Tensor Products and Factorizations of Operator Systems

In this dissertation, we start by studying the operator system maximal tensor product, called max, in [17] from different perspectives. One approach is by the factorization technique used in Banach spaces [11] and operator spaces [5, 31]. Although in [14] it was used in establishing the operator system version of complete positive approximation property, it was not fully utilized in terms of tensor products. From this point of view, we are able to characterize max via approximate completely positive factorization through the matrix algebras.

Motivated by the significant role of self-duality in factorization, we progress to operator systems that are self-dual as matrix-ordered spaces, or in finite-dimensional case, as operator systems. We construct the self-dual operator Hilbert system SOH based on Pisier’s operator Hilbert space OH [29] and prove analogous structural results of SOH. This leads us to create a tensor product of finite-dimensional operator systems via factorization through SOH, denoted by γsoh. We prove various tensorial and nuclearity properties of γsoh, which distinguish γsoh from other known tensor products found in [8, 17, 18]. Then we extend such construction to the infinite-dimensional case and conclude that γsoh indeed defines a new tensor product of operator systems.

The construction of SOH also motivates us to visit the Paulsen system SV of an operator space V (see [26]). We examine some structural questions about SV including the states, matrix-ordered dual, and operator system quotient. We characterize the states on SV , hence lead to proving that the matrix-ordered dual of SV is again an operator system regardless of the operator space V. Finally, we end this dissertation with an exposition to an interesting quotient of SV.