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.     

条件期望作为初等算子在冯-代数上的描述

定义并研究了冯-代数.条件期望基于冯-代数的描述,即作为初等算子的良好性质,会较之一般代数简洁许多.这是因为冯-代数包含一些特殊的子代数.给出了此类代数上置信的初等条件期望的描述及其最小存在的充要条件.并且定义了指标冯-有限条件期望.作为以上结果的推论,得出了条件期望指标有限的充分必要条件和一个重要不等式.     

冯·诺依曼代数交叉积的一点注记

设α是可数离散群G和H的半直积G■_σH在冯·诺依曼代数M上的作用,则β_h=α_((e,h))AdU_h定义了群H在冯·诺依曼代数交叉积M■_αG上的作用β.本文证明了交叉积冯·诺依曼代数M■_α(G■_σH)与(M■_αG)■_βH是*-同构的,因此在一定条件下,冯·诺依曼代数的交叉积满足结合律.     

Pontrjagin空间上算子代数理想的结构

对Π_k空间上一般对称算子代数,给出了对称理想的结构的两个结果．(1)令A是Π_k空间上一般对称算子代数．若M_1∩M_2≠{0},则存在对■~((k))不变的子空间V∈~(k)H~(k),满足M_1∩M_2=F(V) J,这里J=(■),T属于k×k矩阵代数,V=(R){VXX│X∈D},R和R⊥是对*-算子代数A_p~(k)不变的．(2)令A是Π_k空间上一般对称算子代数．设△=M_1∩M_2≠{0}．则M_2：△ U(Q),其中U(Q)是下列元的集(■),这里B∈A_p,q_i是算子代数U到R~⊥的线性映射,并满足条件：q(A B)=Aq(B),A,B∈A_p．     

归纳极限与积C~*-代数的α-比较性

?对于C~*-代数归纳极限A=(lim→)(A_n,φ_(m,n)(其中A_n?A_(n-1)?A且φ_(n,n-1):A_n→A_(n+1)为嵌入映射),若A_n人为具有α-比较的单的含单位元的稳定有限的C~*-代数,则A具有α-比较性;若A_λ(?_λ∈Λ)具有α-比较性,则积C~*-代数(Πλ∈A)A_λ具有α-比较性,特别地,和C~*-代数(λ∈A)A_λ具有α-比较性.     

Toeplitz算子代数的归纳极限

设(G_1,E_1),(G_2,E_2)为两个拟格序群,记■~(E_1),■~(E_2)为相应的Toeplitz算子代数．设■：G_1→G_2为一个保单位的群同态,使得■(E_1)■E_2．本文给出了上述两个Toeplitz算子代数间的自然同态映照成为C~*-代数的单同态的充要条件,刻画了Toeplitz算子代数的归纳极限,证明了任何自由群上的Toeplitz算子代数是顺从的．     

算子代数的性质∏_σ和张量积

本文引入算子代数的性质Ⅱ_σ这一概念,证明了任一von Neumann代数中的套子代数和有限宽度CSL子代数都具有性质Ⅱ_σ.最后得到张量积公式alg_ML_1■alg_NL_2= alg_(M■N)(L_1L_2)成立,这里L_1和L_2分别是von Neumann代数M和N中的有限宽度CSL.     

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.     

矩阵代数的Kadison-Singer格的分类

研究了矩阵代数M_n(C)的KS格,证明了每个生成M_3(C)的KS格都相似于(?)_0或I-(?)_0,其中(?)_0为M_3(C)的一个极大对角投影套和一个赋值全非零的秩1投影所生成的KS格,从而M_3(C)的对角平凡的KS代数都是4维的.同时,还给出了几个生成M_4(C)但非同构的KS格的例子.     

τ-可测正算子生成的交换子的若干不等式

设M是具有正规忠实的半有限迹τ的von Neumann代数,‖.‖ρ是任意非交换Banach函数空间范数,‖.‖是M上的通常范数.证明了若A和B是τ-可测正算子,X∈M,则‖AX-XB‖ρ≤‖X‖‖AB‖ρ.还证明了若A,B是M中的正算子,X是τ-可测算子,则‖AX-XB‖ρ≤max(‖A‖,‖B‖)‖X‖ρ.由此得到了若A∈M是正算子,X是τ-可测正算子,则‖AX-XA‖ρ≤1/2‖A‖‖XX‖ρ.     

On some types of maximal abelian subalgebras

Let H be a Hilbert space and B(H) the algebra of all bounded linear operators on H. It is known that there are two kinds of maximal abelian sub-algebras in B(H), to one of which there exists a unique faithful normal projection of norm one from B(H) and to the other any projection of norm one is singular. Any maximal abelian subalgebra A contains a projection e such that Ae is a maximal abelian subalgebra of B(eH) of the first kind and A(1 − e) is the one of the second kind in B((1 − e)H). This will be generalized to an arbitrary von Neumann algebra together with the existence problem of those kinds of maximal abelian subalgebras.     

Factorization for finite subdiagonal algebras of type 1

Archiv der Mathematik - Let $${\mathfrak {A}}$$ be a type 1 subdiagonal algebra in a finite von Neumann algebra $${\mathcal {M}}$$ with respect to a faithful normal conditional expectation $$\Phi...     

Invariant subspaces and cocycles in nonselfadjoint crossed products

Let G be a compact abelian group with the archimedean totally ordered dual Γ and let $\text{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a one-parameter group {α_γ}_γ?Γ of trace preserving 1-automorphisms of M. In this paper, we investigate the structure of invariant subspaces and cocycles for the subalgebra $\text{L}$₊ of $\text{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group {β_g}_g?G on $\text{L}$ is nonnegative. Our main result asserts that if M is a factor, then $\text{L}$₊ is maximal among the σ-weakly closed subalgebras of $\text{L}$.     

von Neumann代数中CSL子代数上的Jordan(α,β)-导子

设■是Hilbert空间H上的von Neumann代数的CSL子代数.本文证明了,在一定的条件下,■上的Jordan(α,β)-导子是(α,β)-导子,其中α,β是■上的两个自同构.还证明了在没有添加任何条件的情况之下,CSL代数上的任意Jordan(α,β)-导子是(α,β)-导子.另外,讨论了von Neumann代数中的CSL子代数上的n次幂(α,β)-映射.     

Maximality of entropy in finite von Neumann algebras

It is shown that the entropy function H(N ₁,…,N _k ) on finite dimensional von Neumann subalgebras of a finite von Neumann algebra attains its maximal possible value H(⋁ℓ=1k N _ℓ) if and only if there exists a maximal abelian subalgebra A of ⋁ℓ=1k N _ℓ such that A=⋁ℓ=1k(A∩N _ℓ). Oblatum 24-IV-1997 & 6-V-1997     

On maximal injective subalgebras of tensor products of von Neumann algebras

Let Mi be a von Neumann algebra, and Bi be a maximal injective von Neumann subalgebra of Mi, i=1,2. If M₁ has separable predual and the center of B₁ is atomic, e.g., B₁ is a factor, then is a maximal injective von Neumann subalgebra of . This partly answers a question of Popa.     

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.     

Strongly 1-Bounded Von Neumann Algebras

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ₀(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ₀ is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II ₁-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II ₁-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006