A New Class Of Non-self-adjoint Operator Algebras In Matrix Algebras——Kadison-Singer Algebras

Let L0is a Kadison-Singer lattice generated by single-point expand of a nest in matrix algebras, the von Neumann algebras is Matrix algebras Mn(C) and L1is a Kadison-Singer lattice too, its von Neumann algebras is Matrix algebras Mn+1(C). In this paper, the projection set L generated by a set L0(?)L0(or Lo (?) L1) and a rank one projection Pξ which a separating vector is ξ=(1,0,…,0,-1,0,…,0)T (ξ∈C2n or ξ∈C2n+1). We show that L is a reflexive lattice, to study Kadison-Singer characterization.Chapter One introduces some basic knowledge about operator algebras and important conclusion about Kadison-Singer algebras, Moreover, we state the main results of the thesis.Chapter Two we shall give a examples about to Kadison-Singer algebras. We prove that L is a reflexive lattice and L"=Mm(C)(m=2n or2n+1) and when n>2, the reflexive lattice L contain proper sublattice, and it is a Kadison-Singer lattice which L von Neumann is Matrix algebras Mm(C)(m>4), and m>6, it is only one contain the L. In addition, single-point expand of constructed Kadison-Singer lattices which von Neumann are Matrix algebras Mn(C), we prove that their are similar. Given to then other constructed Kadison-Singer lattices which von Neumann are Matrix algebras Mn(C), we prove that their are similar too.Chapter Three summarizes the main result in this thesis and points out several interesting problems.