2003 Fiscal Year Final Research Report Summary
Tracial rank of C^*-crossed products of AF algebras by discrete groups
| Project/Area Number |
14540217
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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 | Ritsumeikan University |
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Principal Investigator |
OSAKA Hiryuki Ritsumeikan Univ., Fac.Science and Engineering, Professor, 理工学部, 教授 (00244286)
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| Co-Investigator(Kenkyū-buntansha) |
KODAKA Kazunori Ryukyu Univ., Fac.Science, Professor, 理学部, 教授 (30221964)
NAGISA Masaru Chiba Univ., Fac.Science, Professor, 理学部, 教授 (50189172)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Operator algebras / C^*-algebras / stable rank / C^*-crossed products / Rokhlin property / The classification Theorem for simple C^*-algebras / AF C^*-algebras |
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| Research Abstract |
1. We define the tracial Rokhlin property for an action on a simple unital C^*-algebra in the joint research with N.C.Phillips, and study its basic properties. This properly is weaker than the Rokhlin property by Kishimoto. When A is a simple unital C^*-algebra of tracial rank zero and α∈Aut(A) has the tracial Rokhlin property, then the crossed product algebra C^*-(Z, A, α) is simple, and has real rank zero, stable rank one, and the Fundamental Comparison Property in the sense of Blackadar. It is still opened if C^*(Z, A, α) has tracial rank zero.
2. We define the cyclic Rokhlin property for an action on a unital C^*-algebra in the joint work with H Lin, which is stronger than the tracial Rokhlin property. We proved that when A is a simple unital C^*-algebra of tracial rank zero and α∈Aut(A) has the cyclic tracial Rokhlin property, then the algebra C^*(Z, A, α) is simple until C^*-algebra of tracial rank zero. Moreover if A is separable with the UCT, then A is a AH algebra. We also proved that if an approximately inner a∈Aut(A) has tracial Rokhlin property, then α has the cyclic tracial Rokhlin property.
3. Let G be a finite group and α an action from G to Aut(UHF). We proved in the joint work with T. Teruya that the crossed product algebra C^*(G, UHF, α) has topological rank more than or equal to 2. It is still opened if C^*(G, A, α) has stable rank 1 for any simple AF algebra A and any action a from G to Aut(A).
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Research Products
(6 results)
