1995 Fiscal Year Final Research Report Summary
Theory of Operator Algebras and Applications to Noncommutative Topological Spaces
| Project/Area Number |
06640281
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Kansai University |
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Principal Investigator |
KUSUDA Masaharu Kansai University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (80195437)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Toru Kansai Univ., Fac.of Eng., Associate Professor, 工学部, 助教授 (20199623)
HIRASHIMA Yasumasa Kansai Univ., Fac.of Eng., Associate Professor, 工学部, 助教授 (80047399)
YAMAMOTO Moboru Kansai Univ., Fac.of Eng., Professor, 工学部, 教授 (80029628)
KURISU Tadashi Kansai Univ., Fac.of Eng., Professor, 工学部, 教授 (00029159)
ISII Keiichi Kansai Univ., Fac.of Eng., Professor, 工学部, 教授 (80029420)
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| Project Period (FY) |
1994 – 1995
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| Keywords | C^<**>-algebra / Von Neumann algebra / State / Factor state / Spectrum / Dual C^<**>-algebra |
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| Research Abstract |
In general, a positive linear functional of norm 1, on a C^<**>-subalgebra, called a state can extend to a state on the whole C^<**>-algebra. The most important theme on extensions of states is extensions of factor ststes. In this research, we have obtained the theorem that every factor state on a separable abelian C^<**>-subalgebra B of a von Neumann algebra M extends uniquely to a pure state of M if and only if B is generated by minimal projections in M.Thus we can clarify the structure of separable abelian C^<**>-subalgebras, of a von Neumann algebra, on which every factor state uniquely extends to a pure state on the von Neumann algebra.
Now let A be a C^<**>-algebra and let A^^<^> be the spectrum of A,i.e., equvalence classes of nonzero irreducible representations of A.Then A^^<^> is a topological space equipped with the Jacobson topology. The reason of importance of A^^<^> in the theory of C^<**>-algebras is why A^^<^>is a rather large space and contains a lot of imformation on the structure of A and why we can see the structure of A through the topology of A^^<^>. Therefore the topology on A^^<^> has been investigated by many researchers. In this research, we have researched conditions for A^^<^> to be discrete, and we have obtained the following result :
Theorem. Let A be a C^<**>-algebra. Then the following conditions are equivalent :
(1) A^^<^> is discrete in the Jacobson topology.
(2) There exists an ideal I of A such that I^^<^> and <(A/I)>^^^ are discrete in the relative topology of A^^<^> and the open projection p in A^<****> satisfying I=A^<****>p A is a multiplier for A.
(3) A^^<^> is a T_1-space in the Jacobson topology and every open central projection in A^<****> is a multiplier for A.
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