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Automorphisms of operator algebras and quantum measures

算子代数和量子测度的自同构

基本信息

批准号：
10640199
负责人：
SAITO Kazuyuki
金额：
$ 0.77万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640199/
关键词：
automorphisms von Neumann algebras monotone complete C^*-algebras AFD-factors outer automorphisms operator algebras C^*-algebras quantum measures 因子環 C^*_- 環

项目摘要

We showed that when M is a von Neumann algebra with a non atomic center then an easy argument can be given to establish the boundedness of completely additive quantum measures ou M.In particular, if {n_j} is a sequence of positive integers and, for each j, A_j ia an abelian von Neumann algebra. with no minimal projections, and M_j is the algebra of n_j by n_j matrices then Σ_j【symmetry】(M_j 【cross product】A_j) is a von Neumann algebra not covered by the Dorofeev-Shertsnev theorem but one to which our results apply. By combining the results obtained here with their deep theorems (specialized to factors) the best possible result is obtained.Let M be a von Neumann algebra which does not have any direct summand isomorphic to the algebra of n by n matrices (for n an integer greater than 1). Then every completely additive quantum measure on M is bounded.2. Let B be any monotone complete C^*-algebra and let G be any locally compact separable Hausdorff group. We gave necessary and sufficient c … More onditions on the (B, G) for the existence of an action α of G on B as a group of *-automorphisms in such a way that (B, G, α) is an admissible dynamical system. Roughly, it is a monotone complete C^*-dynamical system (B, G, α) for which we can construct a monotone complete cross-product B x_α G with the canonical embedding of B.Furthermore, when G is abelian, we can define a dual action of G in such a way that the duality principle of Takesaki is valid.3. We constructed non-trivial examples of admissible monotone complete C^*-dynamic systems. In particular, we constructed such a system where G is the additive group R of real numbers or the Torus T, and where B is the Generic Dynamics Factor A.4. Let Out(A) = Aut(A)/Inn(A) be the outer automorphism group of A.Then, for each integer p with p 【greater than or equal】 2 and each complex number γ with γ^p=1, we constructed periodic automorphisms of A with Connes' outer conjugacy invariant (p, γ) of outer periodicity.5. For any countable discrete group G, it is shown that G can be isomorphically embedded in Out(A). Less

我们证明了当M是一个非原子中心的von Neumann代数时，可以给出一个简单的论证来建立M上完全可加量子测度的有界性.特别地，如果{n_j}是一个正整数序列，且对每个j，A_j是一个交换von Neumann代数.若M_j是n_j乘n_j矩阵的代数，则M_j（A_j）是一个不被Dorofeev-Shertsnev定理覆盖的von Neumann代数，但我们的结果适用于它。将所得结果与它们的深定理（专门针对因子）相结合，得到了可能的最佳结果：设M是一个vonNeumann代数，它没有任何与n × n矩阵代数（n为大于1的整数）同构的直和项。则M上的每一个完全可加量子测度都是有界的.设B是任意单调完备C^*-代数，G是任意局部紧可分Hausdorff群.我们给出了必要且充分的C ...更多信息在（B，G）上存在G在B上的作用α作为一组 *-自同构的条件，使得（B，G，α）是一个容许动力系统。粗略地说，它是一个单调完全C^*-动力系统（B，G，α），对于它，我们可以构造一个单调完全叉积B x_α G与B的正则嵌入.我们构造了可容许单调完备C^*-动力系统的非平凡例子。特别地，我们构造了这样一个系统，其中G是真实的数的加法群R或环面T，并且其中B是通用动力学因子A。设Out（A）= Aut（A）/Inn（A）为A的外自同构群，对任意p ≥ 2的整数p和任意γ^p=1的复数γ，构造了A的具有Connes外共轭不变量（p，γ）的周期自同构.对任意可数离散群G，证明了G可以同构嵌入Out（A）.少