完全分配CSL代数上的中心化子

设L是可分Hilbert空间上的完全分配交换子空间格,A是Alg L的子代数并且包含AlgL的全体有限秩算子.主要结果是:(1)A上的中心化子是拟空间的;(2)AlgL上的Jordan中心化子是中心化子;(3)当L是套时,AlgL上的Lie中心化子可表示成一个中心化子与一个可加泛函之和的形式,该泛函作用在形如AB-BA的算子上为零.     

有限秩算子的表示(Ⅰ)

定义了子空间格代数的(弱闭双边)模,对有限维Hilbert空间的强自反子空间格代数的模及原子Boolean格代数的模中的有限秩算子进行了讨论,得到有限秩算子一定可以表示为秩1算子的和.     

完全分配CSL代数上的中心化子

设L是可分Hilbert空间上的完全分配交换子空间格,A是Alg L的子代数并且包含Alg L的全体有限秩算子.主要结果是:(1)A上的中心化子是拟空间的;(2)Alg L上的Jordan中心化子是中心化子;(3)当L是套时,Alg L上的Lie中心化子可表示成一个中心化子与一个可加泛函之和的形式,该泛函作用在形如AB-BA的算子上为零.     

J-子空间格代数上中心化子和广义导子的刻画

设L是Banach空间X上的J-子空间格,AlgL是相应的(J-子空间格代数.设φ:AlgL→AlgL是可加映射,对每个K∈(J)(L),dimK≥2.该文证明了下列表述等价:(1)φ是中心化子;(2)φ满足AB=0■φ(A)B=Aφ(B)=0;(3)φ满足AB+BA=0■φ(A)B+φ(B)A=Aφ(B)+Bφ(A)=0;(4)φ满足ABC+CBA=0■φ(A)BC+φ(C)BA=ABφ(C)+CBφ(A)=0.作为应用,得到AlgL上在零点广义可导的可加映射的完全刻画.     

一类Kadison-Singer代数的上同调

设C为无限维可分Hilbert空间H上的套N和秩一投影P_ξ所生成的完备格,其中P_ξ表示H到非零向量ξ生成一维子空间上的正交投影.假设ξ为由N生成的von Neumann代数N″的分离向量,本文证明L是个Kadison-Singer格,从而相应的不变子空间格代数Alg(L)是个Kadison-Singer代数.此外,本文刻画Alg(L)的中心和模交换子,证明Alg(L)到其自身内的每个有界导子都是内的,以及Alg(L)的系数在B(H)内的任意n阶上同调群H~n(Alg(L),B(H))都是平凡的,n≥1.     

有限秩算子的表示（I）

定义了子空间格代数的（弱闭双边）模,对有限维Hilbert空间的强自反子空间格代数的模及原子Boolean格代数的模中的有限秩算子进行了讨论,得到了有限秩算子一定可以表示为秩1算子的和。     

格上TL理想的性质及生成

本文利用完备Brouwer格L及L上的无穷V-分配t-模定义分配格M上的TL理想，讨论一些基本性质。并给出由L子集生成的TL理想的计算公式。     

子空间格代数上的局部Lie导子

引入了局部Lie导子的概念,研究了AlgL上的局部Lie导子,其中L是Banach空间X上的子空间格且X≠X_,得到了关于AlgL上局部Lie导子的两个重要结论.     

具有d-Koszul子模滤的分次模

引入了弱d-Koszul模,它是d-Koszul模的一种自然推广.设A是d-Koszul代数,M是有限生成的分次A-模,则M是弱d-Koszul模当且仅当M具有子模滤 0(∪) U0(∪)U1(∪)…(∪)Up=M,使得所有的A-模 Ui/Ui-1是d-Koszul模.设M为一个弱d-Koszul模,则作为分次Ext*A(A0,A0)-模,其Koszul对偶,ε(M)=Ext*A(M,A0)是由0次生成的.     

具有d-Koszul子模滤的分次模

引入了弱d-Koszul模,它是d-Koszul模的一种自然推广．设A是d-Koszul代数,M是有限生成的分次A-模,则M是弱d-Koszul模当且仅当M具有子模滤:0(?)U0(?)U1(?)…(?)Up=M,使得所有的A-模Ui／Ui-1是d-Koszul模．设M为一个弱d-Koszul模,则作为分次ExtA*(A0,A0)-模,其Koszul对偶:ε(M)=ExtA*(M,A0)是由0次生成的．     

Jordan Modules and Jordan Ideals of Reflexive Algebras

Let ${\mathcal{L}}$ be a completely distributive subspace lattice on a Banach space and alg ${\mathcal{L}}$ the associated reflexive algebra. Suppose that the following $$\mbox{Condition A:}\dim(F/F\wedge F_-)\ne1\;\; \mbox{for all}\;\;F\in\mathcal{L}$$ holds; note that if ${\mathcal{L}}$ is an atomic Boolean subspace lattice, this condition means that every atom of ${\mathcal{L}}$ has dimension at least two. It is shown that every reflexive Jordan Alg ${\mathcal{L}}$ -module is an associative Alg ${\mathcal{L}}$ -module. We give an example which shows that if the Condition A is removed, then the conclusion is not necessarily true. Moreover, we prove that all reflexive Jordan ideals of Alg ${\mathcal{L}}$ are associative ideals in the case that no the Condition A is assumed. The same conclusions hold for weakly closed Jordan modules and weakly closed Jordan ideals if the rank one subalgebra of Alg ${\mathcal{L}}$ is weakly dense in Alg ${\mathcal{L}}$ .     

Reflexive algebras with finite width lattices: Tensor products,cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ¹-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg($\text{L}$₁ ? $\text{L}$₂) = Alg $\text{L}$₁ ? Alg $\text{L}$₂, where each $\text{L}$_i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each $\text{L}$_i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if $\text{L}$ is a finite-width commutative subspace lattice and $\text{K}$ is the set of compact operators then the quasitriangular algebra Alg $\text{L}$ + $\text{K}$ is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.     

Commutants modulo the compact operators of certain CSL algebras II

The commutant modulo compacts, or essential commutant, of a reflexive algebra with commutative subspace lattice is a C^* algebra which is the sum of the compact operators in L(H) and a C^* subalgebra of the core. We give a characterization of the essential commutant of a separably acting CSL algebra in terms of properties of the spectral measure of an operator in the intersection of the essential commutant and the core. This is used to determine some sufficient conditions on the lattice for when the essential commutant is norm generated by the projections it contains.     

Local derivations and local automorphisms

In this paper, we show that if is a completely distributive commutative subspace lattice or a -subspace lattice, then the space of all bounded derivations of is reflexive. We also study when local automorphisms on some algebras are automorphisms.     

完全分配格代数中秩一子代数的弱稠密性和套代数的一个特征

如果　Ａ是　Ｈｉｌｂｅｒｔ　空间上的完全分配格代数，　　那么Ａ中秩一算子生成的子代数在　Ａ中弱稠密，　当且仅当，Ａ在迹尖算子空间中的一次和二次预零化子的弱闭包是自反的；如果Ａ是套代数，那么ＬａｔＡ是极大套，当且仅当，Ａ的包含Ａ－的每个弱闭子空间是自反的，其中     

A characterization of generalized derivations of JSL algebras

Let AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : AlgL → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B)for all A, B ∈ AlgL with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.     

Reflexivity of bilattices

We study reflexivity of bilattices. Some examples of reflexive and non-reflexive bilattices are given. With a given subspace lattice ? we may associate a bilattice Σ_?. Similarly, having a bilattice Σ we may construct a subspace lattice Σ_?. Connections between reflexivity of subspace lattices and associated bilattices are investigated. It is also shown that the direct sum of any two bilattices is never reflexive.     

Jordan Derivations of Reflexive Algebras

Let ${\mathcal L}Let L{\mathcal L} be a subspace lattice on a Banach space X and suppose that ú{L ? L: L_- < X}=X{\vee\{L\in\mathcal L: L_- < X\}=X} or ${\land\{L_- : L \in \mathcal L, L>(0)\}=(0)}${\land\{L_- : L \in \mathcal L, L>(0)\}=(0)} . Then each Jordan derivation from AlgL{\mathcal L} into B(X) is a derivation. This result can apply to completely distributive subspace lattice algebras, J{\mathcal J} -subspace lattice algebras and reflexive algebras with the non-trivial largest or smallest invariant subspace.     

Coproducts of bounded distributive lattices: cancellation

Let denote the coproduct of the bounded distributive lattices L and M. At the 1981 Banff Conference on Ordered Sets, the following question was posed: What is the largest class L of finite distributive lattices such that, for every non-trivial Boolean lattice B and every implies ? In this note, the problem is solved. Received March 2, 1999; accepted in final form July 10, 2000.