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 Title: Non commutative version of arithmetic geometric mean inequality and crossed product of ternary ring of operators Author(s): Albar, Wafaa Abdullah Director of Research: Ruan, Zhong-Jin Doctoral Committee Chair(s): Boca, Florin Doctoral Committee Member(s): Junge, Marius; Li, Xiaochun Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Arithmetic geometric mean inequality (AGM) Random matrices Ternary ring of operators (TRO) Crossed product of ternary ring of operators (TROs) 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. Issue Date: 2017-07-11 Type: Text URI: http://hdl.handle.net/2142/98206 Rights Information: Copyright 2017 Wafaa Albar Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08
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