Subadditivity Inequalities in von Neumann Algebras and Characterization of Tracial Functionals

We examine under which assumptions on a positive normal functional φ on a von Neumann algebra, and a Borel measurable function f: R ⁺ → R with f(0) = 0 the subadditivity inequality φ (f(A+B)) ≤ φ(f(A))+φ (f (B)) holds true for all positive operators A, B in . A corresponding characterization of tracial functionals among positive normal functionals on a von Neumann algebra is presented. O.E. Tikhonov - Supported by the Russian Foundation for Basic Research (grant no. 01-01-00129) and the scientific program Universities of Russia – Basic Research (grant no. UR.04.01.061).     

L 2-Cohomology for Von Neumann Algebras

We study L ²-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L ²-cohomology and show how the study of the first L ²-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of such extensions. In particular, we show that the first L ²-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C*-algebra. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest. Received: March 2006 Revision: October 2006 Accepted: October 2006     

von Neumann代数的T-局部导子

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.     

Preduals of von Neumann Algebras

Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra A is the third dual of some Banach space, then the space A is isometrically isomorphic to the second dual of some von Neumann algebra A and the von Neumann algebra A is uniquely determined by its enveloping von Neumann algebra (up to von Neumann algebra isomorphism) and is the unique second predual of A (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders.     

因子von Neumann代数上的非线性Lie导子

设M是作用在维数大于2的复可分Hilbert空间,■上的因子von Neumann代数.证明了因子von Neumann代数M上的每一个非线性Lie导子具有形式A→ψ(A)+h(A)I,其中:.M→M是可加的导子,h:M→C是非线性映射且对所有A,B∈M,有h(AB-BA)=0.     

Von Neumann代数中的套子代数

本文主要讨论因子Ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上的线性满等距和自伴导子．证明了因子Ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上的每个线性满等距是同构乘酉算子或者是反同构乘酉算子；给出了其上自伴导子是内导子的条件并得到有限因子 Ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上的每个自伴导子都是内导子．     

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．     

因子von Neumann代数中套子代数上的Jordan同构

研究了因子yon Neumann代数中套子代数上的Jordan同构,证明了套子代数algMβ和algMγ之间的每一个Jordan同构φ:要么是同构;要么是反同构.     

von Neumann代数中套子代数上的Lie导子

本文对因子von Neumann代数中套子代数上的线性映射L:alg_Mβ→M满足L(AB—BA)=L(A)B-BL(A)+AL(B)-L(B)A( A,B∈alg_Mβ)进行了刻划,证明了存在线性函数h:alg_Mβ→C;且对任意A,B∈alg_Mβ,有h(AB—BA)=0和算子T∈M,使得对任意X∈alg_Mβ,都有L(X)=XT-TX+h(X)I.     

von Neumann代数上非线性强保交换映射

设M是作用在维数大于2的复可分Hilbert空间上的因子von_Neumann代数.本文证明了M上的每个非线性强保交换满射Φ都具有形式:存在常数λ∈{-1,1)和非线性函数h:M→C使得对任意A∈M,有Φ(A)=λA+h(A)I.     

一些von Neumann代数的自由熵维数

In this paper, we estimate the free entropy dimension of the group yon Neumann algebra(L)(Zt), which is less than 1/t,2 ≤ t ≤ ∞. This data is identical with the free dimension defined by Dykema.     

因子von Neumann代数上的k-Jordan*映射

设A和B是两个因子yon Neumann代数,k是n次单位根.证明了任意的A,B∈A,非线性双射Φ:A→B满足Φ(k(AB+BA*))=k(Φ(A)Φ(B)+Φ(B)Φ(A)*)当且仅当Φ是*-环同构.     

Von Neumann algebras and linear independence of translates

For and , define and if , define . It has been conjectured that if , then is linearly independent over ; one motivation for this problem comes from Gabor analysis. We shall prove that is linearly independent if and is contained in a discrete subgroup of , and as a byproduct we shall obtain some results on the group von Neumann algebra generated by the operators . Also, we shall prove these results for the obvious generalization to .

     

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．     

交换von Neumann代数的局部n-上循环

证明交换von Neumann代数到其单位Banach双模内的每个正则局部n-上循环都是n-上循环．     

因子von Neumann代数上的非线性混合Lie三重可导映射

本文通过经典的可导映射,运用矩阵分块的方法,证明了因子von Neumann代数■上的每一个非线性混合Lie三重可导映射都是可加的*-导子.     

因子Von Neumann代数中套子代数上的局部3-上循环

证明了因子von Neumann代数中的套子代数到其单位对偶双模内的每个弱连续的局部3-上循环都是3-上循环.     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

因子von Neumann代数上完全保持 Jordan1-$*$-零积的映射

令$H$和$K$是无限维复Hilbert空间, $\mathcal{A},\mathcal{B}$分别是$H$和$K$上的因子von Neumann代数.结果表明每一个从$\mathcal{A}$到$\mathcal{B}$完全保Jordan1-$*$-零积的满射都是线性$*$-同构或者共轭线性$*$-同构的非零常数倍.