Study of noncommutative probability and noncommutative entropy

近年来算子代数发展的两个主要趋势是由V.F.R. Jones和D. Voiculescu与此相关，本文研究了非对易概率论和非对易熵理论。此外，我们还研究了算子和矩阵的模不等式和迹不等式。所获得的结果总结如下：1.当群G作用在有限指标因子的包含N <$M上时，将交叉积N × G <$M × M的标准不变量与N <$M的标准不变量进行比较.此外，在与Masaki Izumi（京都大学）的联合工作中，在子因子情形下研究了一般融合代数的顺从性和强顺从性.与D. Petz（匈牙利）。讨论了各种约束条件下一元自由熵的极大化问题，系统地研究了随机矩阵经验特征值密度的大偏差原理。此外，我们还建立了定义在非交换随机变量的自伴、非自伴和酉三种自由熵之间的关系，并将其应用于自由熵的可加性和极大化问题.对Hilbert空间算子（特别是矩阵）的模不等式和迹不等式进行了一系列的研究。在与安藤刚（北星学园大学）的联合工作中，我们研究了什么是矩阵（或迹）保持器不等式。与Ando和Okubo（北海道教育大学）一起讨论了两个矩阵重积的迹不等式。与Hideki Kosaki（九州大学）联合，我们得到了模不等式，改进了酉不变模的算术-几何平均不等式。

Two major trends in recent development of operator algebras are subfactor theory initiated by V.F.R. Jones and free probability theory created by D. Voiculescu. Related with these we studied noncommutative probability theory and noncommutative entropy theory in the present research. Also, we studied norm inequalities and trace inequalities for operators and matrices. Results obtained are summarized in the following :1. When a group G is acting on an inclusion N ⊂ M of factors with finite index, the standard invariant of the crossed product N × G ⊂ M × M is compared with that of N ⊂ M. Moreover, in a joint work with Masaki Izumi (Kyoto Univ.), amenability and strong amenability of general fusion algebras are investigated on the model of the subfactor case.2. Large deviation principle for random matrices and free entropy were studied jointly with D. Petz (Hungary). We discussed maximization problems for one-variable free entropy under various constraints, and systematically investigated the large deviation principle for the empirical eigenvalue density of random matrices. Moreover, we established the relation among three types of free entropies defined for noncommutative random variables which are selfadjoint, non-selfadjoint and unitary, respectively, and applied it to the additivity and the maximization problems of free entropy.3. A series of investigations were made for norm inequalities and trace inequalities for Hilbert space operators (in particular matrices). In a joint work with Tsuyoshi Ando (Hokusei Gakuen Univ.), we investigated what is the matrix (or trace) Holder inequality. Jointly with Ando and Okubo (Hokkaido Univ. of Education) we discuss trace inequalities for multiple products of two matrices. Jointly with Hideki Kosaki (Kyushu Univ.), we obtained norm inequalities refining the arithmetic-geometric mean inequality for unitarily invariant norms.