| Co-Investigator(Kenkyū-buntansha) |
WATANABE Keiichi Faculty of Science, Associate Professor, 理学部, 助教授 (50210894)
HATORI Osamu Graduate School of Science and Technology, Associate Professor, 大学院・自然科学研究科, 助教授 (70156363)
SEIKIGAWA Kouei Faculty of Science, Professor, 理学部, 教授 (60018661)
TANAKA Kensuke Faculty of Science, Professor, 理学部, 教授 (70018258)
IZUCHI Keiji Faculty of Science, Professor, 理学部, 教授 (80120963)
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| Research Abstract |
In the theory of operator algebras and operator theory, the structure of invariant subspaces is considered as the notion of Hilbert modules over function algebras. At first, we deeply studied the theory of Hilbert module over function algbras by Douglas-Paulsen. After that, we investigated the theory of Hubert modules over operator algebras. In particular, we studied the stucture of subdiagonal algebras which are important of nonself-adjoint operator algebras. Applying the Tomita-Takesaki theory, we showed that every maximal subdiagonal algebra is always invariant with respect to the modular automorphisms. As the corollary, we showed that every subdiagonal algebra is a nest algebra with the atomic nest. Further, we considered the factorization theorem of operators around maximal subdiagonal algebras as the generalization of inner-outer factorization in function algebras. In particular, the universal factorization property is not alway valid. However, every maximal subdiagonal algebra has the partial factorization property.
On the other hand, from the viewpoint of function algebras, we studied the structure of invariant subspaces of L^2 (T^2), the property of commutators and so on. Further, we applied the optimization theory and infomation theory. We had some results about these topics.
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