Algebraic analogs of the Connes spectrum
Algebraic analogs of the Connes spectrum
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DOI：
10.1016/0021-8693(88)90284-0
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发表时间：
1988-05
影响因子：
0.9
通讯作者：
S. Montgomery;D. Passman
中科院分区：
文献类型：
--
作者：
S. Montgomery;D. Passman
本文得到了vonNeumann代数和C-代数交叉积的某些已知结果的代数类似。这里特别感兴趣的是由A.康乃斯对第三类因素的分类。此外，我们关注的是某些冯诺依曼代数结果中所载的工作康纳斯，竹崎，和Nakagami和某些，更代数，C-代数结果的工作Olesen，佩德森，岸本。设G是有限群，A是G-分次环，1。则存在A的扩张环，由这个结构确定，称为A乘G的smash积。这个覆盖环来自于Hopf代数的理论，我们用A# G表示它。在本文的前半部分，我们引入并研究了G的Connes子群和强Connes子群，并将它们与A# G * 的理想结构联系起来。特别地，我们得到了A# G是素的或单的判据.在另一个方向上，设A是一个K-代数，G是A的K-自同构的有限交换群。若域K含有适当的单位根，则斜群环AG有一个自然Smash积结构A# G，其中G = Hom（G，K′）是G的对偶群.问题是根据G在A上的作用，明确地确定G的Connes子群。这是实现在第二部分的文件下的额外的假设，A是一个素环。关于我们结果的算子代数来源的更详细的讨论将在附录中给出。
In this paper we obtain algebraic analogs of certain known results on crossed products of von Neumann algebras and of C∗-algebras. Of particular interest here is the Connes spectrum, introduced by A. Connes in his classification of factors of type III. In addition, we are concerned with certain von Neumann algebra results contained in the work of Connes, Takesaki, and Nakagami and with certain, more algebraic, C∗-algebra results from the work of Olesen, Pederson, and Kishimoto. Let G be a finite group and let A be a G-graded ring with 1. Then there exists an extension ring of A, determined by this structure, called the smash product of A by G∗. This overring comes from the theory of Hopf algebras and we denote it by A# G∗. In the first half of this paper, we introduce and study the Connes and strong Connes subgroups of G and we relate them to the ideal structure of A# G∗. In particular we obtain criteria for A# G∗ to be prime or simple. In a different direction, suppose A is a K-algebra and G is a finite abelian group of K-automorphisms of A. If the field K contains appropriate roots of unity, then the skew group ring AG has a natural smash product structure A# G ̂∗ where G ̂= Hom (G, K′) is the dual group of G. The problem is to explicitly determine the Connes subgroup of G ̂ in terms of the action of G on A. This is achieved in the second half of the paper under the additional assumption that A is a prime ring. A more detailed discussion of the operator algebra sources of our results will be given in the Appendix.