arXiv 1001.01692010-01math.OA

有限von Neumann algebras的混合子代数

Mixing subalgebras of finite von Neumann algebras

Jan Cameron, Junsheng Fang, Kunal Mukherjee

备注：: 24 pages, minor typos corrected

摘要：

Jolissaint和Stalder引入了有限von Neumann代数的von Neumann子代数的混合和弱混合的定义。本文研究了具有这些混合性质的子代数的各种代数和分析性质。证明了vonNeumann代数的混合包含的一些基本结果，建立了vonNeumann子代数的混合性质与正规化子之间的联系。混合子代数的特殊情况下所产生的包含可数离散群发现应用遍历理论，特别是一个新的推广经典定理的哈尔莫斯的自同构的一个紧凑的阿贝尔群。对于有限vonNeumann代数$M$及其vonNeumann子代数$A$，$B$，引入了$B\subseteq M$相对于$A$的弱混合概念.我们证明了$B\subset M$相对于$A$的弱混合等价于以下性质：如果$x\in M$且存在有限个元素$x_1,...,x_n\in M$使得$Ax\subset \sum_{i=1}^nx_iB$，则$x\in B$ .最后，我们的文件与混合子代数所产生的合并自由产品和交叉产品建设的进一步的例子分类。

英文摘要：

Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical theorem of Halmos on the automorphisms of a compact abelian group. For a finite von Neumann algebra $M$ and von Neumann subalgebras $A$, $B$ of $M$, we introduce a notion of weak mixing of $B\subseteq M$ relative to $A$. We show that weak mixing of $B\subset M$ relative to $A$ is equivalent to the following property: if $x\in M$ and there exist a finite number of elements $x_1,...,x_n\in M$ such that $Ax\subset \sum_{i=1}^nx_iB$, then $x\in B$. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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