{"id":{"repo_id":"houston","oai_identifier":"oai:uh-ir.tdl.org:10657/10237"},"canonical_url":"https://search.dev.ndltd.org/etd/houston/oai:uh-ir.tdl.org:10657/10237","repository":{"repo_id":"houston","name":"University of Houston","base_url":"https://uh-ir.tdl.org/server/oai/request"},"display":{"title":"Structure of Intermediate C*-subalgebras of discrete group actions","abstract":"This thesis deals with the structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$, either of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ or of the type $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$. We begin by investigating the ideal structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$ of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ for commutative unital $\\Gamma$-simple $\\Gamma$-$C^*$-algebras $\\mathcal{A}$. In particular, we show that if $\\Gamma$ is a $C^*$-simple group, then every such intermediate $C^*$-sub-algebra $\\mathcal{B}$ is simple. Continuing our perusal, we find examples of inclusions $C_{\\lambda}^*(\\Gamma)\\subseteq \\mathcal{A}\\rtimes_r\\Gamma$ for which every intermediate $C^*$-sub-algebra $\\mathcal{B}$ of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ is a crossed product. We show that for a large class of actions $\\Gamma\\curvearrowright\\mathcal{A}$ of $C^*$-simple groups $\\Gamma$ on unital $C^*$-algebras $\\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-sub-algebra $\\mathcal{B}$, $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic $C^*$-simple group $\\Gamma$ on a von Neumann algebra $\\mathcal{M}$ with separable predual, every intermediate vNa $\\mathcal{N}$, $L(\\Gamma)\\subseteq\\mathcal{N}\\subseteq\\mathcal{M}\\rtimes\\Gamma$ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$ of the form $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$ for an inclusion of unital $\\Gamma$-simple $\\Gamma$-$C^*$-algebras $C(Y)\\subset C(X)$. We introduce a notion of generalized Powers&apos; averaging and show that it is equivalent to the simplicity of the crossed product $C(X)\\rtimes_r\\Gamma$. As an application, we show that every intermediate $C^*$-sub-algebras $\\mathcal{B}$, $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$ is simple whenever $C(Y)\\rtimes_r\\Gamma$ is simple.","abstract_html":"This thesis deals with the structure of intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebras $\\mathcal{B}$, either of the form <span class=\"etd-inline-math\">C<sub>\\lambda</sub><sup>*</sup>(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes<sub>r</sub>\\Gamma</span> or of the type <span class=\"etd-inline-math\">C(Y)\\rtimes<sub>r</sub>\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes<sub>r</sub>\\Gamma</span>. We begin by investigating the ideal structure of intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebras $\\mathcal{B}$ of the form <span class=\"etd-inline-math\">C<sub>\\lambda</sub><sup>*</sup>(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes<sub>r</sub>\\Gamma</span> for commutative unital $\\Gamma$-simple $\\Gamma$-<span class=\"etd-inline-math\">C<sup>*</sup></span>-algebras $\\mathcal{A}$. In particular, we show that if $\\Gamma$ is a <span class=\"etd-inline-math\">C<sup>*</sup></span>-simple group, then every such intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebra $\\mathcal{B}$ is simple. Continuing our perusal, we find examples of inclusions <span class=\"etd-inline-math\">C<sub>\\lambda</sub><sup>*</sup>(\\Gamma)\\subseteq \\mathcal{A}\\rtimes<sub>r</sub>\\Gamma</span> for which every intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebra $\\mathcal{B}$ of the form <span class=\"etd-inline-math\">C<sub>\\lambda</sub><sup>*</sup>(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes<sub>r</sub>\\Gamma</span> is a crossed product. We show that for a large class of actions $\\Gamma\\curvearrowright\\mathcal{A}$ of <span class=\"etd-inline-math\">C<sup>*</sup></span>-simple groups $\\Gamma$ on unital <span class=\"etd-inline-math\">C<sup>*</sup></span>-algebras $\\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebra $\\mathcal{B}$, <span class=\"etd-inline-math\">C<sub>\\lambda</sub><sup>*</sup>(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes<sub>r</sub>\\Gamma</span>, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic <span class=\"etd-inline-math\">C<sup>*</sup></span>-simple group $\\Gamma$ on a von Neumann algebra $\\mathcal{M}$ with separable predual, every intermediate vNa $\\mathcal{N}$, $L(\\Gamma)\\subseteq\\mathcal{N}\\subseteq\\mathcal{M}\\rtimes\\Gamma$ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebras $\\mathcal{B}$ of the form <span class=\"etd-inline-math\">C(Y)\\rtimes<sub>r</sub>\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes<sub>r</sub>\\Gamma</span> for an inclusion of unital $\\Gamma$-simple $\\Gamma$-<span class=\"etd-inline-math\">C<sup>*</sup></span>-algebras $C(Y)\\subset C(X)$. We introduce a notion of generalized Powers&amp;apos; averaging and show that it is equivalent to the simplicity of the crossed product <span class=\"etd-inline-math\">C(X)\\rtimes<sub>r</sub>\\Gamma</span>. As an application, we show that every intermediate <span class=\"etd-inline-math\">C<sup>*</sup></span>-sub-algebras $\\mathcal{B}$, <span class=\"etd-inline-math\">C(Y)\\rtimes<sub>r</sub>\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes<sub>r</sub>\\Gamma</span> is simple whenever <span class=\"etd-inline-math\">C(Y)\\rtimes<sub>r</sub>\\Gamma</span> is simple.","abstract_has_math":true,"creators":["Amrutam, Tattwamasi"],"institution":"University of Houston","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Kalantar, Mehrdad"],"committee_chairs":[],"committee_members":["Blecher, David P.","Ott, William","Skalski, Adam G."],"year":2021,"date_issued":"2021-05","date_published":"2021-05","updated_at":"2026-07-24T02:32:47Z","subjects":["Crossed products, C*-algebras"],"languages":["eng"],"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. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s)."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10657/10237","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kalantar, Mehrdad"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Blecher, David P.","Ott, William","Skalski, Adam G."]},{"key":"dc:creator","label":"Author","values":["Amrutam, Tattwamasi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-06-30T22:36:52Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-06-30T22:36:52Z"]},{"key":"dc:date.issued","label":"Date","values":["2021-05"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Houston"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Crossed products, C*-algebras"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s)."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10657/10237"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis deals with the structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$, either of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ or of the type $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$. We begin by investigating the ideal structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$ of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ for commutative unital $\\Gamma$-simple $\\Gamma$-$C^*$-algebras $\\mathcal{A}$. In particular, we show that if $\\Gamma$ is a $C^*$-simple group, then every such intermediate $C^*$-sub-algebra $\\mathcal{B}$ is simple. Continuing our perusal, we find examples of inclusions $C_{\\lambda}^*(\\Gamma)\\subseteq \\mathcal{A}\\rtimes_r\\Gamma$ for which every intermediate $C^*$-sub-algebra $\\mathcal{B}$ of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ is a crossed product. We show that for a large class of actions $\\Gamma\\curvearrowright\\mathcal{A}$ of $C^*$-simple groups $\\Gamma$ on unital $C^*$-algebras $\\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-sub-algebra $\\mathcal{B}$, $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic $C^*$-simple group $\\Gamma$ on a von Neumann algebra $\\mathcal{M}$ with separable predual, every intermediate vNa $\\mathcal{N}$, $L(\\Gamma)\\subseteq\\mathcal{N}\\subseteq\\mathcal{M}\\rtimes\\Gamma$ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$ of the form $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$ for an inclusion of unital $\\Gamma$-simple $\\Gamma$-$C^*$-algebras $C(Y)\\subset C(X)$. We introduce a notion of generalized Powers&apos; averaging and show that it is equivalent to the simplicity of the crossed product $C(X)\\rtimes_r\\Gamma$. As an application, we show that every intermediate $C^*$-sub-algebras $\\mathcal{B}$, $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$ is simple whenever $C(Y)\\rtimes_r\\Gamma$ is simple."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Structure of Intermediate C*-subalgebras of discrete group actions"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kalantar, Mehrdad"],"dc:contributor.committeemember":["Blecher, David P.","Ott, William","Skalski, Adam G."],"dc:creator":["Amrutam, Tattwamasi"],"dc:date.accessioned":["2022-06-30T22:36:52Z"],"dc:date.available":["2022-06-30T22:36:52Z"],"dc:date.issued":["2021-05"],"dc:description.abstract":["This thesis deals with the structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$, either of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ or of the type $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$. We begin by investigating the ideal structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$ of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ for commutative unital $\\Gamma$-simple $\\Gamma$-$C^*$-algebras $\\mathcal{A}$. In particular, we show that if $\\Gamma$ is a $C^*$-simple group, then every such intermediate $C^*$-sub-algebra $\\mathcal{B}$ is simple. Continuing our perusal, we find examples of inclusions $C_{\\lambda}^*(\\Gamma)\\subseteq \\mathcal{A}\\rtimes_r\\Gamma$ for which every intermediate $C^*$-sub-algebra $\\mathcal{B}$ of the form $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$ is a crossed product. We show that for a large class of actions $\\Gamma\\curvearrowright\\mathcal{A}$ of $C^*$-simple groups $\\Gamma$ on unital $C^*$-algebras $\\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-sub-algebra $\\mathcal{B}$, $C_{\\lambda}^*(\\Gamma)\\subseteq\\mathcal{B}\\subseteq\\mathcal{A}\\rtimes_r\\Gamma$, is a crossed product. On the von Neumann algebraic side, we show that for every non-faithful action of a acylindrically hyperbolic $C^*$-simple group $\\Gamma$ on a von Neumann algebra $\\mathcal{M}$ with separable predual, every intermediate vNa $\\mathcal{N}$, $L(\\Gamma)\\subseteq\\mathcal{N}\\subseteq\\mathcal{M}\\rtimes\\Gamma$ is a crossed product vNa. Finally, we inquire into the ideal structure of intermediate $C^*$-sub-algebras $\\mathcal{B}$ of the form $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$ for an inclusion of unital $\\Gamma$-simple $\\Gamma$-$C^*$-algebras $C(Y)\\subset C(X)$. We introduce a notion of generalized Powers&apos; averaging and show that it is equivalent to the simplicity of the crossed product $C(X)\\rtimes_r\\Gamma$. As an application, we show that every intermediate $C^*$-sub-algebras $\\mathcal{B}$, $C(Y)\\rtimes_r\\Gamma\\subseteq\\mathcal{B}\\subseteq C(X)\\rtimes_r\\Gamma$ is simple whenever $C(Y)\\rtimes_r\\Gamma$ is simple."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/10657/10237"],"dc:language.iso":["eng"],"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. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s)."],"dc:subject":["Crossed products, C*-algebras"],"dc:title":["Structure of Intermediate C*-subalgebras of discrete group actions"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["University of Houston"]},"updated_at":"2026-07-24T02:32:47Z"}