2000 Fiscal Year Final Research Report Summary
Automorphisms of operator algebras and quantum measures
| Project/Area Number |
10640199
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
SAITO Kazuyuki Mathematical Institute, Tohoku University, Associate Professor, 大学院・理学研究科, 助教授 (60004397)
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| Project Period (FY) |
1998 – 2000
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| Keywords | automorphisms / von Neumann algebras / monotone complete C^*-algebras / AFD-factors / outer automorphisms / operator algebras / C^*-algebras / quantum measures |
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| 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 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
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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
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