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

在上面描述的标题下，首席研究员在算子代数的非交换或“量子”世界中发现了第一批量子三角形，与上面描述的共同研究者合作，与Tsuyoshi Ando教授合作。（北溶胶学院大学经济系，札幌）和Shuhei Wada教授（木更津国立技术大学）作为前一年讨论的运营商之间的基本关系深化的结果。在国立庆北大学数学系举行的KJWG-1998（韩国-日本几何学研讨会）上，首席研究员就量子三角形的结果和量子三角学的结果发表了演讲。首席研究员发现的量子三角形既不位于通常的平面上，也不位于欧几里德空间上，而是位于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的每个点（在这种情况下是运算符）。为了证明存在性，安藤教授和首席研究员开发的算子方法的一般理论起着至关重要的作用。特别是，使用R. J. Duffin发现的平行加法，（美国梅隆大学;当时）或调和平均，首席研究员证明了一般位置上每个点都存在量子锐角三角形。此外，根据安藤教授的建议，使用另一种算子平均，首席研究员证明了一般位置上的每个点都存在量子直角三角形，这使我们能够发展量子三角学的一般理论，类似于通常的平面三角学。相对于平面三角学对三角形的精细结构如九点圆定理的证明，量子三角学有望发现/证明量子三角形的这种精细结构。

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.