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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.

在我国，对琼斯指数理论及其子因子分析的研究已经形成了一支活跃的研究队伍。该小组的主要成员从各种不同的角度研究这些主题，如Ocneanu理论、环路群的表示、双模、III型因子的结构分析、遍历理论和张量范畴。在当前资金的支持下举行了三次会议。虽然在当前资助期间，成员们研究了算子理论和算子代数中的各种各样的主题，但提出的主题的主要成果如下：(i) Longo-Rehren子因子（与量子双模的概念密切相关）和相关双模的融合规则得到澄清。（ii）基于子因子分析，建立了Kac代数经各种环的一定变形理论，使得对低维Kac代数进行分类成为可能。（三）获得了iii型因子结构分析和轨道等效概念的许多“子因子版本”，iii型因子中_0型因子的子因子结构变得相当透明。（iv）（强）适应性（子因子分类所需）的概念在融合代数和张量范畴等许多设置中得到澄清。(v)通过双模方法发现了许多Cuntz-Krieger型C^*-代数的实现，增加了对这些代数的认识。（vi）成功地建立了C^*-代数的（非交换）Rohlin性质，从而使AF和AT代数的自同构研究取得了很大进展。