Study on subfactors

子因素研究

基本信息

批准号：
09304017
负责人：
KOSAKI Hideki
金额：
$ 10.11万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A).
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

项目摘要

There is an active group of researchers of the Jones index theory and related subfactor analysis in our country. Main members in this group studied these subject matters from a variety of different viewpoints such as Ocneanu theory, representations of loop groups, bimodules, structure ananysis on type III factors, ergodic theory, and tensor category. Three conferences were held by support of the current funding. Although a wide variety of subjects in operator theory and operator algebras was investigated by the members in the duration of the current funding, main achievements on the proposed subject are as follows : (i) Longo-Rehren subfactors (closely releted to the notion of a quantum double) and the fusion rule of relevant bimodules were clarified. (ii) A certain deformation theory for Kac algebras via various cocycles was established based on subfactor analysis, and it now becomes possible to classify low-dimensional Kac algebras. (iii) Many "subfactor versions" of structure analysis of type III factors and the notion of orbit equivalence were obtained, and structure of subfactors in type III_0 factors became quite transparent. (iv) The notion of (strong) amenability (required for classification of subfactors) was clarified in many settings such as fusion algebras and tensor categories. (v) Many realizations of Cuntz-Krieger type C^*-algebras were found via bimodule approach, and some new knowledge was added to the understanding of these algebras. (vi) The (non-commutative) Rohlin property for C^*-algebras was successfully formulated, and consequently study on automorphisms of AF and AT algebras has advanced considerably.

我国有一群活跃的琼斯指数理论研究人员和相关的子因素分析。该小组的主要成员从各种不同的观点，例如Ocneanu理论，循环组的表示，双模型，关于III型因子的结构分析，赤道理论和Tensor类别研究了这些主题。通过当前资金的支持，举行了三场会议。尽管成员在当前资助期间研究了操作者理论和操作员代数的各种受试者，但对拟议受试者的主要成就如下：（i）longo-rehren子因子（密切相关的量子双重概念）和相关bimodules的融合规则。（ii）基于亚比例分析建立了通过各种共生的KAC代数的某些变形理论，现在可以对低维kac代数进行分类。（iii）获得了III型因子的结构分析和轨道等效概念的许多“子因子版本”，III_0型因子中的亚比例的结构变得非常透明。（iv）在许多情况下，例如融合代数和张量类别，阐明了（强）舒适性的概念（Summ）舒适性（子因子分类所需的概念）。（v）通过Bimodule方法发现了Cuntz-Krieger类型C^*的许多实现，并在对这些代数的理解中添加了一些新知识。（vi）成功制定了C^* - 代数的（非交通性的）Rohlin特性，因此研究了AF和代数的自动形态，已大大提高。