{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98206"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98206","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Non commutative version of arithmetic geometric mean inequality and crossed product of ternary ring of operators","abstract":"\"This thesis is structured into two parts. In the first two chapters, we prove the non commutative version of the Arithmetic Geometric Mean (AGM) inequality (this is a joint work with Mingyue Zhao and Maruis Junge). We start Chapter 2 by giving some background about the partition and M{\\\"\"o}bius function. We then prove the two main theorems: The AGM inequality for the norm and for the order. In Chapter 3, we provide some applications from random matrices such as Wishart random matrices, vector-valued moments of convex bodies, and freely independent operators. The second part is about a ternary ring of operators (TRO). After giving a quick survey for the work of Todorov on the operator space version of Zettl's decomposition theorem, we introduce crossed products of ternary ring of operators (the full crossed product and the reduced crossed product). We also prove that $V\\rtimes_{\\alpha^{V}}G$ as the off-diagonal corner of the $C^*$-algebra $A(V)\\rtimes_{\\alpha^{A(V)}}G$. Equivalently, we have the $*$-isomorphism between the two linking $C^*$-algebras, i.e. $A(V\\rtimes_{\\alpha^{V}}G)=A(V)\\rtimes_{\\alpha^{A(V)}}G.$ By using this identity, we obtain that if the group $G$ is amenable, some local properties for TRO's preserve with the crossed product. We also provide a counter example which shows that if the linking $C^*$-algebras $A(V)$ and $A(W)$ are $*$-isomorphic or if their diagonal components are $*$-isomorphic, then their TRO's are not isomorphic. Similar example will be applied for $W^*$-TRO's.\"","abstract_html":"&quot;This thesis is structured into two parts. In the first two chapters, we prove the non commutative version of the Arithmetic Geometric Mean (AGM) inequality (this is a joint work with Mingyue Zhao and Maruis Junge). We start Chapter 2 by giving some background about the partition and M{\\&quot;&quot;o}bius function. We then prove the two main theorems: The AGM inequality for the norm and for the order. In Chapter 3, we provide some applications from random matrices such as Wishart random matrices, vector-valued moments of convex bodies, and freely independent operators. The second part is about a ternary ring of operators (TRO). After giving a quick survey for the work of Todorov on the operator space version of Zettl&#x27;s decomposition theorem, we introduce crossed products of ternary ring of operators (the full crossed product and the reduced crossed product). We also prove that <span class=\"etd-inline-math\">V\\rtimes<sub>&alpha;<sup>V</sup></sub>G</span> as the off-diagonal corner of the <span class=\"etd-inline-math\">C<sup>*</sup></span>-algebra <span class=\"etd-inline-math\">A(V)\\rtimes<sub>&alpha;<sup>A(V)</sup></sub>G</span>. Equivalently, we have the $*$-isomorphism between the two linking <span class=\"etd-inline-math\">C<sup>*</sup></span>-algebras, i.e. <span class=\"etd-inline-math\">A(V\\rtimes<sub>&alpha;<sup>V</sup></sub>G)=A(V)\\rtimes<sub>&alpha;<sup>A(V)</sup></sub>G.</span> By using this identity, we obtain that if the group $G$ is amenable, some local properties for TRO&#x27;s preserve with the crossed product. We also provide a counter example which shows that if the linking <span class=\"etd-inline-math\">C<sup>*</sup></span>-algebras $A(V)$ and $A(W)$ are $*$-isomorphic or if their diagonal components are $*$-isomorphic, then their TRO&#x27;s are not isomorphic. Similar example will be applied for <span class=\"etd-inline-math\">W<sup>*</sup></span>-TRO&#x27;s.&quot;","abstract_has_math":true,"creators":["Albar, Wafaa Abdullah"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ruan, Zhong-Jin","Boca, Florin","Junge, Marius","Li, Xiaochun"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T17:45:09Z","date_published":"2017-09-29T17:45:09Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Arithmetic geometric mean inequality (AGM)","Random matrices","Ternary ring of operators (TRO)","Crossed product of ternary ring of operators (TROs)"],"languages":["en"],"rights":["Copyright 2017 Wafaa Albar"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98206","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ruan, Zhong-Jin","Boca, Florin","Junge, Marius","Li, Xiaochun"]},{"key":"dc:creator","label":"Author","values":["Albar, Wafaa Abdullah"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T17:45:09Z","2020-03-03T10:15:22Z","2017-07-11","2017-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Arithmetic geometric mean inequality (AGM)","Random matrices","Ternary ring of operators (TRO)","Crossed product of ternary ring of operators (TROs)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Wafaa Albar"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98206"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"This thesis is structured into two parts. In the first two chapters, we prove the non commutative version of the Arithmetic Geometric Mean (AGM) inequality (this is a joint work with Mingyue Zhao and Maruis Junge). We start Chapter 2 by giving some background about the partition and M{\\\"\"o}bius function. We then prove the two main theorems: The AGM inequality for the norm and for the order. In Chapter 3, we provide some applications from random matrices such as Wishart random matrices, vector-valued moments of convex bodies, and freely independent operators. The second part is about a ternary ring of operators (TRO). After giving a quick survey for the work of Todorov on the operator space version of Zettl's decomposition theorem, we introduce crossed products of ternary ring of operators (the full crossed product and the reduced crossed product). We also prove that $V\\rtimes_{\\alpha^{V}}G$ as the off-diagonal corner of the $C^*$-algebra $A(V)\\rtimes_{\\alpha^{A(V)}}G$. Equivalently, we have the $*$-isomorphism between the two linking $C^*$-algebras, i.e. $A(V\\rtimes_{\\alpha^{V}}G)=A(V)\\rtimes_{\\alpha^{A(V)}}G.$ By using this identity, we obtain that if the group $G$ is amenable, some local properties for TRO's preserve with the crossed product. We also provide a counter example which shows that if the linking $C^*$-algebras $A(V)$ and $A(W)$ are $*$-isomorphic or if their diagonal components are $*$-isomorphic, then their TRO's are not isomorphic. Similar example will be applied for $W^*$-TRO's.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2019-08-01","The student, Wafaa Albar, accepted the attached license on 2017-07-10 at 13:21.","The student, Wafaa Albar, submitted this Dissertation for approval on 2017-07-10 at 14:12.","This Dissertation was approved for publication on 2017-07-11 at 09:23.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11369 on 2017-09-29 at 10:46:52","Made available in DSpace on 2017-09-29T17:45:09Z (GMT). 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In the first two chapters, we prove the non commutative version of the Arithmetic Geometric Mean (AGM) inequality (this is a joint work with Mingyue Zhao and Maruis Junge). We start Chapter 2 by giving some background about the partition and M{\\\"\"o}bius function. We then prove the two main theorems: The AGM inequality for the norm and for the order. In Chapter 3, we provide some applications from random matrices such as Wishart random matrices, vector-valued moments of convex bodies, and freely independent operators. The second part is about a ternary ring of operators (TRO). After giving a quick survey for the work of Todorov on the operator space version of Zettl's decomposition theorem, we introduce crossed products of ternary ring of operators (the full crossed product and the reduced crossed product). We also prove that $V\\rtimes_{\\alpha^{V}}G$ as the off-diagonal corner of the $C^*$-algebra $A(V)\\rtimes_{\\alpha^{A(V)}}G$. Equivalently, we have the $*$-isomorphism between the two linking $C^*$-algebras, i.e. $A(V\\rtimes_{\\alpha^{V}}G)=A(V)\\rtimes_{\\alpha^{A(V)}}G.$ By using this identity, we obtain that if the group $G$ is amenable, some local properties for TRO's preserve with the crossed product. We also provide a counter example which shows that if the linking $C^*$-algebras $A(V)$ and $A(W)$ are $*$-isomorphic or if their diagonal components are $*$-isomorphic, then their TRO's are not isomorphic. Similar example will be applied for $W^*$-TRO's.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2019-08-01","The student, Wafaa Albar, accepted the attached license on 2017-07-10 at 13:21.","The student, Wafaa Albar, submitted this Dissertation for approval on 2017-07-10 at 14:12.","This Dissertation was approved for publication on 2017-07-11 at 09:23.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11369 on 2017-09-29 at 10:46:52","Made available in DSpace on 2017-09-29T17:45:09Z (GMT). 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