Non-commutative weak*-continuous operator extensions
| dc.contributor.advisor | Zimmer, Beate | |
| dc.contributor.author | Flores, Luis Carlos | |
| dc.contributor.committeeMember | Blecher, David | |
| dc.contributor.committeeMember | Tintera, George | |
| dc.date.accessioned | 2018-10-19T15:35:42Z | |
| dc.date.available | 2018-10-19T15:35:42Z | |
| dc.date.issued | 2018-05 | |
| dc.description.abstract | Extension theorems such as the Hahn-Banach Extension Theorem are a central idea of functional analysis. In 1962, Gleason and Whitney proved an extension theorem for weak*-continuous, linear functionals on H^∞ (D) to positive, weak*-continuous functionals on L^∞ (T). Hoffman and Rossi in 1967 provided a related, albeit different, extension theorem: weak*-continuous characters on a unital subalgebra A of L^∞ (T) can be extended to positive functionals on L^∞ (T). They demonstrated that A+A^* being weak*-dense in H^∞ (D) was not necessary for a weak*-continuous character to have a positive, weak*-continuous extension. Arveson is credited with a non-commutative version of the Hahn-Banach Extension Theorem: a completely positive, linear map from a unital, self-adjoint subspace X of a C^*-algebra A into B(H), where H is an arbitrary Hilbert space, can be extended to a completely positive map from A into B(H). In this work, we look at the methods Hoffman and Rossi used to develop their extension theorem, and investigate if they can be used in a non-commutative setting. We provide a non-commutative generalization of their Extended Krein-Smulian Theorem, which was a central part of their proof. We also demonstrate that some special subalgebras of the non-commutative space, M_2 (L^∞ (m)) will have weak*-continuous functionals on them with weak*-continuous and positive extensions on all of M_2 (L^∞ (m)). Furthermore, we generalize the work of Hoffman and Rossi in a non-commutative fashion to some weak*-continuous characters on singly generated dual algebras. | en_US |
| dc.description.college | College of Science and Engineering | en_US |
| dc.description.department | Mathematics and Statistics | en_US |
| dc.format.extent | 37 pages | en_US |
| dc.identifier.uri | https://tamucc-ir.tdl.org/handle/1969.6/87014 | |
| dc.language.iso | en_US | en_US |
| dc.rights | This material is made available for use in research, teaching, and private study, pursuant to U.S. Copyright law. The user assumes full responsibility for any use of the materials, including but not limited to, infringement of copyright and publication rights of reproduced materials. Any materials used should be fully credited with its source. All rights are reserved and retained regardless of current or future development or laws that may apply to fair use standards. Permission for publication of this material, in part or in full, must be secured with the author and/or publisher. | en_US |
| dc.subject | Dual Algebras | en_US |
| dc.subject | Hoffman-Rossi | en_US |
| dc.subject | Operator Spaces | en_US |
| dc.subject | von Neumann Algebras | en_US |
| dc.subject | Weak*-Continuous Character | en_US |
| dc.title | Non-commutative weak*-continuous operator extensions | en_US |
| dc.type | Text | en_US |
| dc.type.genre | Thesis | en_US |
| thesis.degree.discipline | Mathematics | en_US |
| thesis.degree.grantor | Texas A & M University--Corpus Christi | en_US |
| thesis.degree.level | Masters | en_US |
| thesis.degree.name | Master of Science | en_US |