A research of non-commutative inequalities among operator algebras using a computer algebra

1998 Fiscal Year Final Research Report Summary

• Project/Area Number: 09640167
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year 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)
• Co-Investigator(Kenkyū-buntansha): 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
• Keywords: operator / non-commutativity / inequality / computer algebra / quantum triangle / quantum trigonometry

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 co-investigators 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 KJWG-1998 (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 (non-commutative) 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 (Carnegie-Mellon University ; at that time) or the harmonic mean the head investigator gave a proof for the existence of quantum acute-angled 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 right-angled 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 nine-point-circle theorem, the quantum trigonometry found in this investigation is expected to find/prove such fine structure of quantum triangles.