Automorphisms of operator algebras and quantum measures

我们证明了当M是一个具有非原子中心的von Neumann代数时，可以给出一个简单的论证来建立M中完全加性量子测度的有界性。特别是，如果{n_j}是一个正整数序列，并且对于每个j， A_j是一个阿贝尔冯诺伊曼代数。并且M_j是n_j由n_j矩阵构成的代数，那么Σ_j【对称】（M_j【叉积】A_j）是一个冯·诺伊曼代数，它没有被Dorofeev-Shertsnev定理所涵盖，但我们的结果适用于它。通过将这里得到的结果与他们的深层定理（专门针对因子）相结合，可以得到最好的结果。设M是一个冯诺依曼代数，它与n × n矩阵的代数（对于n大于1的整数）没有任何直接求和同构。那么M上的每一个完全加性量子测度都是有界的。设B是任意单调完备C^*代数，设G是任意局部紧可分Hausdorff群。我们给出了（B， G）作为*-自同构群上G的作用α存在的充分必要条件，从而证明（B， G, α）是一个可容许的动力系统。粗略地说，它是一个单调完备的C^*动力系统（B， G, α），我们可以用B的正则嵌入构造一个单调完备的叉积B x_α G。此外，当G是阿贝尔时，我们可以用Takesaki对偶原理成立的方式定义G的对偶作用。构造了可容许单调完全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是实数的加性群R或环面T，其中B是一般动力学因子a。设Out(A) = Aut(A)/Inn(A)为A的外自同构群，然后，对于p【大于等于】2的每一个整数p和γ^p=1的每一个复数γ，我们构造了具有外周期cones外共轭不变量（p, γ）的A的周期自同构。对于任意可数离散群G，证明G可以同构嵌入到Out(A)中。少

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