The classical Carathéodory interpolation problem is the following: let n be a natural number, a0, a1, . . . , aN be complex numbers, and D the unit disk. When does there exist an analytic function F : D → C and complex numbers aN+1, aN+2, . . . such that F(z) = a0 + a1z + a2z2 + . . . + aNzN + aN+1zN+1 + . . . and ||F||∞ < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.
Arias presented generalizations of the original commutant lifting theorem relating to the full Fock space. Popescu then refined the approach by introducing domain algebras. While Arias and Popescu focused on module maps from one noncommutative domain algebra to itself, we present a unitarily equivalent representation which allows us to easily generalize their results to module maps between different noncommutative domain algebras. We use a renorming technique to prove that as long as the “formal identity map” is bounded, we can lift module maps between different noncommutative domains generated by finite positive regular free holomorphic functions. Finally, we apply the lifting theorem to create projective resolutions.
","abstract_html":"<p>The classical Carathéodory interpolation problem is the following: let n be a natural number, a<sub>0</sub>, a<sub>1</sub>, . . . , a<sub>N</sub> be complex numbers, and <strong>D</strong> the unit disk. When does there exist an analytic function F : <strong>D</strong> → <strong>C</strong> and complex numbers a<sub>N+1</sub>, a<sub>N+2</sub>, . . . such that F(z) = a<sub>0</sub> + a<sub>1</sub>z + a<sub>2</sub>z<sup>2</sup> + . . . + a<sub>N</sub>z<sup>N</sup> + a<sub>N+1</sub>z<sup>N+1</sup> + . . . and ||F||<sub>∞</sub> < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.</p> <p>Arias presented generalizations of the original commutant lifting theorem relating to the full Fock space. Popescu then refined the approach by introducing domain algebras. While Arias and Popescu focused on module maps from one noncommutative domain algebra to itself, we present a unitarily equivalent representation which allows us to easily generalize their results to module maps between different noncommutative domain algebras. We use a renorming technique to prove that as long as the “formal identity map” is bounded, we can lift module maps between different noncommutative domains generated by finite positive regular free holomorphic functions. Finally, we apply the lifting theorem to create projective resolutions.</p>","abstract_has_math":false,"creators":["Von Stroh, Jonathan"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Alvaro Arias, Ph.D.","James Haglar","Frédéric Latrémolière","Nicholas Ormes","Ramakrishna Thurimella"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-01-01T08:00:00Z","date_published":"2010-01-01T08:00:00Z","updated_at":"2026-07-24T02:02:19Z","subjects":["Commutant lifting","Fock space","Interpolation","Multivariable operator theory","Noncommutative domain","Weighted shifts","Mathematics","Physical Sciences and Mathematics"],"languages":["en"],"rights":["Copyright is held by the author. User is responsible for all copyright compliance.
"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.du.edu/etd/677","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Alvaro Arias, Ph.D.","James Haglar","Frédéric Latrémolière","Nicholas Ormes","Ramakrishna Thurimella"]},{"key":"dc:creator","label":"Author","values":["Von Stroh, Jonathan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2001-01-01T08:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Commutant lifting","Fock space","Interpolation","Multivariable operator theory","Noncommutative domain","Weighted shifts","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright is held by the author. User is responsible for all copyright compliance.
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Arias presented generalizations of the original commutant lifting theorem relating to the full Fock space. Popescu then refined the approach by introducing domain algebras. While Arias and Popescu focused on module maps from one noncommutative domain algebra to itself, we present a unitarily equivalent representation which allows us to easily generalize their results to module maps between different noncommutative domain algebras. We use a renorming technique to prove that as long as the “formal identity map” is bounded, we can lift module maps between different noncommutative domains generated by finite positive regular free holomorphic functions. Finally, we apply the lifting theorem to create projective resolutions.
"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Lifting Module Maps Between Different Noncommutative Domain Algebras"]}]}],"canonical_facts":{"dc:contributor":["Alvaro Arias, Ph.D.","James Haglar","Frédéric Latrémolière","Nicholas Ormes","Ramakrishna Thurimella"],"dc:creator":["Von Stroh, Jonathan"],"dc:date.available":["2001-01-01T08:00:00Z"],"dc:description.abstract":["The classical Carathéodory interpolation problem is the following: let n be a natural number, a0, a1, . . . , aN be complex numbers, and D the unit disk. When does there exist an analytic function F : D → C and complex numbers aN+1, aN+2, . . . such that F(z) = a0 + a1z + a2z2 + . . . + aNzN + aN+1zN+1 + . . . and ||F||∞ < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.
Arias presented generalizations of the original commutant lifting theorem relating to the full Fock space. Popescu then refined the approach by introducing domain algebras. While Arias and Popescu focused on module maps from one noncommutative domain algebra to itself, we present a unitarily equivalent representation which allows us to easily generalize their results to module maps between different noncommutative domain algebras. We use a renorming technique to prove that as long as the “formal identity map” is bounded, we can lift module maps between different noncommutative domains generated by finite positive regular free holomorphic functions. Finally, we apply the lifting theorem to create projective resolutions.
"],"dc:format":["application/pdf"],"dc:identifier":["https://digitalcommons.du.edu/etd/677"],"dc:language":["en"],"dc:rights":["Copyright is held by the author. User is responsible for all copyright compliance.
"],"dc:subject":["Commutant lifting","Fock space","Interpolation","Multivariable operator theory","Noncommutative domain","Weighted shifts","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Lifting Module Maps Between Different Noncommutative Domain Algebras"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T02:02:19Z"}