Unital Operator Spaces And Approximately Unital Operator Systems

The thesis is divided into four parts. Chapter 1 is a preliminary chapters, in which we introduce the reader to basic definitions and elementary properties. In Chapter 2. we characterize unital operator space via a notion of matrix numerical index. Using this, we prove that the quotient of a unital operator spaces by a complete M-ideals is a unital operator space, and that both (CB(A),I_A) and (CB(M_*),IM*) are unital oper-ator spaces (when A is a C*-algebra and M is a von-Neumann algebra). On the other hand, we consider tensor products of quasi-unital operator spaces and construct a "uni-tal tensor product" from a given operator space tensor products. We also show that for a normed space X, (minL(X),Ix) is a unital operator space if and only if X has numerical index 1. With this one can construct many new unital operator spaces from well-known normed spaces. Finally, we will prove that (V, u) and (V, u*) are unital operator spaces, if V= span{e, u, u*} is a 3-dimensional operator system in a unital C*-algebra with the "generator" u being an isometry. The main theme of Chapter 3 is to consider’approximately unital operator systems’. The goals are to prove a version of the Choi-Effros theorem for these systems:to introduce a process to obtain an approxi-mately unital operator systems from a given matrix ordered vector space with a proper approximate order unit; to study the second duals of these objects and to prove that a C*-algebra can be characterized as an approximately unital operator system that is also an approximately unital matrix ordered *-algebra. Chapter 4 is devoted to the computa-tion of the complete positive numerical indices of quasi-operator systems. We will show that{n_cb~+(V):V is matrix regular operator space}= [1/2,1] and that the injective tensor product of two quasi-operator systems is again a quasi-operator system.