Research On A Class Of Operators In The Hyperfinite ?₁ Factor

In recent years,the hyperfinite ? factor was extensively studied.In the paper,we study the class of operators uf(v)in R,where f is a bounded Lesbegue measurable function on the unit circle S1.let u,v be two generators of R such that u*u = v*v = 1 and vu = e2?i?uv for an irrational number ?.and many von Neumann algebras can be generated by two unitary operators u and v with u2 = v3 = 1.In chapter 3,we obtain the spectrum and spectrum radius of uf(v)by using Birkhoff's Er-godic theorem and the unique ergodicity of the irrational rotation.Moreover,we obtain spectrum of uf(v)by using the averaging technique.where f is a continuous function on S1.In chapter 4,we study the von Neumann algebra generated by uf(v).We show that if the zero set of f(z)?L?(S1,m)has Lebesgue measure zero,then W*(uf(v))is an irreducible subfactor of R with index n for some positive integer n.In chapter 5,we study the invariant subspace problem of uf(v)relative to R.By calculate the Brown measure of uf(v).We will show that the Brown measure of uf(v)(in R)is the Haar measure on ?(f(v))S1 for all f?L?(S1,m).As a corollary of Haagerup and Schultz'z result,if?(f(v))>0,for example f is a polynomial,then uf(v)has a continuous family of hyperinvariant subspaces affiliated with R.On the other hand,if ?(f(v))= 0 we show that the known methods are unable to determine whether or not the operator uf(v)has a nontrivial,closed,invariant subspace affiliated with R.In chapter 6,we show that the C*-algebra generated by uf(v)and the identity operator is closely related to the generalized universal irrational rotation algebra.Precisely,we will prove the following result.Let Y be the zero points of f(z).If Y satisfy ?n(Y)?Y=? for any integer n?0,where ?(z)= e2?i?z,then C*(uf(v),1)is a generalized universal irrational rotation C*-algebra.Furthermore,if |f|(z)is not a periodic function,then C*(uf(v),1)(?)A?,|f|2.As a corollary,we will show that if A?,? is a simple C*-algebra,then A?,? is generated by an element uf(v)and the identity operator for some f(z)? C(S1).In chapter 7,we prove that many von Neumann algebras can be generated by two unitary operators u and v with u2 = v3 = 1.As applications,three important classes of ?1 factors possessing this property are those with Property ?,those with a Cartan masa,and non-prime ones.