A research of noncommutative inequalities among operator algebras using a computer algebra
Project/Area Number  09640167 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
解析学

Research Institution  Toyama University 
Principal Investigator 
KUDO Fumio Toyama University, Department of Mathematics, Professor, 理学部, 教授 (90101188)

CoInvestigator(Kenkyūbuntansha) 
KOUYAMA Naoto Toyama University, Department of Mathematics, Professor, 理学部, 助手 (10293284)
MIZUNO Tooru Toyama University, Department of Mathematics, Professor, 理学部, 講師 (10018997)
SUZUKI Masaaki Toyama University, Department of Mathematics, Professor, 理学部, 教授 (10037236)
KAZAMAKI Norihiko Toyama University, Department of Mathematics, Professor, 理学部, 教授 (50004396)
YOSHIDA Norio Toyama University, Department of Mathematics, Professor, 理学部, 教授 (80033934)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1998 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1997 : ¥1,900,000 (Direct Cost : ¥1,900,000)

Keywords  operator / noncommutativity / inequality / computer algebra / quantum triangle / quantum trigonometry / 非可換不等式 / Webページ / cgiシステム / 自動投稿システム / Operator Trivia 
Research Abstract 
Under the title described above, the head investigator found first several quantum triangles in the non commutative or 'quantum' world of operator algebras, in collaboration with the coinvestigators described above, with Prof. Tsuyoshi Ando (Faculty of Economics, Hokusel Gakuen University, Sapporo) and with Prof. Shuhei Wada (Kisarazu National College of Technology) as a deepened result of the fundamental relations among operators discussed in the previous year. The head investigator gave a lecture on the result of quantum triangles together with that of quantum trigonometry at the KJWG1998 (Korea Japan Workshop on Geometry) held at the Department of Mathematics, Kyunpook National University. The quantum triangle found by the head investigator lies neither on the usual plane nor on the Euclidean space but on the (noncommutative) algebra of bounded linear operators on a Hilbert space. The point is that it does exist for each points (operators, in this case) in general position. To show the existence, the general theory of operator means developed by Prof. Ando and the head investigator plays an essential role. Especially, using the parallel addition found by R.J.Duffin (CarnegieMellon University ; at that time) or the harmonic mean the head investigator gave a proof for the existence of quantum acuteangled triangles for each points in general position. Furthermore, by a suggestion of Prof. Ando, using another operator mean, the head investigator gave a proof for the existence of quantum rightangled triangles for each points in general position, which enables us to develop the general theory of quantum trigonometry analogously to the usual plane trigonometry. In comparison with the fact that the plane trigonometry gives a proof for the fine structure of triangles such as ninepointcircle theorem, the quantum trigonometry found in this investigation is expected to find/prove such fine structure of quantum triangles.

Report
(3results)
Research Products
(3results)