Denseness Of The Migration Operator, And P - Haar Unitary Yuan And Ks - Algebra

We denote by B(H) the algebra of all bounded linear operators on a Hilbert spaceH. With regards to the transitive algebra question. Kadison suggested the idea thatsome self-adjoint maximal abelian subalgebra of B(H) and some elements not in thesubalgebra might generate a non-trivial transitive algebra. Arveson proved in [1] thatKadison’s original idea does not work. That is, if A is a transitive subalgebra of B(H)which contains a self-adjoint maximal abelian von Neumann algebra, then A is strong-operator dense in B(H). Inspired by the invariant subspace problem afliated with a vonNeumann algebra [6], Fang, Hadwin and Ravichandran [4] studied the transitive algebragenerated by some set of operators in a fnite von Neumann algebra and its commutant.we study the transitivity in a factor of type Π1. we also study p Haar unitary and theproblems of Kadison-Singer algebras.This thesis consists of three chapters:In chapter one we study the transitive algebra question by considering the invariantsubspace problem relative to von Neumann algebras. Let Gi={e, ui}, i∈N, ui2=ebe a cyclic group and its order is two. G=i∈NGi. Let L(G) be a von Neumannalgebra, L(G)=R(G). We prove that the von Neumann algebra generated by {λu1u2+λu3u4, λu5u6+λu7u8} and its commutation is strong-operator dense in B(H).We also provethat the algebra generated by {λu1u2, λu3u4} and its commutation is strong-operator densein B(H).In chapter two we exhibit the distribution of p Haar unitary U and the distribution of the self-adjoint element U+U. We prove the existence of the p Haarunitary in Mn(C).In chapter three we give some examples of Kadison-Singer algebras with diagonalequal to CI. The corresponding Kadison-Singer lattices is obtained through one pointextension of nests.