SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of Id_(M)= vu : M→uB(H)→vM with u normal, unital, positive and v completely contractive. As a corollary, if M has a separable predual, M is isomorphic(as a Banach space) to B(l2). For instance this applies(rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since B(H) fails the approximation property(due to Szankowski) there are M ’s(namely B(H)^(**) and certain finite examples defined using ultraproducts) that are not seemingly injective.Moreover, for M to be seemingly injective it suffices to have the above factorization of I dM through B(H) with u, v positive(and u still normal).

T-local Derivations of von Neumann Algebras

In this paper, we introduce the concept of T-local derivations and obtain the main result: each T-local derivation of a von Neumann algebra A into a dual A-bimodule M is a T-derivation, where T is an endomorphism of A to A.

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L∞(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L2(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .

Von Neumann Algebras Associated with the Group SL2（R）

Let M and N be the yon Neumann algebras induced by the rational action of the group SL2（R） and its subgroup P on the upper half plane H. We have shown that N is spatial isomorphic to the group von Neumann algebra Lp and characterized M and its commutant M＇ and gotten a generalization of the Mautner＇s lemma. It is also shown that the Berezin operator commutates with the Laplaeian operator.

*-Lie Derivable Mappings on Von Neumann Algebras

In this paper,we prove that every*-Lie derivable mapping on a von Neu-mann algebra with no central abelian projections can be expressed as the sum of anadditive*-derivation and a mapping with image in the center vanishing at commuta-tors.

A Weak* Similarity Degree Characterization for Injective von Neumann Algebras

In this note, we show that avon Neumann algebra M is injective if and only if the weak＊ similarity degree d.（M） ≤ 2.

Similarity Degrees for the Crossed Product of von Neumann Algebras

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.

ON TOPOLOGICAL REFLEXIVITY OF THE SPACES OF DERIVATIONS ON OPERATOR ALGEBRAS

It is proved that the spaces of derivations on some operator algebras are topologically reflexive in the weak operator topology.

L^(p,∞)(M)上的乘法算子和复合算子（英文）

We give some properties of the composition and multiplication operators on L^(p,∞)(M), where M is a semifinite von Neumann algebra with a normal semifinite faithful trace τ.

冯诺依曼代数中的广义波算子

Let M⊆B(H)be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weightτwhere B(H)is the set of all bounded linear operators on Hilbert space H.The main purpose of this article is to introduce generalized weak wave operators Wf_(±),generalized weak abelian wave operators ■ and generalized stationary wave operators U_(±) in M and then to explore the relation among W_(±),■,U_(±) and generalized wave operators W_(±).

Nonlinear A^(*)B+B^(*)A Type Derivations on^(*)-Algebras

Let A be a unital^(*)-algebra with the unit I and a nontrivial projection P∈A.Suppose that A satisfies XAP=0 implies X=0 and XA(I-P)=0 implies X=0.In this paper,we prove thatΦis a nonlinear A^(*)B+B^(*)A type derivation on A if and only ifΦis an additive^(*)-derivation.This is then applied to prime^(*)-algebra,von Neumann algebras with no central summands of type I1.factor von Neumann algebras and standard operator algebras.

Non-commutative Kolmogorov Inequality and Zygmund Inequality

In this paper,we discuss some properties of distribution functions of τ-measurable operators and obtain non-commutative analogue of Kolmogorov inequality and Zygmund inequality.

LOCAL 2-COCYCLES

The aim of this paper is to show that every local 2-cocycle of a von Neumann algebra R with coefficients in S (a unital dual R-bimodule) is a 2-cocycle.

L_p-CONTINUITY OF NONCOMMUTATIVE CONDITIONAL EXPECTATIONS

Let （M,τ） be a noncommutative probability space, （Mn）n≥l a sequence of von Neumann subalgebras of M and N a von Neumann subalgebra of M. We introduce the notions of It-approach and orthogonal approach for （Mn）n≥1 and prove that ε（x｜Mn）Lp→ε（x｜N） for any x ∈ Lp（M） （1 ≤ p 〈 ∞） if and only if （Mn）n≥1 τ-approaches and orthogonally approaches N.

Reducing subspaces of multiplication operators on function spaces

This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multiplication operators on the Bergman space.

On Monotone Product of Operator Algebras

In this note, we give complete descriptions of the structure of the monotone product of two yon Neumann algebras and two C^＊-algebras. We show that the monotone product of two simple yon Neumann algebras and C^＊-algebras aren＇t simple again. We also show that the monotone product of two hyperfinite yon Neumann algebras is again hyperfinite and determine the type of the monotone product of two factors.

On Strong Kadison–Singer Algebras

We show that many Kadison-Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. Diagonal-trivial maximal non self-adjoint subalgebras of matrix algebras with lower dimensions are classified.

The Pressure in Operator Algebras

We introduce two notions of the pressure in operator algebras, one is the pressure Pα（π, T） for an automorphism α of a unital exact C^＊-algebra A at a self-adjoint element T in A with respect to a faithful unital ＊-representation π the other is the pressure Pτ,α（T） for an automorphism α of a hyperfinite von Neumann algebra M at a self-adjoint element T in M with respect to a faithful normal α-invariant state τ. We give some properties of the pressure, show that it is a conjugate invaxiant, and also prove that the pressure of the implementing inner automorphism of a crossed product A×α Z at a self-adjoint operator T in A equals that of α at T.

On Maximal Abelian Self-adjoint Subalgebras of Factors of Type II_1

In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.

Strong Skew Commutativity Preserving Maps on Rings with Involution

Let R be a unital ＊-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP= 0 implies A = 0 and AR（I - P） = 0 implies A = 0. In this paper, it is shown that a surjective map Ф： R →R is strong skew commutativity preserving （that is, satisfies Ф（A）Ф（B） - Ф（B）Ф（A）＊ ： AB- BA＊ for all A, B ∈R） if and only if there exist a map f ： R → ZSz（R） and an element Z ∈ ZS（R） with Z^2 =I such that Ф（A） =ZA ＋ f（A） for all A ∈ R, where ZS（R） is the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime ＊-rings and von Neumann algebras with no central summands of type I1 are characterized.