries on Noncommutative Symmetric Spaces

Doklady Mathematics - Let $$\mathcal{M}$$ be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ (or else, an atomic von Neumann algebra with all...     

Martingale-type convergence of modular automorphism groups on von Neumann algebras

The strong convergence of modular automorphism groups and Connes cocycle derivatives is discussed under an increasing or decreasing net of von Neumann subalgebras and faithful semifinite normal weights on a von Neumann algebra.     

Commutativity of projections and characterization of traces on Von Neumann algebras

We find new necessary and sufficient conditions for the commutativity of projections in terms of operator inequalities. We apply these inequalities to characterize a trace on von Neumann algebras in the class of all positive normal functionals. We obtain some characterization of a trace on von Neumann algebras in terms of the commutativity of products of projections under a weight.     

von Neumann代数中套子代数上的Lie同构

本文给出因子von Neumann代数中的幂等算子在广义Lie积下的一个刻画; 得到因子von Neumann代数中套子代数的幂等算子在Lie积下的一个特征．作为应用, 研究了因子von Neumann代数中套子代数上的Lie同构,并证明因子von Neumann 代数中套子代数之间的Lie同构,要么是同构与广义迹之和,要么是负反同构与广义迹之和．     

Commutativity of projectors and trace characterization on von Neumann algebras. I

We obtain the necessary and sufficient conditions for commutativity of projectors in terms of operator inequalities. We apply these conditions for the trace characterization on von Neumann algebras in the class of all positive normal functionals. We also propose a trace characterization on von Neumann algebras in terms of the commutation of products of projectors under the weight sign.     

von Neumann代数中套子代数的摄动

本文主要讨论ｖｏｎ　Ｎｅｕｍａｎｎ代数中套子代数的摄动．给出了因子ｖｏｎ　Ｎｅｕｍａｎｎ代数中套相似的一个充分条件．证明了任何因子ｖｏｎ　Ｎｅｕｍａｎｎ代数中相邻的套子代数经由一个邻近于单位元的可逆算子是相似的．     

Vertex-Compressed Subalgebras of a Graph von Neumann Algebra

In Cho (Acta Appl. Math. 95:95–134, 2007 and Complex Anal. Oper. Theory 1:367–398, 2007), we introduced Graph von Neumann Algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via graph-representations, which are groupoid actions. In Cho (Acta Appl. Math. 95:95–134, 2007), we showed that such crossed product algebras have the amalgamated reduced free probabilistic properties, where the reduction is totally depending on given directed graphs. Moreover, in Cho (Complex Anal. Oper. Theory 1:367–398, 2007), we characterize each amalgamated free blocks of graph von Neumann algebras: we showed that they are characterized by the well-known von Neumann algebras: Classical group crossed product algebras and (operator-valued) matricial algebras. This shows that we can provide a nicer way to investigate such groupoid crossed product algebras, since we only need to concentrate on studying graph groupoids and characterized algebras. How about the compressed subalgebras of them? i.e., how about the inner (cornered) structures of a graph von Neumann algebra? In this paper, we will provides the answer of this question. Consequently, we show that vertex-compressed subalgebras of a graph von Neumann algebra are characterized by other graph von Neumann algebras. This gives the full characterization of the vertex-compressed subalgebras of a graph von Neumann algebra, by other graph von Neumann algebras.     

On a property ofL p spaces on semifinite von Neumann algebras

A characterization of the traces in a broad class of weights on von Neumann algebras is obtained. A new property of the domain ideals of these traces is proved. In the semifinite case, a relation for a faithful normal trace is established. This result is new even for the algebra of all bounded operators on a Hilbert space. Applications of the main result to the structure theory of von Neumann algebras and to the Köthe duality theory for ideal spaces of Segal measurable operators are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 185–190, August, 1998.The author wishes to express his deep gratitude to Professor A. N. Sherstnev for setting the problem and for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00025.     

On characterization of integrable sesquilinear forms

We give a necessary and sufficient condition for a sesquilinear form to be integrable with respect to a faithful normal state on a von Neumann algebra.     

正交酉元列在有限von Neumann代数的迹自由积中的应用

令M_1为一个有限的von Neumann代数,τ_1为其上的一个忠实正规迹态.我们将证明,如果M_1中存在一列两两正交的酉元列{u_k:k∈N},则对任意具有忠实正规迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.作为推论可以得出,如果M_1有一个von Neumann子代数N不包含最小投影,则对任意具有忠实迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.     

Comp osition and Multiplication Op erators on Lp,∞(M)

We give some properties of the composition and multiplication operators on Lp,∞(M), where M is a semifinite von Neumann algebra with a normal semifinite faithful traceτ.     

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...