Project/Area Number  09640144 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Iwate University 
Principal Investigator 
MIURA Yasuhide Iwate University, Faculty of Humanities and Social Sciences, Professor, 人文社会科学部, 教授 (20091647)

CoInvestigator(Kenkyūbuntansha) 
ONISHSI Yoshihiro Iwate University, Faculty of Humanities and Social Sciences, Associate Professor, 人文社会科学部, 助教授 (60250643)
ODAI Yoshitaka Iwate University, Faculty of Humanities and Social Sciences, Associate Professor, 人文社会科学部, 助教授 (10204215)
ISHIKAWA Yoichiro Iwate University, Faculty of Humanities and Social Sciences, Associate Professor, 人文社会科学部, 助教授 (80004000)
ISHIKAWA Akihiko Iwate University, Faculty of Humanities and Social Sciences, Professor, 人文社会科学部, 教授 (00084377)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,500,000 (Direct Cost : ¥2,500,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥1,500,000 (Direct Cost : ¥1,500,000)

Keywords  matrix ordered Hilbert space / selfdual cone / completely positive map / type of von Neumann algebra / operator inequality / 行列順序ヒルベルト空間 / 自己共役錐体 / 完全正値写像 / ノイマン環の型 / 作用素不等式 / ノイマン環 / 不等式 
Research Abstract 
We considered the question how the algebraic structure of a operator algebra determines or is determined by the structure of its underlying Hilbert space. In order to give the answer to this problem, we have introduced the notion of the complete order isomorphisms and complete orthogonal decomposition isomorphisms on the matrix ordered Hilbert space with a family of selfdual cones. Matrix ordered Hilbert spaces are the appropriate objects to which completely positive maps apply and enabled us to handle noncommutative order. In this research, we have proven that a completely positive projection on the Hilbert space induces a conditional expectation on a von Neumann algebra, and a complete order isomorphism induces an automorphism of the von Neumann algebra. We have then investigated the relationship between the type of von Neumann algebras and completely positive maps, and characterized the matrix ordered standard forms of finite von Neumann algebras. We have also introduced the notion
… More
of the order of operators on a Hilbert space based on the positivity with respect to the selfdual cone. The different point of this order from the usual order of operators is the compatibility with the products. We have investigated the fundamental properties on the operator inequalities with respect to this order, and sharpened the arithmetic and geometric mean inequality in the category of the ordered Hilbert space associated with a finite dimensional commutative von Neumann algebra. Furthermore, we have obtained some results in the geometry and the number theory. One of them is the investigation of the relationship between the sphere theorem and the Dehn lemma. The rank of the group of relative units of a finite Galois extension of the algebraic number field has then been calculated. It is necessary to determine the finite groups all whose abelian subgroups are cyclic. We have described by cyclotomic polynomials quite simply a mechanical system of onedimensional hard core chain which consists of infinite many particles with two different alternating mass Finally, we have generalized and given very generally applicable proof of the formula of Grant, which is a generalization of product formulae for division values of periods of the exponential function or each elliptic function with complex multiplication in a cyclotomic field, to every hyperelliptic function of cyclotomic type. Less
