A Class Of Non Self-Adjoint Subalgebras Of Matrix Algebras

Let L be the lattice of projections generated by the maximal diagonal projection nest on the matrix algebra Mn(C) and the projection where{Eij:i,j=1,2,…,n}is the canonical matrix units of Mn(C). In this thesis, we characterize some properties of the reflexive algebra Alg(L) determined by L The thesis consists of five sections.In Section 1, we give a brief introduction and some basics on operator algebras.In Section 2, we prove that L is a Kadison-Singer lattice which generates the matrix algebra Mn(C), and thus the associated reflexive algebra Alg(L) is a Kadison-Singer algebra.In Section 3, we study some properties of Alg(L) such as its dimension, Hamel base and the center. We also describe two-sided ideals of the Kadison-Singer algebra.In Section 4, we study derivations on Alg(L), and prove that every deriva-tion from Alg(L) into itself is inner.In Section 5, we study automorphisms on Alg(L), and show that each automorphism of Alg(L) is spatial.