Von Neumann Algebras and Hilbert Quantales

When A is a von Neumann algebra, the set of all weakly closed linear subspaces forms a Gelfand quantale, Max_w A. We prove that Max_w A is a von Neumann quantale for all von Neumann algebras A. The natural morphism from Max_w A to the Hilbert quantale on the lattice of weakly closed right ideals of A is, in general, not an isomorphism. However, when A is a von Neumann factor, its restriction to right-sided elements is an isomorphism and this leads to a new characterization of von Neumann factors.     

Lie Triple Derivations on von Neumann Algebras

Let A be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ :A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] +[[a, b], δ(c)] for any a, b, c∈ A with ab = 0(resp. ab = P, where P is a fixed nontrivial projection in A), then there exist an additive derivation d from A into itself and an additive map f :A → ZA vanishing at every second commutator [[a, b], c] with ab = 0(resp.ab = P) such that δ(a) = d(a) + f(a) for any a∈ A.     

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.     

von Neumann代数上的局部可乘Lie n-导子

A₁,…,A_n的（n-1）-换位子记为p_n（A₁,…,A_n）.令M是von Neumann代数,n ≥ 2是任意正整数,L：M → M是一个映射.本文证明了,若M不含I₁型中心直和项,且L满足L（p_n（A₁,…,A_n））=∑_k=1ⁿ p_n（A₁,…,A_k-1,L（A_k）,A_k+1,…,A_n）对所有满足条件A₁A₂=0的A₁,A₂,…,A_n ∈ M成立,则L（A）=φ（A）+f（A）对所有A ∈ M成立,其中φ：M → M和f：M → Z（M）（M的中心）是两个映射,且满足φ在P_iMP_j上是可加导子,f（p_n（A₁,A₂,…,A_n））=0对所有满足A₁A₂=0的A₁,A₂,…,A_n ∈ P_iMP_j成立（1 ≤ i,j ≤ 2）,P₁ ∈ M是core-free投影,P₂=I-P₁;若M还是因子且n ≥ 3,则L满足条件L（p_n（A₁,A₂,…,A_n））=∑_k=1ⁿ p_n（A₁,…,A_k-1,L（A_k）,A_k+1,…,A_n）对所有满足A₁A₂A₁=0的A₁,A₂,…,A_n ∈ M成立当且仅当L（A）=φ（A）+h（A）I对所有A ∈ M成立,其中φ是M上的可加导子,h是M上的泛函且满足h（p_n（A₁,A₂,…,A_n））=0对所有满足条件A₁A₂A₁=0的A₁,A₂,…,A_n ∈ M成立.     

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.     

von Neumann 代数上保持混合Jordan三重积的非线性映射

本文证明了在没有中心交换投影的von Neumann代数上的一个双映射$\Phi$如果保持混合Jordan三重积, 则$\Phi(I)\Phi$是一个线性*-同构和一个共轭线性*-同构的和, 其中$\Phi(I)$是中心自伴元素且$\Phi(I)^{2}=I$. 同时给出了因子von Neumann 代数上保持混合Jordan三重积映射的结构.     

The Bass Conjecture and Group von Neumann Algebras

The precise relationship between the Bass conjecture for the Hattori–Stallings trace for projective ZG-modules and the map from reduced K-theory of ZG to reduced K-theory of the von Neumann algebra, NG, of G, of G is determined. As a consequence it is shown this map is zero for all groups G. It is also shown that the map induced on K-theory from the inclusion of NG to the ring of closed, densely defined operators affiliated to NG is an isomorphism. Together with the above result, this gives some positive evidence for the validity of the Division Ring Conjecture for torsion free groups.     

Hochschild Cohomology and Representation-Finite Algebras

Using Grothendieck's semicontinuity theorem for half-exact functors,we derive two semicontinuity results on Hochschild cohomology.We apply these to show that the first Hochschild cohomogy groupof the mesh algebra of a translation quiver over a domain vanishesif and only if the translation quiver is simply connected. Wethen establish an exact sequence relating the first Hochschildcohomology group of an algebra to that of the endomorphism algebraof a module and apply it to study the first Hochschild cohomologygroup of an Auslander algebra. Our main result shows that fora finite-dimensional and representation-finite algebra algebraA over an algebraically closed field with Auslander algebra the following conditions are equivalent:

1. (1)A admits no outer derivation;
2. (2) admits no outer derivations;
3. (3) A is simply connected;
4. (4) is strongly simply connected.

. 2000 Mathematics Subject Classification 16E30, 16G30.     

Leibniz Homology of Extended Lie Algebras

Leibniz homology is a noncommutative homology theory for Lie algebras. In this paper, we compute low-dimensional Leibniz homology of extended Lie algebras.     

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.     

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 .

     

交换环上辛代数的导子代数

对含幺交换环上辛代数的导子代数做了详细的描述.     

交换环上辛代数的导子代数(英文)

对含幺交换环上辛代数的导子代数做了详细的描述.     

Artin Algebras with Loops but No Outer Derivations

We construct families of Artin algebras over fields of arbitrary characteristic that contain a loop in their ordinary quiver but admit no nontrivial outer derivation. This refuses the long-held belief that such algebras should not exist.     

二元广义外代数的Hochschild同调群

设Aq=k／(x2,xy+qyx,y2)是含有两个变量的广义外代数,基于Buch- weitz等人构造的极小投射双模解,广义外代数的各阶Hochschild同调群的维数被清晰地计算．     

Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus

It is proved that if a normal semifinite weight on a von Neumann algebra satisfies the inequality for any selfadjoint operators in , then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality is refined and reinforced.