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

在上述标题下，首席研究员与上述共同研究者，与Tsuyoshi Ando教授（札幌北馆学园大学经济学院）和Shuhei Wada教授（Kisarazu国立技术学院）合作，在算子代数的非交换或“量子”世界中首次发现了几个量子三角形，作为前一年讨论的算子之间基本关系的深化结果。在庆北大学数学系举行的kkjwg -1998（韩日几何研讨会）上，首席研究员就量子三角形和量子三角学的结果进行了演讲。首席研究员发现的量子三角形既不在通常平面上，也不在欧几里得空间上，而是在希尔伯特空间上有界线性算子的（非交换）代数上。关键是它确实存在于一般位置上的每个点（这里是算子）。为了证明这一理论的存在，由安藤教授和首席研究员提出的算子手段的一般理论起了至关重要的作用。特别是，首席研究员利用R.J.Duffin（当时卡内基梅隆大学）发现的平行加法或调和平均值，证明了一般位置上每个点的量子锐角三角形的存在性。此外，在Ando教授的建议下，首席研究员使用另一种算子均值证明了一般位置上每个点的量子直角三角形的存在性，这使我们能够类似于通常的平面三角学来发展量子三角学的一般理论。与平面三角学证明了如九点圆定理等三角形的精细结构相比，本研究中发现的量子三角学有望发现/证明量子三角形的精细结构。

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.