群在von Neumann代数上作用的自由和遍历性注记

NOTES ON FREE AND ERGODIC ACTIONS OF GROUPS ON VON NEUMANN ALGEBRAS

本文研究了群在von Neumann代数上作用的自由性和遍历性问题.利用投影和群SL2(R)的Iwasawa分解,得到了可数离散群在交换von Neumann代数上作用的自由性的等价刻画,证明了SL2(R)在上半平面H上有理作用导出的SL2(R)在极大交换von Neumann代数A={Mf:f∈L2(H,dxdy/y2)}上的作用α是遍历的,但不是自由的.

In this article, we study the free and ergodic action of groups on von Neumann algebras. By using the projections and the Iwasawa decomposition of the group SL2（R）, we char- acterize the free action of a countable discrete group on an abelian von Neumann algebra and show that the action of SL2（R） on the abelian von Neumann algebra A = {Mf ： f ∈ L2（H,dxdy/y2）} induced by the rational action of the group on the upper-half plane H is ergodic but not free.