Some Problems Related To Property ?

In order to prove the isomorphism of type ?1 factors,von Neumann and Murray introduced the invariant-property ? in 1943.Let M be a type ?1 factor,? be a faithful normal tracial state on M.For any x1,:x2,…xn?M and arbitrary ?>0,there is a unitary u?u(M),?(u)=0,such that ?[xk,u]?2<?(k=1,…,n),then M is said to have property ?.In this paper,we study some questions related to the property ?.Let N=(?)GN1,where N1 is a finite von Neumann algebra and G is a countable discrete group.The action ?:G?Aut(N)is a Bernoulli action,then N(?)? G is a type ?1 factor.The paper is divided into two parts:In the first part,we prove that N(?)? Z has property ? by the definition of property? and some auxiliary lemmasLet M be a separable type ?1 von Neumann algebra with the property ?.In the second part,we prove that M is singly generated by the techniques of matrix and the methods in algebra and analysis.