{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-1561"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-1561","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"The Segal-Bargmann transform on inductive limits of compact symmetric spaces","abstract":"We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces $\\{L^2(M_n)^{K_n}\\}_n$ where $\\{M_n = U_n/K_n\\}_n$ is a propagating sequence of symmetric spaces of compact type with the assumption that $U_n$ is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on $L^2(M_n)^{K_n}, \\ n = 1,2,...$. For each $n$, let $\\widehat{U_n}$ be the set of equivalence classes of irreducible unitary representations of $U_n$ and let $\\widehat{U_n/K_n} \\subseteq \\widehat{U_n}$ be the set of $K_n$-spherical representations. The definition of the propagation gives a nice property allowing us to embed $\\widehat{U_n/K_n}$ into $\\widehat{U_m/K_m}$ for $m \\geq n$ in a natural way. With these embeddings, we can produce the unitary embeddings from $L^2(M_n)^{K_n}$ into $L^2(M_m)^{K_m}$ for $m \\geq n$. Hence, the direct limit of the Hilbert spaces $\\{L^2(M_n)^{K_n}\\}_n$ is obtained in the category of Hilbert spaces and unitary embeddings and we can construct the Segal-Bargmann transform on the resulting limit in a canonical way.","abstract_html":"We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces <span class=\"etd-inline-math\">\\{L<sup>2</sup>(M<sub>n</sub>)<sup>K<sub>n</sub></sup>\\}<sub>n</sub></span> where <span class=\"etd-inline-math\">\\{M<sub>n</sub> = U<sub>n</sub>/K<sub>n</sub>\\}<sub>n</sub></span> is a propagating sequence of symmetric spaces of compact type with the assumption that <span class=\"etd-inline-math\">U<sub>n</sub></span> is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on <span class=\"etd-inline-math\">L<sup>2</sup>(M<sub>n</sub>)<sup>K<sub>n</sub></sup>,  n = 1,2,...</span>. For each $n$, let <span class=\"etd-inline-math\">\\widehat{U<sub>n</sub>}</span> be the set of equivalence classes of irreducible unitary representations of <span class=\"etd-inline-math\">U<sub>n</sub></span> and let <span class=\"etd-inline-math\">\\widehat{U<sub>n</sub>/K<sub>n</sub>} \\subseteq \\widehat{U<sub>n</sub>}</span> be the set of <span class=\"etd-inline-math\">K<sub>n</sub></span>-spherical representations. The definition of the propagation gives a nice property allowing us to embed <span class=\"etd-inline-math\">\\widehat{U<sub>n</sub>/K<sub>n</sub>}</span> into <span class=\"etd-inline-math\">\\widehat{U<sub>m</sub>/K<sub>m</sub>}</span> for $m \\geq n$ in a natural way. With these embeddings, we can produce the unitary embeddings from <span class=\"etd-inline-math\">L<sup>2</sup>(M<sub>n</sub>)<sup>K<sub>n</sub></sup></span> into <span class=\"etd-inline-math\">L<sup>2</sup>(M<sub>m</sub>)<sup>K<sub>m</sub></sup></span> for $m \\geq n$. Hence, the direct limit of the Hilbert spaces <span class=\"etd-inline-math\">\\{L<sup>2</sup>(M<sub>n</sub>)<sup>K<sub>n</sub></sup>\\}<sub>n</sub></span> is obtained in the category of Hilbert spaces and unitary embeddings and we can construct the Segal-Bargmann transform on the resulting limit in a canonical way.","abstract_has_math":true,"creators":["Wiboonton, Keng"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T02:57:49Z","subjects":["heat equation","compact symmetric spaces","inductive limits","Hilbert spaces of holomorphic functions"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07062009-114849","https://repository.lsu.edu/gradschool_dissertations/562"],"render_values":[{"text":"etd-07062009-114849","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/562","href":"https://repository.lsu.edu/gradschool_dissertations/562","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.562","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Wiboonton, Keng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2009-06-23"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:09:14Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["heat equation","compact symmetric spaces","inductive limits","Hilbert spaces of holomorphic functions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07062009-114849","10.31390/gradschool_dissertations.562","https://repository.lsu.edu/gradschool_dissertations/562"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces $\\{L^2(M_n)^{K_n}\\}_n$ where $\\{M_n = U_n/K_n\\}_n$ is a propagating sequence of symmetric spaces of compact type with the assumption that $U_n$ is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on $L^2(M_n)^{K_n}, \\ n = 1,2,...$. For each $n$, let $\\widehat{U_n}$ be the set of equivalence classes of irreducible unitary representations of $U_n$ and let $\\widehat{U_n/K_n} \\subseteq \\widehat{U_n}$ be the set of $K_n$-spherical representations. The definition of the propagation gives a nice property allowing us to embed $\\widehat{U_n/K_n}$ into $\\widehat{U_m/K_m}$ for $m \\geq n$ in a natural way. With these embeddings, we can produce the unitary embeddings from $L^2(M_n)^{K_n}$ into $L^2(M_m)^{K_m}$ for $m \\geq n$. Hence, the direct limit of the Hilbert spaces $\\{L^2(M_n)^{K_n}\\}_n$ is obtained in the category of Hilbert spaces and unitary embeddings and we can construct the Segal-Bargmann transform on the resulting limit in a canonical way."]},{"key":"dc:title","label":"Title","values":["The Segal-Bargmann transform on inductive limits of compact symmetric spaces"]}]}],"canonical_facts":{"dc:creator":["Wiboonton, Keng"],"dc:date":["2009-06-23"],"dc:date.available":["2022-05-12T23:09:14Z"],"dc:description.abstract":["We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces $\\{L^2(M_n)^{K_n}\\}_n$ where $\\{M_n = U_n/K_n\\}_n$ is a propagating sequence of symmetric spaces of compact type with the assumption that $U_n$ is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on $L^2(M_n)^{K_n}, \\ n = 1,2,...$. For each $n$, let $\\widehat{U_n}$ be the set of equivalence classes of irreducible unitary representations of $U_n$ and let $\\widehat{U_n/K_n} \\subseteq \\widehat{U_n}$ be the set of $K_n$-spherical representations. The definition of the propagation gives a nice property allowing us to embed $\\widehat{U_n/K_n}$ into $\\widehat{U_m/K_m}$ for $m \\geq n$ in a natural way. With these embeddings, we can produce the unitary embeddings from $L^2(M_n)^{K_n}$ into $L^2(M_m)^{K_m}$ for $m \\geq n$. Hence, the direct limit of the Hilbert spaces $\\{L^2(M_n)^{K_n}\\}_n$ is obtained in the category of Hilbert spaces and unitary embeddings and we can construct the Segal-Bargmann transform on the resulting limit in a canonical way."],"dc:identifier":["etd-07062009-114849","10.31390/gradschool_dissertations.562","https://repository.lsu.edu/gradschool_dissertations/562"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["heat equation","compact symmetric spaces","inductive limits","Hilbert spaces of holomorphic functions"],"dc:title":["The Segal-Bargmann transform on inductive limits of compact symmetric spaces"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:57:49Z"}