群在von Neumann代数上作用的自由和遍历性注记

摘要

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 characterize 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.%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Yon 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.