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

在上述标题下，首席研究员与上述合作研究人员、安藤刚教授(札幌Hokusel Gakuen大学经济系)和和田秀平教授(Kisarazu国立理工学院)合作，首先在非对易或“量子”的算子代数世界中发现了几个量子三角形，这是前一年讨论的算子之间基本关系深化的结果。在庆福国立大学数学系举行的KJWG-1998(韩日几何研讨会)上，首席研究人员就量子三角形和量子三角的结果进行了演讲。首席研究人员发现的量子三角形既不在通常的平面上，也不在欧几里德空间上，而是在希尔伯特空间上的有界线性算子的(非交换)代数上。问题是，它对于一般位置中的每个点(在本例中为运算符)确实存在。为了证明它的存在性，由Ando教授和首席研究员发展的算子平均的一般理论起着至关重要的作用。特别是，利用R.J.Duffin(当时的卡内基-梅隆大学)发现的平行加法或调和平均，首席研究员给出了一般位置上每个点存在量子锐角三角形的证明；此外，在Ando教授的建议下，使用另一种算子平均，首席研究员给出了一般位置上每个点存在量子直角三角形的证明，这使得我们能够类似于通常的平面三角来发展量子三角的一般理论。与平面三角给出了三角形精细结构的证明(如九点圆定理)相比，本研究发现的量子三角有望发现/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.