The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions
The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions
复制标题
单射因子自同构群的结构与离散交换群作用的余循环共轭
DOI：
10.1007/bf02392758
复制
发表时间：
1992
期刊：
影响因子：
3.7
通讯作者：
M. Takesaki
中科院分区：
文献类型：
--
作者：
Yasuyuki Kawahigashi;C. Sutherland;M. Takesaki
本文首次证明了cones关于一类内射因子的近似内自同构和中心平凡自同构的声明，并给出了一类离散阿贝儿群或有限群对一类唯一内射因子的作用的一个直到循环共轭的分类，从而完成了该类群对一类内射因子作用分类的最后一步。自同构群的研究是理解冯·诺依曼代数结构的有力方法。Connes在[4,6,7,8]中出色地发展了这种方法。Jones[15]和Ocneanu[18]遵循Connes[4,6]的思路，在唯一的近似有限维（AFD）因子II1上完成了离散可服从群体动作的分类。他们的工作也为III型病例提供了有用的工具。Sutherland-Takesaki[20]给出了III~， 0~< 2< i型AFD因子上的离散可服从群作用的分类，通过他们和Ocneanu的工作，两类特殊自同构的重要性变得清晰起来。这类是因子v~的近似内自同构Int （d~）和中心平凡自同构Cnt （~ t）。Connes[5]宣布了这些类对III型AFD因子的表征，但此后十多年来一直没有得到证明，尽管该结果被用于Connes[8]的引理2 (a)中，该引理与Haagerup[13]一起建立了1111型AFD因子的唯一性，并在上述论文[20]中也得到了证明。在Connes [5， section 3.8]中没有证明的描述如下。（参见[11]和[4]的符号。）
The purpose of this paper is to give a proof of Connes' announcement on approximately inner automorphisms and centrally trivial automorphisms of an injective factor of type III for the first time, and to provide a classification, up to cocycle conjugacy, of actions of a discrete abelian or finite group on the unique injective factor of type III1, which completes the final step of classification of actions of such groups on injective factors.The study of automorphism groups has been a powerful method for understanding the structure of von Neumann algebras. Connes magnificently developed this approach in [4, 6, 7, 8]. Jones [15] and Ocneanu [18] followed the line of Connes [4, 6] and completed the classification of discrete amenable group actions on the unique approximately finite dimensional (AFD) factor of type II1. Their work also provides useful tools for the case of type III. Sutherland-Takesaki [20] gave a classification of discrete amenable group actions on AFD factors of type III~, 0~< 2< I. Through their and Ocneanu's work, importance of two special classes of automorphisms became clear. The classes are the approximately inner automorphisms Int (d~) and the centrally trivial automorphisms Cnt (~ t) of a factor v~. Connes [5] announced a characterization of these classes for AFD factors of type III, but the proof has been unavailable for more than ten years since then, though this result was used in Lemma 2 (a) of Connes [8], which together with Haagerup [13] established the uniqueness of AFD factors of type 1111, and also in the above-mentioned paper [20]. The characterization, announced in Connes [5, section 3.8] without proof, is as follows.(See [11] and [4] for notations.)