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.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2016

Author dc:creator

• Ng, Wai Hin 1987-

Advisor dc:contributor.advisor

• Paulsen, Vern I.

Committee members dc:contributor.committeemember

• Blecher, David P.
• Tomforde, Mark
• Smith, Roger

• Tensor products
• Factorization

Statement dc:rights

• The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/10657/3186

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/3186