Study of noncommutative probability and noncommutative entropy

非交换概率和非交换熵的研究

基本信息

批准号：
09640152
负责人：
HIAI Fumio
金额：
$ 1.86万
依托单位：
TOHOKU UNIVERSITY (1998-1999)Ibaraki University (1997)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640152/
关键词：
operator algebras noncommutative probability theory free probability theory noncommutative entropy free entropy random matrix large deviation principle norm inequality エントロピーアメナビリティ行列

项目摘要

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.

近期运营商代数开发的两个主要趋势是V.F.R.发起的亚比例理论。琼斯和自由概率理论由D. voiculescu创建。与这些相关的我们研究了本研究中的非交通概率理论和非交通性熵理论。此外，我们研究了运营商和物质的规范不平等和追踪不平等现象。所获得的结果总结在以下内容中：1。当G组对具有有限指数的因子的纳入n⊂M时，将交叉产品N×G g gg⊂m×M的标准不变性与N⊂MM. M. Masaki Izumi（Masaki izumi（Kyoto Univ。）的共同作品进行了比较。与D. Petz（匈牙利）共同研究了随机物品和自由熵的大偏差原理。我们讨论了在各种约束下对单变量自由熵的最大化问题，并系统地研究了随机物品的经验特征值密度的大偏差原理。此外，我们分别为非交换性随机变量定义的三种类型的自由熵之间建立了关系，这些变量分别是自偏，非偏爱和单一的，并将其应用于自由熵的添加性和最大化问题3。对希尔伯特太空经营者（特别是物质）的规范不平等和痕迹不平等进行了一系列投资。在与Tsuyoshi Ando（Hokusei Gakuen Univ）的联合合作中，我们研究了矩阵（或痕量）持有人不平等的是什么。与Ando和Okubo（北海道大学）共同讨论了两种物品的多种产品的痕迹不平等。我们与Hideki Kosaki（Kyushu Univ。）共同获得了规范不平等，以完善单位不变规范的算术几何平均值。