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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, ZhongJin 
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 UrbanaChampaign 
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, vectorvalued 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 offdiagonal 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:  20170711 
Type:  Text 
URI:  http://hdl.handle.net/2142/98206 
Rights Information:  Copyright 2017 Wafaa Albar 
Date Available in IDEALS:  20170929 
Date Deposited:  201708 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois