Automorphisms of operator algebras and quantum measures
Project/Area Number 
10640199

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Tohoku University 
Principal Investigator 
SAITO Kazuyuki Mathematical Institute, Tohoku University, Associate Professor, 大学院・理学研究科, 助教授 (60004397)

Project Period (FY) 
1998 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1999: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)

Keywords  automorphisms / von Neumann algebras / monotone complete C^*algebras / AFDfactors / outer automorphisms / operator algebras / C^*algebras / quantum measures / 因子環 / C^*_ 環 
Research Abstract 
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 DorofeevShertsnev 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 crossproduct 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 nontrivial 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

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Research Products
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